http://www3.villanova.edu/maple/misc/frenetellipse.htmThe ellipse and its evolute and the distance problemFritz Hartmann and Bob JantzenDepartment of Mathematical Sciences, Villanova University23-sep-2004->6-sep-2009-> 2021
Maple 13.0[about 1.5 MB with output removed, about 12 MB when executed]Abstract. Inspired by a simple max/min distance problem from a calculus course, we were naturally led to explore the geometry of the evolute of an ellipse, ignorant of many classical results that we later found by web/library searching. This is a beautiful example of how, empowered by a computer algebra system, one can follow one's nose so to speak in uncovering elegant mathematical structure that would otherwise be unreachable in practice. A knowledge of parametrized plane curves and their geometry as covered in a typical multivariable calculus course is assumed, including the osculating circle which is not typically covered but which requires little extra effort. It is not necessary to understand the MAPLE graphing code: its purpose is to visualize the geometry and allow user interaction.
This worksheet only takes about a minute to execute completely without the distance plot section, which takes another minute by itself it I can get it to work again.LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JVEqU3RhcnRUaW1lRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEqJmNvbG9uZXE7RicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y9LyUpc3RyZXRjaHlHRj0vJSpzeW1tZXRyaWNHRj0vJShsYXJnZW9wR0Y9LyUubW92YWJsZWxpbWl0c0dGPS8lJ2FjY2VudEdGPS8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkwtRiw2JVEldGltZUYnRi9GMi1JKG1mZW5jZWRHRiQ2JC1GIzYlLUYsNiNRIUYnLyUrZXhlY3V0YWJsZUdGPUY5RjlGWkY5a calculus max/min distance problem
leads to counting the normalsSuppose one is asked to find the point on an ellipse closest to (or farthest from) a given point in the plane of the ellipse. Parametrizing the ellipse, this becomes an ordinary max/min problem for a function of a single variable that one might consider in a calculus course. The connecting line segment from the given point to a point on the ellipse where the distance is extremal must clearly be perpendicular to the tangent line there and is therefore a normal to the ellipse (see the next section). So one can then ask the more general question: how many points on the ellipse satisfy this condition as one moves the point around in the plane, i.e., how many normals can be drawn to the ellipse from a given point in the plane? These points where the normals contact the ellipse (the "feet" of the normal line segments) are all critical points of the distance function between the point on the ellipse and the given point.
In the special case of a circle, there are always two such points, which are the intersection with the circle of the line containing the center of the circle and the given point. For the center of an ellipse, the 4 intersections of the ellipse with its minor and major axes all satisfy this condition. Circle, ellipse, ..., if a point is at the center of an osculating circle of an ellipse, then it automatically satisfies this condition at the point of tangency, so perhaps the locus of all such osculating circle centers (the evolute of the ellipse) has something to do with this problem. A computer algebra system allows one to explore this rather effectively.describing an 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 calculus books have a section or appendix describing the conic sections. The larger of the two parameters of the standard equation of the ellipse centered at the origin is called the major semiaxis, while the smaller is called the minor semiaxis. The distance from the center along the major axis to the two foci of the ellipse is the focus distance (usually called LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiY0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=); dividing it by the major semi-axis defines the fractional distance along the semiaxis of the focus compared to the semiaxis itself, called the eccentricity of the ellipse, taking values between 0 and 1.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The sum of the distances of any point on the ellipse from the two foci is a constant equal to twice the major semiaxis, which defines the ellipse geometrically.aside: the algebra confirming this remarkA little algebra verifies this remark. For example, assuming LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJiRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiPEYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUYsNiVRImFGJ0Y0RjdGPkYrRj4=, the two foci are at LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkobWZlbmNlZEdGJDYmLUYjNiYtSSNtaUdGJDYlUSJjRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiLEYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4zMzMzMzMzZW1GJy1JI21uR0YkNiRRIjBGJ0Y+Rj5GPi8lJW9wZW5HUSJbRicvJSZjbG9zZUdRIl1GJ0Y+ and 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, so the sum of the distances between them and a point LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkobWZlbmNlZEdGJDYmLUYjNiYtSSNtaUdGJDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiLEYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4zMzMzMzMzZW1GJy1GMTYlUSJ5RidGNEY3Rj5GPi8lJW9wZW5HUSJbRicvJSZjbG9zZUdRIl1GJ0Y+ is: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Putting the second radical on the right hand side and squaring: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 everything on the left hand side and simplifying, then dividing by 4: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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2JS1JJm1mcmFjR0YkNigtRiM2JS1GLDYlUSIlRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvRj1RJ25vcm1hbEYnLUYjNiUtSSNtbkdGJDYkUSI0RidGQkY/RkIvJS5saW5ldGhpY2tuZXNzR1EiMUYnLyUrZGVub21hbGlnbkdRJ2NlbnRlckYnLyUpbnVtYWxpZ25HRk8vJSliZXZlbGxlZEdGQUY/RkJGK0Y/RkI=and then again isolating the radical on the right hand side and squaring, then using the definition of 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 and simplifying everything on the left hand side leads to the standard equation but with all terms on the left hand side: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Choosing LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJiRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiPEYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUYsNiVRImFGJ0Y0RjdGPkYrRj4= aligns the major axis with the LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=-axis, but one can fix LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiPUYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUkjbW5HRiQ2JFEiMUYnRj5GPkYrRj4= to set the distance scale for the problem and reduce the number of parameters in the problem by one (a simplification), which means that depending on whether LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiYkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= is a proper or improper fraction the major axis is horizontal or vertical. The single parameter LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiYkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= can assume any positive real value, but only proper fractional values would be needed to study all ellipses, fixing the major axis to be horizontal.The ellipse is easily parametrized using the trigonometric identity 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 by identifying the left hand side terms in this identity with the corresponding terms in the standard equation of the 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 the parameter LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= taking values from 0 to LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1JI21uR0YkNiRRIjJGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictSSNtb0dGJDYtUTEmSW52aXNpYmxlVGltZXM7RidGNS8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGPi8lKXN0cmV0Y2h5R0Y+LyUqc3ltbWV0cmljR0Y+LyUobGFyZ2VvcEdGPi8lLm1vdmFibGVsaW1pdHNHRj4vJSdhY2NlbnRHRj4vJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZNLUYsNiVRJSZwaTtGJy8lJ2l0YWxpY0dGPkY1RjVGK0Y1 during one counterclockwise loop around the ellipse from the positive LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=-axis starting point.The Calculus Max/Min Distance ProblemIf we let 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 denote the parametrized ellipse, and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkobWZlbmNlZEdGJDYmLUYjNiYtSSNtaUdGJDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiLEYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4zMzMzMzMzZW1GJy1GMTYlUSJ5RidGNEY3Rj5GPi8lJW9wZW5HUSJbRicvJSZjbG9zZUdRIl1GJ0Y+ any given point in the plane, then the square of the distance between the given point and the point on the ellipse is the self dot product of the difference vector 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 :LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYqLUklbXN1cEdGJDYlLUYjNictSSNtaUdGJDYlUSlEaXN0YW5jZUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RMCZBcHBseUZ1bmN0aW9uO0YnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZRLUkobWZlbmNlZEdGJDYkLUYjNiUtRjE2JVEidEYnRjRGNy8lK2V4ZWN1dGFibGVHRkJGPkY+RmZuRj4tSSNtbkdGJDYkUSIyRidGPi8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRictRjs2LVEiPUYnRj5GQEZDRkVGR0ZJRktGTS9GUFEsMC4yNzc3Nzc4ZW1GJy9GU0Zjby1GVTYkLUYjNigtRjE2JVEiUkYnRjRGN0ZULUY7Ni1RKCZtaW51cztGJ0Y+RkBGQ0ZFRkdGSUZLRk0vRlBRLDAuMjIyMjIyMmVtRicvRlNGYHAtRlU2Ji1GIzYnLUYxNiVRInhGJ0Y0RjctRjs2LVEiLEYnRj5GQC9GREY2RkVGR0ZJRktGTUZPL0ZTUSwwLjMzMzMzMzNlbUYnLUYxNiVRInlGJ0Y0RjdGZm5GPkY+LyUlb3BlbkdRIltGJy8lJmNsb3NlR1EiXUYnRmZuRj5GPi1GOzYtUSJ+RidGPkZARkNGRUZHRklGS0ZNRk9GUi1GOzYtUSkmbWlkZG90O0YnRj5GQEZDRkVGR0ZJRktGTS9GUFEsMC4xNjY2NjY3ZW1GJy9GU0ZfckZlb0ZmbkY+and the derivative of this function of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= is twice the inner product of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJSRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEwJkFwcGx5RnVuY3Rpb247RicvRjhRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZCLyUpc3RyZXRjaHlHRkIvJSpzeW1tZXRyaWNHRkIvJShsYXJnZW9wR0ZCLyUubW92YWJsZWxpbWl0c0dGQi8lJ2FjY2VudEdGQi8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRlEtSShtZmVuY2VkR0YkNiQtRiM2JC1GLDYlUSJ0RidGNEY3Rj5GPkY+RitGPg== and the difference vector 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so it is zero when the difference vector is perpendicular to the tangent vector 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 (the "orthogonality condition"). Thus the critical points of the distance function (the same as those of its square) are those points on the ellipse where the difference vector is along a normal line to the ellipse.We are interested in the more general problem of finding all the critical points of the distance function, each of which corresponds to a normal line from the ellipse passing through the given point.visualizing the problemJust to get an idea of the situation, we build up an animation of the connecting line segment between the given point and the point on the ellipse as it moves around the ellipse. Slowing down the animation helps see roughly where the critical points of the distance function occur during this motion, clearly evident in the corresponding plot of the distance function below. The horizontal and vertical semi-axes are fixed to be LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiPUYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUkjbW5HRiQ2JFEiMUYnRj5GPkYrRj4= and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiYkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=, while the given point is located with polar coordinates LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkobWZlbmNlZEdGJDYmLUYjNiYtSSNtaUdGJDYlUSJyRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiLEYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4zMzMzMzMzZW1GJy1GMTYlUSgmdGhldGE7RicvRjVGQkY+Rj5GPi8lJW9wZW5HUSJbRicvJSZjbG9zZUdRIl1GJ0Y+. One can vary these remaining 3 parameters and re-execute the animation to explore what happens in different situations. There is no need to understand the guts of the plot. We build up the pieces of the final plot one by one, using an animated line segment to sweep around the ellipse from the given point.The parametrized ellipse and given fixed point are represented as: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Note that the parameter LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnRjIvRjZRJ25vcm1hbEYn agrees with the polar angle of the point on the ellipse only on the coordinate axes: 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, except for the degenerate case of a circle.We pick some concrete parameter values to make a figure, the final result of which is an animation to be played: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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JVElcGx0ZUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RKiZjb2xvbmVxO0YnL0YzUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGPS8lKXN0cmV0Y2h5R0Y9LyUqc3ltbWV0cmljR0Y9LyUobGFyZ2VvcEdGPS8lLm1vdmFibGVsaW1pdHNHRj0vJSdhY2NlbnRHRj0vJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZMLUYsNiVRIiVGJ0YvRjItRjY2LVEiOkYnRjlGO0Y+RkBGQkZERkZGSEZKRk0vJStleGVjdXRhYmxlR0Y9Rjk=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The difference vector isLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYuLUkjbWlHRiQ2JVEiUkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYlLUYsNiVRInRGJ0YvRjIvJStleGVjdXRhYmxlR1EmZmFsc2VGJy9GM1Enbm9ybWFsRidGQC1JI21vR0YkNi1RKCZtaW51cztGJ0ZALyUmZmVuY2VHRj8vJSpzZXBhcmF0b3JHRj8vJSlzdHJldGNoeUdGPy8lKnN5bW1ldHJpY0dGPy8lKGxhcmdlb3BHRj8vJS5tb3ZhYmxlbGltaXRzR0Y/LyUnYWNjZW50R0Y/LyUnbHNwYWNlR1EsMC4yMjIyMjIyZW1GJy8lJ3JzcGFjZUdGVi1GQzYtUSJ+RidGQEZGRkhGSkZMRk5GUEZSL0ZVUSYwLjBlbUYnL0ZYRmduLUY2NiYtRiM2Jy1GLDYlUSJ4RidGL0YyLUZDNi1RIixGJ0ZARkYvRklGMUZKRkxGTkZQRlJGZm4vRlhRLDAuMzMzMzMzM2VtRictRiw2JVEieUYnRi9GMkY9RkBGQC8lJW9wZW5HUSJbRicvJSZjbG9zZUdRIl1GJy1GQzYtUSI9RidGQEZGRkhGSkZMRk5GUEZSL0ZVUSwwLjI3Nzc3NzhlbUYnL0ZYRmNwLUY2NiYtRiM2Li1GLDYlUSJhRidGL0YyLUZDNi1RMSZJbnZpc2libGVUaW1lcztGJ0ZARkZGSEZKRkxGTkZQRlJGZm5GaG4tRiw2JVEkY29zRicvRjBGP0ZARjVGYG8tRiw2JVEiYkYnRi9GMkZccS1GLDYlUSRzaW5GJ0ZicUZARjUtRiw2I1EhRidGPUZARkBGaW9GXHBGaXEtRkM2LVEqJnVtaW51czA7RidGQEZGRkhGSkZMRk5GUEZSRlRGVy1GNjYmLUYjNiwtRiw2JVEickYnRi9GMkZccS1GIzYnRl9xLUZDNi1RMCZBcHBseUZ1bmN0aW9uO0YnRkBGRkZIRkpGTEZORlBGUkZmbkZobi1GNjYkLUYjNiUtRiw2JVEoJnRoZXRhO0YnRmJxRkBGPUZARkBGPUZARmBvRmNyRlxxLUYjNidGZnFGaHJGW3NGPUZARmlxRj1GQEZARmlvRlxwRj1GQA==whose length is the distance function.The plot of this distance function (red) and its derivative (blue) is easy: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For the initial choice of parameters there are 4 critical points of the distance function and hence 4 normals passing through the given point. Increasing LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEickYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= at the same angle for the same ellipse eventually decreases this to 2 (try LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJyRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiPUYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUkjbW5HRiQ2JFEkMC41RidGPkY+RitGPg==).The distance function squared and its derivative for the max/min problem are: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the latter of which is a condition of the form 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 (the "orthogonality condition"), which can be converted into a quartic equation in either LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSRjb3NGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUkjbW9HRiQ2LVEwJkFwcGx5RnVuY3Rpb247RidGNy8lJmZlbmNlR0Y2LyUqc2VwYXJhdG9yR0Y2LyUpc3RyZXRjaHlHRjYvJSpzeW1tZXRyaWNHRjYvJShsYXJnZW9wR0Y2LyUubW92YWJsZWxpbWl0c0dGNi8lJ2FjY2VudEdGNi8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRk4tSShtZmVuY2VkR0YkNiQtRiM2JC1GLDYlUSJ0RicvRjVRJXRydWVGJy9GOFEnaXRhbGljRidGN0Y3RjdGK0Y3 or LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSRzaW5GJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUkjbW9HRiQ2LVEwJkFwcGx5RnVuY3Rpb247RidGNy8lJmZlbmNlR0Y2LyUqc2VwYXJhdG9yR0Y2LyUpc3RyZXRjaHlHRjYvJSpzeW1tZXRyaWNHRjYvJShsYXJnZW9wR0Y2LyUubW92YWJsZWxpbWl0c0dGNi8lJ2FjY2VudEdGNi8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRk4tSShtZmVuY2VkR0YkNiQtRiM2JC1GLDYlUSJ0RicvRjVRJXRydWVGJy9GOFEnaXRhbGljRidGN0Y3RjdGK0Y3 as shown below, unless the given point lies on one of the axes of the ellipse (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), in which case only two of the three terms survive and one can factor a sine or cosine out of the equation to immediately solve it for the LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=-values corresponding to the two points on the ellipse symmetrically located across that axis.loading the VectorCalculus and LinearAlgebra packagesWhen this worksheet was created, we relied on the curve geometry commands of the VecCalc package and the linear algebra package linalg, but later Maple introduced its own VectorCalculus package and a new LinearAlgebra package based on a new Vector/Matrix structure. After years of upgrading, we decided to abandon this in favor of Maple's built in VectorCalculus package so that no special effort is needed to execute this worksheet.We thank the VecCalc people for the initial inspiration:Version 8 package of Arthur Belmonte and Philip B. Yasskin at Texas A&M University [downloadable as a zipped file from http://calclab.math.tamu.edu/maple/veccalc/]
We use VectorCalculus package and LinearAlgebra package and "Vectors" instead of lists and "Matrices" instead of "matrices".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We only need the commands for the curvature, unit tangent and unit normal of the curve:
Curvature(R(t),t)
TNBFrame(R(t), 'output'=['T'] )
TNBFrame(R(t), 'output'=['N'] )
The VectorCalculus package is needed to use prime derivative notation with Maple Vector functions. The plottools package is used for plotting circles easily.animated ellipse with evoluteThe horizontal semi-axis of the ellipse will be fixed to be LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiPUYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUkjbW5HRiQ2JFEiMUYnRj5GPkYrRj4=, while the vertical semi-axis will be our positive parameter LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiYkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=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 terms of which the eccentricity LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiZUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= of the ellipse (with values in the interval [0,1) ) is given byLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYvLUkjbWlHRiQ2JVEiYUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RKiZjb2xvbmVxO0YnL0YzUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGPS8lKXN0cmV0Y2h5R0Y9LyUqc3ltbWV0cmljR0Y9LyUobGFyZ2VvcEdGPS8lLm1vdmFibGVsaW1pdHNHRj0vJSdhY2NlbnRHRj0vJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZMLUkjbW5HRiQ2JFEiMUYnRjktRjY2LVEiOkYnRjlGO0Y+RkBGQkZERkZGSEZKRk0tSSdtc3BhY2VHRiQ2Ji8lJ2hlaWdodEdRJjAuMGV4RicvJSZ3aWR0aEdRJjAuMGVtRicvJSZkZXB0aEdGZW4vJSpsaW5lYnJlYWtHUShuZXdsaW5lRictRjY2LVEifkYnRjlGO0Y+RkBGQkZERkZGSC9GS0Zobi9GTkZobi1GLDYlUSJlRidGL0YyLUY2Ni1RIj1GJ0Y5RjtGPkZARkJGREZGRkhGSkZNLUYsNiVRKnBpZWNld2lzZUYnRi9GMi1JKG1mZW5jZWRHRiQ2JC1GIzY2LUklbXN1cEdGJDYlRistRiM2JS1GUDYkUSIyRidGOS8lK2V4ZWN1dGFibGVHRj1GOS8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRictRjY2LVEiPEYnRjlGO0Y+RkBGQkZERkZGSEZKRk0tRmJwNiUtRiw2JVEiYkYnRi9GMkZmcEZbcS1GNjYtUSIsRidGOUY7L0Y/RjFGQEZCRkRGRkZIRmFvL0ZOUSwwLjMzMzMzMzNlbUYnLUkmbWZyYWNHRiQ2KC1GIzYlLUkmbXNxcnRHRiQ2Iy1GIzYnRmFxLUY2Ni1RKCZtaW51cztGJ0Y5RjtGPkZARkJGREZGRkgvRktRLDAuMjIyMjIyMmVtRicvRk5GanItRmJwNiVGK0ZmcEZbcUZpcEY5RmlwRjktRiM2JUZjcUZpcEY5LyUubGluZXRoaWNrbmVzc0dGUi8lK2Rlbm9tYWxpZ25HUSdjZW50ZXJGJy8lKW51bWFsaWduR0Zkcy8lKWJldmVsbGVkR0Y9RmZxRmNxRmZvRitGZnEtRlA2JEZdcUY5RmZxRmFxRl5xRmFwRmZxLUYjNiUtRl1yNigtRmJyNiMtRiM2J0Zcc0ZmckZhcUZpcEY5LUYjNiVGK0ZpcEY5RmBzRmJzRmVzRmdzRmlwRjktRiw2I1EhRidGaXBGOUY5LUY2Ni1RIjtGJ0Y5RjtGaXFGQEZCRkRGRkZIRmFvRk1GaXBGOQ==LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYvLUkjbWlHRiQ2JVElcGxvdEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYoLUY2NiYtRiM2KC1GLDYjUSFGJy1GIzYnLUYsNiVRJHJoc0YnRi9GMi1JI21vR0YkNi1RMCZBcHBseUZ1bmN0aW9uO0YnL0YzUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGTi8lKXN0cmV0Y2h5R0ZOLyUqc3ltbWV0cmljR0ZOLyUobGFyZ2VvcEdGTi8lLm1vdmFibGVsaW1pdHNHRk4vJSdhY2NlbnRHRk4vJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0Znbi1GNjYkLUYjNiUtRiw2JVEiJUYnRi9GMi8lK2V4ZWN1dGFibGVHRk5GSkZKRmFvRkotRkc2LVEiLEYnRkpGTC9GUEYxRlFGU0ZVRldGWUZlbi9GaW5RLDAuMzMzMzMzM2VtRictSSNtbkdGJDYkUSIxRidGSkZhb0ZKRkovJSVvcGVuR1EiW0YnLyUmY2xvc2VHUSJdRidGY28tRiM2KC1GLDYlUSJiRidGL0YyLUZHNi1RIj1GJ0ZKRkxGT0ZRRlNGVUZXRlkvRmZuUSwwLjI3Nzc3NzhlbUYnL0ZpbkZccS1GIzYnLUZqbzYkUSIwRidGSi1GRzYtUSMuLkYnRkpGTEZPRlFGU0ZVRldGWS9GZm5RLDAuMjIyMjIyMmVtRidGaG4tRmpvNiRRIjhGJ0ZKRmFvRkpGPkZhb0ZKRj5GYW9GSkZKLUZHNi1RIjtGJ0ZKRkxGZm9GUUZTRlVGV0ZZRmVuRl1xLUknbXNwYWNlR0YkNiYvJSdoZWlnaHRHUSYwLjBleEYnLyUmd2lkdGhHRmduLyUmZGVwdGhHRmNyLyUqbGluZWJyZWFrR1EobmV3bGluZUYnLUZHNi1RIn5GJ0ZKRkxGT0ZRRlNGVUZXRllGZW5GaG4tRiw2JVEiYUYnRi9GMi1GRzYtUSomY29sb25lcTtGJ0ZKRkxGT0ZRRlNGVUZXRllGW3FGXXEtRkc2LVEiJ0YnRkpGTEZPRlFGU0ZVRldGWS9GZm5RLDAuMTExMTExMWVtRidGaG5GXnNGZHMtRkc2LVEiOkYnRkpGTEZPRlFGU0ZVRldGWUZbcUZdcUZhb0ZKIn the guts of the procedure below, CC(t) is the position vector of the evolute, the first C standing for the Curve and the second C standing for the Center of the osculating circle. The code cannibalizes the example from the frenet.mws worksheet of the VecCalc package which animates the Frenet frame, here instead tracing out the evolute at the same time as the ellipse is traced out. Again there is no need to understand the guts of the procedure.definition of animation procedures EllipseEvolute(b), EllipseEvoluteWithCircle(b)
and plot procedure EllipseEvolutePlot(b)We will test out the new combined commands with the following 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The position vector of the center of curvature (center of the osculating circle) is a distance equal to the radius of curvature (in turn the reciprocal of the curvature LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JlEia0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkobWZlbmNlZEdGJDYlLUYjNiUtRiw2JlEidEYnRi9GMkY1RjIvRjZRJ25vcm1hbEYnRjJGQEYyRkA=) along the unit normal LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEiTkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnRjIvRjZRJ25vcm1hbEYn to the curve from the tip of the position vector of the curve:
CC(t) = R(t) +1/k(t) N(t).
We create a Maple Vector function for this position vector: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We make a more general procedure.We now create a list of 24 frame display plots, each of which plots the ellipse and corresponding points of its evolute with the connecting line segment between them (along the normal line) from LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JlEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LlEiPUYnRjIvRjZRJ25vcm1hbEYnLyUmZmVuY2VHRjQvJSpzZXBhcmF0b3JHRjQvJSlzdHJldGNoeUdGNC8lKnN5bW1ldHJpY0dGNC8lKGxhcmdlb3BHRjQvJS5tb3ZhYmxlbGltaXRzR0Y0LyUnYWNjZW50R0Y0LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTi1JI21uR0YkNiVRIjBGJ0YyRjxGMkY8 up to the current running value of the parameter LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnRjIvRjZRJ25vcm1hbEYn. Then we manually create an animation in the parameter LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnRjIvRjZRJ25vcm1hbEYn with the plot option insequence = true which displays each of these frames in sequence. This procedure takes the semiaxis parameter LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JlEiYkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LlEifkYnRjIvRjZRJ25vcm1hbEYnLyUmZmVuY2VHRjQvJSpzZXBhcmF0b3JHRjQvJSlzdHJldGNoeUdGNC8lKnN5bW1ldHJpY0dGNC8lKGxhcmdlb3BHRjQvJS5tb3ZhYmxlbGltaXRzR0Y0LyUnYWNjZW50R0Y0LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTkYyRjw=as its single input.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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYqLUkjbWlHRiQ2JVEvRWxsaXBzZUV2b2x1dGVGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSShtZmVuY2VkR0YkNiQtRiM2Jy1GLDYjUSFGJy1GIzYmRjotSSZtc3FydEdGJDYjLUYjNiUtSSNtbkdGJDYkUSIyRicvRjNRJ25vcm1hbEYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRidGSEZKRkhGOkZKRkhGSEY6LUknbXNwYWNlR0YkNiYvJSdoZWlnaHRHUSYwLjBleEYnLyUmd2lkdGhHUSYwLjBlbUYnLyUmZGVwdGhHRlIvJSpsaW5lYnJlYWtHUShuZXdsaW5lRictRk42JkZQRlNGVi9GWVElYXV0b0YnRjpGSkZINext we add the osculating circle to this animation.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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVE5RWxsaXBzZUV2b2x1dGVXaXRoQ2lyY2xlRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkobWZlbmNlZEdGJDYkLUYjNictRiw2I1EhRictRiM2JS1JJm1zcXJ0R0YkNiMtRiM2JS1JI21uR0YkNiRRIjJGJy9GM1Enbm9ybWFsRicvJStleGVjdXRhYmxlR1EmZmFsc2VGJ0ZIRkpGSEY6RkpGSEZIRkpGSA==For the final plot picture we define a simple plot: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Here is the particular case in which the horizontal semi-axes of the evolute coincide with those of the ellipse. The annoying error message for now is not understood. For this initial choice of the parameter LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJiRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiPUYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUkmbXNxcnRHRiQ2Iy1JI21uR0YkNiRRIjJGJ0Y+Rj5GK0Y+, the evolute fits exactly inside the ellipse. It helps to slow down the animation. Recall that LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiYUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRIjFGJ0Y5Rjk= is assumed throughout.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 inside or outside of the ellipse Since the horizontal semi-axis is fixed to be LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiYUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRIjFGJ0Y5Rjk=, when the curvature is 1/2 so that the radius of curvature is 2, the osculating circle of its endpoints has its center at the other endpoint. In general, the curvature must be 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 for the radius of curvature to be LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1JI21uR0YkNiRRIjJGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictSSNtb0dGJDYtUTEmSW52aXNpYmxlVGltZXM7RidGNS8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGPi8lKXN0cmV0Y2h5R0Y+LyUqc3ltbWV0cmljR0Y+LyUobGFyZ2VvcEdGPi8lLm1vdmFibGVsaW1pdHNHRj4vJSdhY2NlbnRHRj4vJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZNLUYsNiVRImFGJy8lJ2l0YWxpY0dRJXRydWVGJy9GNlEnaXRhbGljRidGNUYrRjU= so that the center of curvature is at the opposite end of the ellipse along that axis: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When 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, the evolute touches the ellipse at the tips of its horizontal semiaxes, and then for greater values, it extends outside the ellipse along the minor axis. This assumes LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiPEYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUYsNiVRImJGJ0Y0RjdGPkYrRj4=, ignoring the vertical axis, but for small enough proper fractional values of 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, then the evolute exits the ellipse in the vertical direction. This is discussed below in more detail using the equation of the evolute.LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEvRWxsaXBzZUV2b2x1dGVGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSShtZmVuY2VkR0YkNiQtRiM2JS1JJm1mcmFjR0YkNigtRiM2JS1JI21uR0YkNiRRIjFGJy9GM1Enbm9ybWFsRicvJStleGVjdXRhYmxlR1EmZmFsc2VGJ0ZDLUYjNiUtSSZtc3FydEdGJDYjLUZANiRRIjJGJ0ZDRkVGQy8lLmxpbmV0aGlja25lc3NHRkIvJStkZW5vbWFsaWduR1EnY2VudGVyRicvJSludW1hbGlnbkdGVC8lKWJldmVsbGVkR0ZHRkVGQ0ZDRkVGQw==normal lines which pass through a given pointNext we need to add in the extras needed to plot both the ellipse and its evolute, together with the points on the ellipse satisfying the orthogonality condition, namely the condition that the normal from those points passes through the given point.locating the normals through a given point: step one, the angle made by the normalThe cosine of the angle LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEpJnZhcnBoaTtGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnRjI= between the connecting line segment and the tangent line is the normalized dot product of the difference vector and the tangent vector, i.e., their dot product divided by their lengths: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This expression for the cosine of the angle between the displacement vector and the tangent vector is zero when the orthogonality condition is satisfied, and its values range between LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JS1JI21vR0YkNi1RKiZ1bWludXMwO0YnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGOi8lKXN0cmV0Y2h5R0Y6LyUqc3ltbWV0cmljR0Y6LyUobGFyZ2VvcEdGOi8lLm1vdmFibGVsaW1pdHNHRjovJSdhY2NlbnRHRjovJSdsc3BhY2VHUSwwLjIyMjIyMjJlbUYnLyUncnNwYWNlR0ZJLUkjbW5HRiQ2JFEiMUYnRjVGNUYrRjU= and 1 so its plots are nicely bounded for an easier graphical search for the zeros.Here is the case of a circle with LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiPUYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUYsNiVRImJGJ0Y0RjdGPkYrRj4= = 1 and a point halfway between its center and its intersection with the positive LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=-axis. There are 2 zeros of the orthogonality condition corresponding to the ends of the horizontal diameter of the circle where the cosine of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JlEnJiM5NjY7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUwZm9udF9zdHlsZV9uYW1lR1EoMkR+TWF0aEYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Y1 is zero LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRidGK0YrLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYn:LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2J0YrLUYjNictRiw2JVElc3Vic0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RMCZBcHBseUZ1bmN0aW9uO0YnL0Y6USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGRC8lKXN0cmV0Y2h5R0ZELyUqc3ltbWV0cmljR0ZELyUobGFyZ2VvcEdGRC8lLm1vdmFibGVsaW1pdHNHRkQvJSdhY2NlbnRHRkQvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZTLUkobWZlbmNlZEdGJDYkLUYjNigtRlc2Ji1GIzYrRistRiM2Jy1GLDYlUSJiRidGNkY5LUY9Ni1RIj1GJ0ZARkJGRUZHRklGS0ZNRk8vRlJRLDAuMjc3Nzc3OGVtRicvRlVGYm8tSSNtbkdGJDYkUSIxRidGQC8lK2V4ZWN1dGFibGVHRkRGQC1GPTYtUSIsRidGQEZCL0ZGRjhGR0ZJRktGTUZPRlEvRlVRLDAuMzMzMzMzM2VtRictRiM2Jy1GLDYlUScmIzk1MjtGJy9GN0ZERkBGXm8tRmVvNiRRIjBGJ0ZARmhvRkBGam8tRiM2Jy1GLDYlUSJyRidGNkY5Rl5vLUkmbWZyYWNHRiQ2KC1GIzYlRmRvRmhvRkAtRiM2JS1GZW82JFEiMkYnRkBGaG9GQC8lLmxpbmV0aGlja25lc3NHRmdvLyUrZGVub21hbGlnbkdRJ2NlbnRlckYnLyUpbnVtYWxpZ25HRlxyLyUpYmV2ZWxsZWRHRkRGaG9GQEYrRmhvRkBGQC8lJW9wZW5HUSJ8ZnJGJy8lJmNsb3NlR1EifGhyRidGam8tRiw2JVEqQ29zaW5lcGhpRidGNkY5LUZXNiQtRiM2JS1GLDYlUSJ0RidGNkY5RmhvRkBGQEZob0ZARkBGaG9GQEYrRmhvRkAtRj02LVEiO0YnRkBGQkZdcEZHRklGS0ZNRk9GUUZjb0Zob0ZALUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2J0YrLUYjNictRiw2JVEpc2ltcGxpZnlGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYtUTAmQXBwbHlGdW5jdGlvbjtGJy9GOlEnbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkQvJSlzdHJldGNoeUdGRC8lKnN5bW1ldHJpY0dGRC8lKGxhcmdlb3BHRkQvJS5tb3ZhYmxlbGltaXRzR0ZELyUnYWNjZW50R0ZELyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGUy1JKG1mZW5jZWRHRiQ2JC1GIzYlLUYsNiVRIiVGJ0Y2RjkvJStleGVjdXRhYmxlR0ZERkBGQEZobkZARitGaG5GQC1GPTYtUSI7RidGQEZCL0ZGRjhGR0ZJRktGTUZPRlEvRlVRLDAuMjc3Nzc3OGVtRidGaG5GQA==LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2J0YrLUYjNictRiw2JVElcGxvdEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RMCZBcHBseUZ1bmN0aW9uO0YnL0Y6USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGRC8lKXN0cmV0Y2h5R0ZELyUqc3ltbWV0cmljR0ZELyUobGFyZ2VvcEdGRC8lLm1vdmFibGVsaW1pdHNHRkQvJSdhY2NlbnRHRkQvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZTLUkobWZlbmNlZEdGJDYkLUYjNigtRiw2JVEiJUYnRjZGOS1GPTYtUSIsRidGQEZCL0ZGRjhGR0ZJRktGTUZPRlEvRlVRLDAuMzMzMzMzM2VtRictRiM2KC1GLDYlUSJ0RidGNkY5LUY9Ni1RIj1GJ0ZARkJGRUZHRklGS0ZNRk8vRlJRLDAuMjc3Nzc3OGVtRicvRlVGZ28tRiM2KC1JI21uR0YkNiRRIjBGJ0ZALUY9Ni1RIy4uRidGQEZCRkVGR0ZJRktGTUZPL0ZSUSwwLjIyMjIyMjJlbUYnRlQtRiM2Jy1GXHA2JFEiMkYnRkAtRj02LVExJkludmlzaWJsZVRpbWVzO0YnRkBGQkZFRkdGSUZLRk1GT0ZRRlQtRiw2JVEnJiM5NjA7RicvRjdGREZALyUrZXhlY3V0YWJsZUdGREZARitGYHFGQEYrRmBxRkBGK0ZgcUZARkBGYHFGQEYrRmBxRkBGK0ZgcUZAvisualize the angle cosine functionWe show the line from a given point to all points on the ellipse.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graphical determination of one normal: zero of angle cosine functionNext we consider an ellipse with LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJiRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiPUYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUkjbW5HRiQ2JFEkMS41RidGPkY+RitGPg== and vary the distance from the origin of the fixed point, for a fixed angle of 45 degrees: 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, .6 , and then iterate the graph by trial and error to find approximately the radius LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJyRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiPUYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUkjbW5HRiQ2JFEpMC41MDMzODdGJ0Y+Rj5GK0Y+ (to this numerical accuracy) at which the two central roots merge as we move out radially from the origin at this angle: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Now we plot the corresponding cosine functions which are zero at the points where a normal goes through the given point: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 zoom in on the transition point: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 we will be able to calculate this value exactly as the radius at which we cross the evolute moving along this direction.
We can also find the all the roots for a given radius numerically: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If we go out along the horizontal axis of the ellipse, one has a transition from 4 directly to 2 roots at the vertex of the evolute which occurs at LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJyRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiPUYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUkmbXNxcnRHRiQ2Iy1JI21uR0YkNiRRIjJGJ0Y+Rj5GK0Y+ as we will see below: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determining exactly the roots of the angle cosine functionNext we have to come up with a way to automatically produce the points on the ellipse from which the normal lines that pass through the given point originate, namely the roots of the equation which determine the parameter values corresponding to these points: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 one can isolate the sine terms, square, and replace LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JC1JJW1zdXBHRiQ2JS1GLDYlUSRzaW5GJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUkjbW5HRiQ2JFEiMkYnRjovJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnRjpGK0Y6 by LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JC1JJW1zdXBHRiQ2JS1GLDYlUSRjb3NGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUkjbW5HRiQ2JFEiMkYnRjovJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnRjpGK0Y6 to get a fourth degree polynomial in LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSRjb3NGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUkjbW9HRiQ2LVEwJkFwcGx5RnVuY3Rpb247RidGNy8lJmZlbmNlR0Y2LyUqc2VwYXJhdG9yR0Y2LyUpc3RyZXRjaHlHRjYvJSpzeW1tZXRyaWNHRjYvJShsYXJnZW9wR0Y2LyUubW92YWJsZWxpbWl0c0dGNi8lJ2FjY2VudEdGNi8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRk4tSShtZmVuY2VkR0YkNiQtRiM2JC1GLDYlUSJ0RicvRjVRJXRydWVGJy9GOFEnaXRhbGljRidGN0Y3RjdGK0Y3. Solving this polynomial for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSRjb3NGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUkjbW9HRiQ2LVEwJkFwcGx5RnVuY3Rpb247RidGNy8lJmZlbmNlR0Y2LyUqc2VwYXJhdG9yR0Y2LyUpc3RyZXRjaHlHRjYvJSpzeW1tZXRyaWNHRjYvJShsYXJnZW9wR0Y2LyUubW92YWJsZWxpbWl0c0dGNi8lJ2FjY2VudEdGNi8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRk4tSShtZmVuY2VkR0YkNiQtRiM2JC1GLDYlUSJ0RicvRjVRJXRydWVGJy9GOFEnaXRhbGljRidGN0Y3RjdGK0Y3, we can then backsubstitute these values into the expression for the sine to define it so that the pair of cosine and sine values solve the original unsquared equation (squaring introduces extra solutions) and use the two argument arctan function
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
to produce the correct angle between LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbW9HRiQ2LVEqJnVtaW51czA7RicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRLDAuMjIyMjIyMmVtRicvJSdyc3BhY2VHRkMtSSNtaUdGJDYlUScmIzk2MDtGJy8lJ2l0YWxpY0dGNEYvRi8= and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnJiM5NjA7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yy for the entire ellipse, something the usual single variable inverse cosine function cannot do. For example, this value of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEickYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= below was graphically found by trial and error to lie (approximately) on the evolute where two of the four roots coallesce into one and then disappear as you increase LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEickYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=: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Now having found the points on the ellipse satisfying the orthogonality condition, we add the connecting line segments to the above plot of the ellipse and its evolute: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 points inside the evolute, one connects to 4 points on the ellipse, while for points outside the evolute, one connects to 2 points on the ellipse, and to 3 points on the ellipse when the given point is on the evolute, except for the vertices of the evolute, where there are only 2 instead of 3 (namely the normals from the two corresponding semi-axis endpoints).The only problem that occurs if the given point lies on one of the two axes of the ellipse is that the two solutions are lost by the procedure which derails it since it expects 4 real or complex roots (when a root pair is complex the corresponding normals are simply not plotted since only real points can be plotted), so we must restore them by hand with if statements that correct the missing roots from our solution procedure.
When the point is on the vertical axis (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), the two solutions on that axis 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 corresponding to the lost factor 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 are lost by this solution procedure and so must be added in by hand: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When the point is on the horizontal axis (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), the only the factor 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 enters the process because of our choice to eliminate the sine in favor of the cosine, yielding the two solutions on the axis, but the two solutions off the axis (real or complex) are lost: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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2J0YrLUYjNigtRiw2JVEkZXEyRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEjOj1GJy9GOlEnbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkQvJSlzdHJldGNoeUdGRC8lKnN5bW1ldHJpY0dGRC8lKGxhcmdlb3BHRkQvJS5tb3ZhYmxlbGltaXRzR0ZELyUnYWNjZW50R0ZELyUnbHNwYWNlR1EsMC4yNzc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lK2V4ZWN1dGFibGVHRjFGMi1GNjYtUSI6RidGMkY5RjtGPUY/RkFGQ0ZFRkdGSkZVRjI=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 we must instead eliminate the cosine in favor of the sine to get the other two solutions and add by hand the two axis solutions in this case.Now we put these two exceptions into the general algorithm:LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=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Now we are ready to make this into a procedure.making this into a procedure: Ellipser(b,r,theta)
and the multivalued distance functions:
EllipseDistance(b,r,theta) and EllipseDistanceMinMax(b,r,theta)We start with just a plot. Later we animate it. If we find a point on the ellipse for which 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, we are on the vertical axis, while if 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 then we are on the horizontal axis. In each case we must take special action. One must solve a simpler form of the equation on the axis for either the cosine or sine of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnRjIvRjZRJ25vcm1hbEYn and then map the multiple roots to the corresponding values of the other function which satisfy this equation, and then determine the values of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnRjIvRjZRJ25vcm1hbEYn for those pairs. This algorithm is tricky to digest.aside on how this worksVertical Axis Example of how algorithm works.
The equation to be solved for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnRjIvRjZRJ25vcm1hbEYn simplifies to two terms and has two independent solutions for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JlEkc2luRicvJSdpdGFsaWNHUSZmYWxzZUYnLyUrZXhlY3V0YWJsZUdGMS8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictSShtZmVuY2VkR0YkNiUtRiM2JS1GLDYmUSJ0RicvRjBRJXRydWVGJ0YyL0Y1USdpdGFsaWNGJ0YyRjRGMkY0LUYsNiNRIUYnRjJGNA== since LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JlEkY29zRicvJSdpdGFsaWNHUSZmYWxzZUYnLyUrZXhlY3V0YWJsZUdGMS8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictSShtZmVuY2VkR0YkNiUtRiM2JS1GLDYmUSJ0RicvRjBRJXRydWVGJ0YyL0Y1USdpdGFsaWNGJ0YyRjRGMkY0RjJGNA== factors out: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 we solve this for both trig functions and add back in the simple axis solutions for the cosine at the end: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It works!Now we compute the distances from the given point to the four real or complex LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= values produced by this procedure, which when real are ordered by LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSRjb3NGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUkjbW9HRiQ2LVEwJkFwcGx5RnVuY3Rpb247RidGNy8lJmZlbmNlR0Y2LyUqc2VwYXJhdG9yR0Y2LyUpc3RyZXRjaHlHRjYvJSpzeW1tZXRyaWNHRjYvJShsYXJnZW9wR0Y2LyUubW92YWJsZWxpbWl0c0dGNi8lJ2FjY2VudEdGNi8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRk4tSShtZmVuY2VkR0YkNiQtRiM2JC1GLDYlUSJ0RicvRjVRJXRydWVGJy9GOFEnaXRhbGljRidGN0Y3RjdGK0Y3. For now we do not eliminate the complex roots so that there are always 4 entries in the list of solutions for plotting purposes. If there is a variable number, we cannot plot the corresponding 4 distances as four plots whether they are real or complex, while the complex values are simply ignored by plot commands. [We have still not succeeded in doing this yet anyway.]
We simply copy the previous procedure which produces the four real or complex LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= values before calculating the corresponding points on the ellipse and plotting them, then remove the complex values and take the min and max of the real ones. Since the roots of the cosine are ordered, but arctan produces angles in the range LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2J0YrLUYjNiUtSSNtb0dGJDYtUSomdW1pbnVzMDtGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjwvJSlzdHJldGNoeUdGPC8lKnN5bW1ldHJpY0dGPC8lKGxhcmdlb3BHRjwvJS5tb3ZhYmxlbGltaXRzR0Y8LyUnYWNjZW50R0Y8LyUnbHNwYWNlR1EsMC4yMjIyMjIyZW1GJy8lJ3JzcGFjZUdGSy1GLDYlUSUmcGk7RicvJSdpdGFsaWNHRjxGN0Y3LUY0Ni1RIy4uRidGN0Y6Rj1GP0ZBRkNGRUZHRkkvRk1RJjAuMGVtRidGTkY3RitGNw==, the LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=-parameter values are ordered by the LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSRjb3NGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUkjbW9HRiQ2LVEwJkFwcGx5RnVuY3Rpb247RidGNy8lJmZlbmNlR0Y2LyUqc2VwYXJhdG9yR0Y2LyUpc3RyZXRjaHlHRjYvJSpzeW1tZXRyaWNHRjYvJShsYXJnZW9wR0Y2LyUubW92YWJsZWxpbWl0c0dGNi8lJ2FjY2VudEdGNi8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRk4tSShtZmVuY2VkR0YkNiQtRiM2JC1GLDYlUSJ0RicvRjVRJXRydWVGJy9GOFEnaXRhbGljRidGN0Y3RjdGK0Y3 values when real. But when the angle is purely imaginary, the trig values are real and yield a real distance, so we arbitrarily set this distance to be LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiSUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= in order to maintain 4 real or complex 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Okay, now that that works we apply it to the points on the ellipse whose normals pass through the given point: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And while we are at it, we find the minimum and maximum real distances: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 because this is giving "Error, invalid subscript selector" when in plot3d without double right quotes around the function, we define separate min and max distance functions: 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 of various configurations for location of given pointWhen the roots become complex, the corresponding distances are assigned to the value LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiSUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= so there are always four in the list.Just barely inside the evolute:LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2J0YrLUYjNictRiw2JVEpRWxsaXBzZXJGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYtUTAmQXBwbHlGdW5jdGlvbjtGJy9GOlEnbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkQvJSlzdHJldGNoeUdGRC8lKnN5bW1ldHJpY0dGRC8lKGxhcmdlb3BHRkQvJS5tb3ZhYmxlbGltaXRzR0ZELyUnYWNjZW50R0ZELyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGUy1JKG1mZW5jZWRHRiQ2JC1GIzYpLUkjbW5HRiQ2JFEkMS41RidGQC1GPTYtUSIsRidGQEZCL0ZGRjhGR0ZJRktGTUZPRlEvRlVRLDAuMzMzMzMzM2VtRictRmZuNiRRJDAuNUYnRkBGaW4tSSZtZnJhY0dGJDYoLUYjNiUtRiw2JVEnJiM5NjA7RicvRjdGREZALyUrZXhlY3V0YWJsZUdGREZALUYjNiUtRmZuNiRRIjRGJ0ZARltwRkAvJS5saW5ldGhpY2tuZXNzR1EiMUYnLyUrZGVub21hbGlnbkdRJ2NlbnRlckYnLyUpbnVtYWxpZ25HRmdwLyUpYmV2ZWxsZWRHRkRGW3BGQEZARltwRkBGK0ZbcEZALUY9Ni1RIjtGJ0ZARkJGXG9GR0ZJRktGTUZPRlEvRlVRLDAuMjc3Nzc3OGVtRidGW3BGQA==LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYpLUkjbWlHRiQ2I1EhRictSSdtc3BhY2VHRiQ2Ji8lJ2hlaWdodEdRJjAuMGV4RicvJSZ3aWR0aEdRJjAuMGVtRicvJSZkZXB0aEdGNC8lKmxpbmVicmVha0dRKG5ld2xpbmVGJy1GMDYmRjJGNUY4L0Y7USVhdXRvRictRiM2Jy1GLDYlUTBFbGxpcHNlRGlzdGFuY2VGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYtUTAmQXBwbHlGdW5jdGlvbjtGJy9GSlEnbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRlQvJSlzdHJldGNoeUdGVC8lKnN5bW1ldHJpY0dGVC8lKGxhcmdlb3BHRlQvJS5tb3ZhYmxlbGltaXRzR0ZULyUnYWNjZW50R0ZULyUnbHNwYWNlR0Y3LyUncnNwYWNlR0Y3LUkobWZlbmNlZEdGJDYkLUYjNiktSSNtbkdGJDYkUSQxLjVGJ0ZQLUZNNi1RIixGJ0ZQRlIvRlZGSEZXRllGZW5GZ25GaW5GW28vRl5vUSwwLjMzMzMzMzNlbUYnLUZlbzYkUSQwLjVGJ0ZQRmhvLUkmbWZyYWNHRiQ2KC1GIzYlLUYsNiVRJyYjOTYwO0YnL0ZHRlRGUC8lK2V4ZWN1dGFibGVHRlRGUC1GIzYlLUZlbzYkUSI0RidGUEZqcEZQLyUubGluZXRoaWNrbmVzc0dRIjFGJy8lK2Rlbm9tYWxpZ25HUSdjZW50ZXJGJy8lKW51bWFsaWduR0ZmcS8lKWJldmVsbGVkR0ZURmpwRlBGUEZqcEZQLUZNNi1RIjtGJ0ZQRlJGW3BGV0ZZRmVuRmduRmluRltvL0Zeb1EsMC4yNzc3Nzc4ZW1GJ0ZqcEZQLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2KUYrLUknbXNwYWNlR0YkNiYvJSdoZWlnaHRHUSYwLjBleEYnLyUmd2lkdGhHUSYwLjBlbUYnLyUmZGVwdGhHRjYvJSpsaW5lYnJlYWtHUShuZXdsaW5lRictRjI2JkY0RjdGOi9GPVElYXV0b0YnLUYsNiVRNkVsbGlwc2VEaXN0YW5jZU1pbk1heEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYpLUkjbW5HRiQ2JFEkMS41RicvRkpRJ25vcm1hbEYnLUkjbW9HRiQ2LVEiLEYnRlUvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkgvJSlzdHJldGNoeUdGZ24vJSpzeW1tZXRyaWNHRmduLyUobGFyZ2VvcEdGZ24vJS5tb3ZhYmxlbGltaXRzR0Znbi8lJ2FjY2VudEdGZ24vJSdsc3BhY2VHRjkvJSdyc3BhY2VHUSwwLjMzMzMzMzNlbUYnLUZSNiRRJDAuNUYnRlVGVy1JJm1mcmFjR0YkNigtRiM2JS1GLDYlUScmIzk2MDtGJy9GR0ZnbkZVLyUrZXhlY3V0YWJsZUdGZ25GVS1GIzYlLUZSNiRRIjRGJ0ZVRmVwRlUvJS5saW5ldGhpY2tuZXNzR1EiMUYnLyUrZGVub21hbGlnbkdRJ2NlbnRlckYnLyUpbnVtYWxpZ25HRmFxLyUpYmV2ZWxsZWRHRmduRmVwRlVGVUZlcEZVRitGZXBGVQ==Just outside the evolute: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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=Just off center: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On one of the two axes but one has to change to boxed or no axes to see the red line segments to along the coordinate axis: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On the ellipse and an 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the ellipse not on an 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still have to deal with these complex values which are ignored in plot commands.plotting the multivalued distance functions
[warning: this section takes a few minutes total to execute---but no longer works!]Now that we have found the points where the distance is extremized, we can plot the various extremal distances as a function of the given point.LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzY6LUkjbWlHRiQ2JVEocmVzdGFydEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIjpGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1GLDYlUSV3aXRoRidGL0YyLUkobWZlbmNlZEdGJDYkLUYjNiUtRiw2JVEvVmVjdG9yQ2FsY3VsdXNGJ0YvRjIvJStleGVjdXRhYmxlR0Y9RjlGOUY1LUY2Ni1RIn5GJ0Y5RjtGPkZARkJGREZGRkgvRktRJjAuMGVtRicvRk5Gam4tRiw2JVEsQmFzaXNGb3JtYXRGJ0YvRjItRlM2JC1GIzYlLUYsNiVGPUYvRjJGWkY5RjlGNS1JJ21zcGFjZUdGJDYmLyUnaGVpZ2h0R1EmMC4wZXhGJy8lJndpZHRoR0Zqbi8lJmRlcHRoR0Zqby8lKmxpbmVicmVha0dRKG5ld2xpbmVGJ0ZmbkZPLUZTNiQtRiM2JS1GLDYlUS5MaW5lYXJBbGdlYnJhRidGL0YyRlpGOUY5RjVGZm5GTy1GUzYkLUYjNiUtRiw2JVEmcGxvdHNGJ0YvRjJGWkY5RjlGNUZmbkZPLUZTNiQtRiM2JS1GLDYlUSpwbG90dG9vbHNGJ0YvRjJGWkY5RjlGNUZaRjk=reload 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kRGRkZIRmR1RmZ1RmFdbEZqb0Y5RjlGZHlGZ3lGYXZGX3BGXHEtRjY2L1ElZWxpZkYnRlJGVEZWRjtGPkZARkJGREZGRkhGWEZaRlxxRmVxRmJ3RmV3RlxxRmd3Rl9wRlxxRlxxRlxxRmdyRjUtRmZuNiQtRiM2KEZndUZhdUZkc0ZeeEZqb0Y5RjlGXHFGYHgtRmZ0NiUtRmZuNiQtRiM2JUZocUZqb0Y5RjlGaHRGXnVGYndGaXhGal5sRmF2Rl9wRlxxRlxxRlxxRmFyRjUtRmZuNiYtRiM2JkZkdi1GZm42JC1GIzYoRmdyRl1vLUYsNiVGZnNGL0YyRmdzRmpvRjlGOUZqb0Y5RjlGZHlGZ3lGYXZGX3BGXHFGXHFGXHFGXnJGNUZqeS1GZm42JC1GIzYqRmF6RmR6Rmd6LUZmbjYkLUYjNipGZHNGZ3NGYndGYXpGXW9GaHFGam9GOUY5Rl1vRmFyRmpvRjlGOUZhdkZfcEZccUZccUZccUZkckY1Rl5bbEZhW2xGYXZGX3BGXHFGXHFGXHFGZHJGNS1GZm42Ji1GIzYqRmpcbEZdXWxGXW9GZXdGXW9GZF1sRmpvRjlGOUZkeUZneUZhdkZfcEZccS1GNjYvUSVlbHNlRidGUkZURlZGO0Y+RkBGQkZERkZGSEZYRlpGX3BGXHFGXHFGXHFGW3JGNUZqd0ZccUZgeEZjeEZid0ZpeEZjeEZhdkZfcEZccUZccUZccUZeckY1Rlx5RmF2Rl9wRlxxRlxxRlxxRmFyRjVGanlGXXpGYXZGX3BGXHFGXHFGXHFGZHJGNUZeW2xGYVtsRmF2Rl9wRlxxLUY2Ni9RJGVuZEYnRlJGVEZWRjtGPkZARkJGREZGRkhGWEZaRlxxRl93RmF2Rl9wRlxxRmp5LUZmbjYkLUYjNipGW3RGZHotRiw2JVEmZXZhbGZGJ0YvRjItRmZuNiQtRiM2Ji1GLDYlUSlEaXN0YW5jZUYnRi9GMi1GZm42JC1GIzYrRmpuRl1vRmNvRl1vRmZvRl1vRlt0RmpvRjlGOUZqb0Y5RjlGXW9GZHJGam9GOUY5RmF2Rl9wRlxxRmlgbEZccUZPRmpvRjk=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzZdcC1JI21pR0YkNiVRNkVsbGlwc2VEaXN0YW5jZU1pbk1heEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RKiZjb2xvbmVxO0YnL0YzUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGPS8lKXN0cmV0Y2h5R0Y9LyUqc3ltbWV0cmljR0Y9LyUobGFyZ2VvcEdGPS8lLm1vdmFibGVsaW1pdHNHRj0vJSdhY2NlbnRHRj0vJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZMLUY2Ni9RJXByb2NGJy8lJWJvbGRHRjEvRjNRJWJvbGRGJy8lK2ZvbnR3ZWlnaHRHRlVGO0Y+RkBGQkZERkZGSC9GS1EmMC4wZW1GJy9GTkZZLUkobWZlbmNlZEdGJDYkLUYjNiktRiw2JVEiYkYnRi9GMi1GNjYtUSIsRidGOUY7L0Y/RjFGQEZCRkRGRkZIRlgvRk5RLDAuMzMzMzMzM2VtRictRiw2JVEickYnRi9GMkZdby1GLDYlUSgmdGhldGE7RicvRjBGPUY5LyUrZXhlY3V0YWJsZUdGPUY5RjktRiw2I1EhRictSSdtc3BhY2VHRiQ2Ji8lJ2hlaWdodEdRJjAuMGV4RicvJSZ3aWR0aEdGWS8lJmRlcHRoR0ZkcC8lKmxpbmVicmVha0dRKG5ld2xpbmVGJy1GNjYtUSJ+RidGOUY7Rj5GQEZCRkRGRkZIRlhGWi1GNjYvUSZsb2NhbEYnRlJGVEZWRjtGPkZARkJGREZGRkhGWEZaRlxxLUYsNiVRIk1GJ0YvRjJGXW8tRiw2JVEqcmVhbHJvb3R6RidGL0YyRl1vLUYsNiVRImlGJ0YvRjItRjY2LVEiO0YnRjlGO0Zgb0ZARkJGREZGRkhGWEZNRl9wRlxxRmJxRjUtRiw2JVEwRWxsaXBzZURpc3RhbmNlRidGL0YyRmVuLUY2Ni1RIjpGJ0Y5RjtGPkZARkJGREZGRkhGSkZNRl9wRlxxRmVxRjUtRmZuNiYtRiM2JUZccEZqb0Y5RjkvJSVvcGVuR1EiW0YnLyUmY2xvc2VHUSJdRidGYXJGX3BGXHEtRjY2L1EkZm9yRidGUkZURlZGO0Y+RkBGQkZERkZGSEZYRlpGXHFGaHFGXHEtRjY2L1ElZnJvbUYnRlJGVEZWRjtGPkZARkJGREZGRkhGWEZaRlxxLUkjbW5HRiQ2JFEiMUYnRjlGXHEtRjY2L1EjdG9GJ0ZSRlRGVkY7Rj5GQEZCRkRGRkZIRlhGWkZccS1GZXM2JFEiNEYnRjlGXHEtRjY2L1EjZG9GJ0ZSRlRGVkY7Rj5GQEZCRkRGRkZIRlhGWkZfcEZccUZccUZccS1GNjYvUSNpZkYnRlJGVEZWRjtGPkZARkJGREZGRkhGWEZaRlxxLUYsNiVRJmV2YWxiRidGL0YyLUZmbjYkLUYjNigtRiw2JVEjSW1GJ0Zpb0Y5LUZmbjYkLUYjNiZGYnEtRmZuNiYtRiM2JUZocUZqb0Y5RjlGaHJGW3NGam9GOUY5LUY2Ni1RIj1GJ0Y5RjtGPkZARkJGREZGRkhGSkZNLUZlczYkUSIwRidGOUZqb0Y5RjlGXHEtRjY2L1EldGhlbkYnRlJGVEZWRjtGPkZARkJGREZGRkhGWEZaRlxxRmVxRjUtRmZuNiYtRiM2KS1GLDYlUSNvcEYnRi9GMi1GZm42JC1GIzYlRmVxRmpvRjlGOUZdb0ZicUZidUZqb0Y5RjlGaHJGW3NGYXJGX3BGXHFGXHFGXHEtRjY2L1ElZWxzZUYnRlJGVEZWRjtGPkZARkJGREZGRkhGWEZaRlxxLUY2Ni9RI2ZpRidGUkZURlZGO0Y+RkBGQkZERkZGSEZYRlpGYXJGX3BGXHEtRjY2L1Ejb2RGJ0ZSRlRGVkY7Rj5GQEZCRkRGRkZIRlhGWkZhckZfcEZccS1GZm42Ji1GIzYpLUYsNiVRJG1pbkYnRmlvRjktRmZuNiQtRiM2JkZjdkZmdkZqb0Y5RjlGXW8tRiw2JVEkbWF4RidGaW9GOUZqd0Zqb0Y5RjlGaHJGW3NGYXJGX3BGXHEtRjY2L1EkZW5kRidGUkZURlZGO0Y+RkBGQkZERkZGSEZYRlpGXHFGT0Zqb0Y5LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzZkby1JI21pR0YkNiVRM0VsbGlwc2VEaXN0YW5jZU1pbkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RKiZjb2xvbmVxO0YnL0YzUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGPS8lKXN0cmV0Y2h5R0Y9LyUqc3ltbWV0cmljR0Y9LyUobGFyZ2VvcEdGPS8lLm1vdmFibGVsaW1pdHNHRj0vJSdhY2NlbnRHRj0vJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZMLUY2Ni9RJXByb2NGJy8lJWJvbGRHRjEvRjNRJWJvbGRGJy8lK2ZvbnR3ZWlnaHRHRlVGO0Y+RkBGQkZERkZGSC9GS1EmMC4wZW1GJy9GTkZZLUkobWZlbmNlZEdGJDYkLUYjNiktRiw2JVEiYkYnRi9GMi1GNjYtUSIsRidGOUY7L0Y/RjFGQEZCRkRGRkZIRlgvRk5RLDAuMzMzMzMzM2VtRictRiw2JVEickYnRi9GMkZdby1GLDYlUSgmdGhldGE7RicvRjBGPUY5LyUrZXhlY3V0YWJsZUdGPUY5RjktRiw2I1EhRictSSdtc3BhY2VHRiQ2Ji8lJ2hlaWdodEdRJjAuMGV4RicvJSZ3aWR0aEdGWS8lJmRlcHRoR0ZkcC8lKmxpbmVicmVha0dRKG5ld2xpbmVGJy1GNjYtUSJ+RidGOUY7Rj5GQEZCRkRGRkZIRlhGWi1GLDYlUSJNRidGL0YyRjUtRiw2JVEwRWxsaXBzZURpc3RhbmNlRidGL0YyRmVuLUY2Ni1RIjpGJ0Y5RjtGPkZARkJGREZGRkhGSkZNRl9wRlxxLUYsNiVRKnJlYWxyb290ekYnRi9GMkY1LUZmbjYmLUYjNiVGXHBGam9GOUY5LyUlb3BlbkdRIltGJy8lJmNsb3NlR1EiXUYnRmVxRl9wRlxxLUY2Ni9RJGZvckYnRlJGVEZWRjtGPkZARkJGREZGRkhGWEZaRlxxLUYsNiVRImlGJ0YvRjJGXHEtRjY2L1ElZnJvbUYnRlJGVEZWRjtGPkZARkJGREZGRkhGWEZaRlxxLUkjbW5HRiQ2JFEiMUYnRjlGXHEtRjY2L1EjdG9GJ0ZSRlRGVkY7Rj5GQEZCRkRGRkZIRlhGWkZccS1GX3M2JFEiNEYnRjlGXHEtRjY2L1EjZG9GJ0ZSRlRGVkY7Rj5GQEZCRkRGRkZIRlhGWkZfcEZccUZccUZccS1GNjYvUSNpZkYnRlJGVEZWRjtGPkZARkJGREZGRkhGWEZaRlxxLUYsNiVRJmV2YWxiRidGL0YyLUZmbjYkLUYjNigtRiw2JVEjSW1GJ0Zpb0Y5LUZmbjYkLUYjNiZGX3EtRmZuNiYtRiM2JUZockZqb0Y5RjlGX3JGYnJGam9GOUY5LUY2Ni1RIj1GJ0Y5RjtGPkZARkJGREZGRkhGSkZNLUZfczYkUSIwRidGOUZqb0Y5RjlGXHEtRjY2L1EldGhlbkYnRlJGVEZWRjtGPkZARkJGREZGRkhGWEZaRlxxRmhxRjUtRmZuNiYtRiM2KS1GLDYlUSNvcEYnRi9GMi1GZm42JC1GIzYlRmhxRmpvRjlGOUZdb0ZfcUZcdUZqb0Y5RjlGX3JGYnJGZXFGX3BGXHFGXHFGXHEtRjY2L1ElZWxzZUYnRlJGVEZWRjtGPkZARkJGREZGRkhGWEZaRlxxLUY2Ni9RI2ZpRidGUkZURlZGO0Y+RkBGQkZERkZGSEZYRlpGZXFGX3BGXHEtRjY2L1Ejb2RGJ0ZSRlRGVkY7Rj5GQEZCRkRGRkZIRlhGWkZlcUZfcEZccS1GLDYlUSRtaW5GJ0Zpb0Y5LUZmbjYkLUYjNiZGXXZGYHZGam9GOUY5LUY2Ni1RIjtGJ0Y5RjtGYG9GQEZCRkRGRkZIRlhGTUZfcEZccS1GNjYvUSRlbmRGJ0ZSRlRGVkY7Rj5GQEZCRkRGRkZIRlhGWkZccUZPRmpvRjk=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzZfcC1JI21pR0YkNiVRM0VsbGlwc2VEaXN0YW5jZU1heEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RKiZjb2xvbmVxO0YnL0YzUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGPS8lKXN0cmV0Y2h5R0Y9LyUqc3ltbWV0cmljR0Y9LyUobGFyZ2VvcEdGPS8lLm1vdmFibGVsaW1pdHNHRj0vJSdhY2NlbnRHRj0vJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZMLUY2Ni9RJXByb2NGJy8lJWJvbGRHRjEvRjNRJWJvbGRGJy8lK2ZvbnR3ZWlnaHRHRlVGO0Y+RkBGQkZERkZGSC9GS1EmMC4wZW1GJy9GTkZZLUkobWZlbmNlZEdGJDYkLUYjNiktRiw2JVEiYkYnRi9GMi1GNjYtUSIsRidGOUY7L0Y/RjFGQEZCRkRGRkZIRlgvRk5RLDAuMzMzMzMzM2VtRictRiw2JVEickYnRi9GMkZdby1GLDYlUSgmdGhldGE7RicvRjBGPUY5LyUrZXhlY3V0YWJsZUdGPUY5RjktRjY2LVEifkYnRjlGO0Y+RkBGQkZERkZGSEZYRlotSSdtc3BhY2VHRiQ2Ji8lJ2hlaWdodEdRJjAuMGV4RicvJSZ3aWR0aEdGWS8lJmRlcHRoR0ZkcC8lKmxpbmVicmVha0dRKG5ld2xpbmVGJy1GYHA2JkZicEZlcEZncC9GanBRJWF1dG9GJy1GNjYvUSZsb2NhbEYnRlJGVEZWRjtGPkZARkJGREZGRkhGWEZaRlxwLUYsNiVRIk1GJ0YvRjJGXW8tRiw2JVEqcmVhbHJvb3R6RidGL0YyRl1vLUYsNiVRImlGJ0YvRjItRjY2LVEiO0YnRjlGO0Zgb0ZARkJGREZGRkhGWEZNRl9wRl9wRlxwRmNxRjUtRiw2JVEwRWxsaXBzZURpc3RhbmNlRidGL0YyRmVuLUY2Ni1RIjpGJ0Y5RjtGPkZARkJGREZGRkhGSkZNRl9wRlxwRmZxRjUtRmZuNiYtRiM2JS1GLDYjUSFGJ0Zqb0Y5RjkvJSVvcGVuR1EiW0YnLyUmY2xvc2VHUSJdRidGYnJGX3BGXHAtRjY2L1EkZm9yRidGUkZURlZGO0Y+RkBGQkZERkZGSEZYRlpGXHBGaXFGXHAtRjY2L1ElZnJvbUYnRlJGVEZWRjtGPkZARkJGREZGRkhGWEZaRlxwLUkjbW5HRiQ2JFEiMUYnRjlGXHAtRjY2L1EjdG9GJ0ZSRlRGVkY7Rj5GQEZCRkRGRkZIRlhGWkZccC1GaXM2JFEiNEYnRjlGXHAtRjY2L1EjZG9GJ0ZSRlRGVkY7Rj5GQEZCRkRGRkZIRlhGWkZfcEZccEZccEZccC1GNjYvUSNpZkYnRlJGVEZWRjtGPkZARkJGREZGRkhGWEZaRlxwLUYsNiVRJmV2YWxiRidGL0YyLUZmbjYkLUYjNigtRiw2JVEjSW1GJ0Zpb0Y5LUZmbjYkLUYjNiZGY3EtRmZuNiYtRiM2JUZpcUZqb0Y5RjlGXHNGX3NGam9GOUY5LUY2Ni1RIj1GJ0Y5RjtGPkZARkJGREZGRkhGSkZNLUZpczYkUSIwRidGOUZqb0Y5RjlGXHAtRjY2L1EldGhlbkYnRlJGVEZWRjtGPkZARkJGREZGRkhGWEZaRlxwRmZxRjUtRmZuNiYtRiM2KS1GLDYlUSNvcEYnRi9GMi1GZm42JC1GIzYlRmZxRmpvRjlGOUZdb0ZjcUZmdUZqb0Y5RjlGXHNGX3NGYnJGX3BGXHBGXHBGXHAtRjY2L1ElZWxzZUYnRlJGVEZWRjtGPkZARkJGREZGRkhGWEZaRlxwLUY2Ni9RI2ZpRidGUkZURlZGO0Y+RkBGQkZERkZGSEZYRlpGYnJGX3BGXHAtRjY2L1Ejb2RGJ0ZSRlRGVkY7Rj5GQEZCRkRGRkZIRlhGWkZickZfcEZccC1GLDYlUSRtYXhGJ0Zpb0Y5LUZmbjYkLUYjNiZGZ3ZGanZGam9GOUY5RlxyRl9wRlxwLUY2Ni9RJGVuZEYnRlJGVEZWRjtGPkZARkJGREZGRkhGWEZaRlxwRk9Gam9GOQ==LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYtLUkjbWlHRiQ2JVEwRWxsaXBzZURpc3RhbmNlRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkobWZlbmNlZEdGJDYkLUYjNiktSSNtbkdGJDYkUSQuNzVGJy9GM1Enbm9ybWFsRictSSNtb0dGJDYtUSIsRidGPi8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGMS8lKXN0cmV0Y2h5R0ZGLyUqc3ltbWV0cmljR0ZGLyUobGFyZ2VvcEdGRi8lLm1vdmFibGVsaW1pdHNHRkYvJSdhY2NlbnRHRkYvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4zMzMzMzMzZW1GJy1GOzYkUSQwLjVGJ0Y+RkAtSSZtZnJhY0dGJDYoLUYsNiVRJSZwaTtGJy9GMEZGRj4tRiM2JS1GOzYkUSI0RidGPi8lK2V4ZWN1dGFibGVHRkZGPi8lLmxpbmV0aGlja25lc3NHUSIxRicvJStkZW5vbWFsaWduR1EnY2VudGVyRicvJSludW1hbGlnbkdGaW8vJSliZXZlbGxlZEdGRkZib0Y+Rj4tRkE2LVEiO0YnRj5GREZHRklGS0ZNRk9GUUZTL0ZXUSwwLjI3Nzc3NzhlbUYnLUYsNiVRM0VsbGlwc2VEaXN0YW5jZU1heEYnRi9GMkY1Rl5wLUYsNiVRM0VsbGlwc2VEaXN0YW5jZU1pbkYnRi9GMkY1Rl5wRmJvRj4=For some reason Maplesoft did not think it appropriate to plot LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiekYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= versus LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEickYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIixGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRiw2JVEoJnRoZXRhO0YnL0YwRj1GOUY5 in cylindrical coordinates in plot3d, but the help shows how to do this: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We just move the last coordinate first, since trial and error shows that the first variable is plotted versus the latter two: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, it works, plotting LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEiekYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnRjIvRjZRJ25vcm1hbEYn versus LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkobWZlbmNlZEdGJDYlLUYjNictSSNtaUdGJDYmUSJyRicvJSdpdGFsaWNHUSV0cnVlRicvJStleGVjdXRhYmxlR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYuUSIsRidGNy9GO1Enbm9ybWFsRicvJSZmZW5jZUdGOS8lKnNlcGFyYXRvckdGNi8lKXN0cmV0Y2h5R0Y5LyUqc3ltbWV0cmljR0Y5LyUobGFyZ2VvcEdGOS8lLm1vdmFibGVsaW1pdHNHRjkvJSdhY2NlbnRHRjkvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4zMzMzMzMzZW1GJy1GMTYmUScmIzk1MjtGJy9GNUY5RjdGQUY3RkFGN0ZBRjdGQQ==.First the we replot the ellipse and its evolute above which we will plot the minimum and maximum distance functions of the ellipse from a variable point in the plane.
We use the explicit parametrized equations for the evolute from the next section: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If we need to eliminate complex roots but apparently not necessary: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Now we give it a 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I am not sure how to interpret that. It seems wrong. Compare with the minimum distance. This is the long one, taking about 33 sec in Maple 11.02, 2008, oops again that MESH error after this ran successfully many times.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 the following is inconsistent with the previous two plots and I do not understand why at this moment. Will have to play with it later.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 give up here. We had this working once, but cannot recover it. Frustrating.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JSFHBy trial and error we found that wrapping the function in a pair of right quotes to delay evaluation produces a plot but it takes quite some time. explicit equations for the evolute of an ellipseFor plotting the evolute with the ellipse, we did not need its parametrization or equations since its geometric definition in terms of the radius of curvature and normal line was used, but if we wish to understand what the equation of the evolute is, we need to make them explicit and simplify formulas. We get the parametrization of the curve of centers of the osculating circles by going a distance equal to the radius of curvature (reciprocal of the curvature) along the normal vector from the point of tangency to get its position vector: 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Using the basic trigonometric identity, we easily write the unparametrized equation for the ellipse evolute: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where 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.Thus the semi-axes of the ellipse and evolute respectively are: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 we assume that LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiYUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= is the major semiaxis (LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJiRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiPEYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUYsNiVRImFGJ0Y0RjdGPkYrRj4=), then LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJjRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiPEYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUYsNiVRImFGJ0Y0RjdGPkYrRj4= so LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiY0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= times LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkmbWZyYWNHRiQ2KC1GIzYkLUkjbWlHRiQ2JVEiY0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GOFEnbm9ybWFsRictRiM2JC1GMTYlUSJhRidGNEY3RjovJS5saW5ldGhpY2tuZXNzR1EiMUYnLyUrZGVub21hbGlnbkdRJ2NlbnRlckYnLyUpbnVtYWxpZ25HRkYvJSliZXZlbGxlZEdRJmZhbHNlRidGOg== is less than LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiY0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=, so the corresponding semiaxis of the evolute is always strictly smaller than the focal length LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiY0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=. The endpoints of this axis of the evolute are thus always confined between the two foci of the ellipse. On the other hand the axis of the evolute along the minor axis is only less than LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiYkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= if LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2KEYrLUYjNihGKy1GIzYkLUklbXN1cEdGJDYlLUYsNiVRImFGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtbkdGJDYkUSIyRicvRj9RJ25vcm1hbEYnLyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJ0ZFLUkjbW9HRiQ2LVEoJm1pbnVzO0YnRkUvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRlAvJSlzdHJldGNoeUdGUC8lKnN5bW1ldHJpY0dGUC8lKGxhcmdlb3BHRlAvJS5tb3ZhYmxlbGltaXRzR0ZQLyUnYWNjZW50R0ZQLyUnbHNwYWNlR1EsMC4yMjIyMjIyZW1GJy8lJ3JzcGFjZUdGaW4tRiM2JC1GNjYlLUYsNiVRImJGJ0Y7Rj5GQUZHRkVGK0ZFLUZLNi1RIjxGJ0ZFRk5GUUZTRlVGV0ZZRmVuL0ZoblEsMC4yNzc3Nzc4ZW1GJy9GW29GZ29GXG9GK0ZFRitGRQ==, or 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, while the evolute extends outside the ellipse along its minor axis if 0 < 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 . This is nicely shown in an animation, where we set 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 so we don't have to run it backwards to start with a circle: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We can add the foci to this plot as well, using the fact that the focal distance from the center is 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The ellipse and evolute disappear once the evolute appears to pass the foci. Will have to troubleshoot why later. The foci should always be outside the evolute intercepts on the major axis.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If we look for the value of the radius of the point on the evolute in the first quadrant for which LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiPUYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUYsNiVRInlGJ0Y0RjdGPkYrRj4=, then the angle is 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 and the radius is 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 and we find the numerical value exactlly that we found above by graphical searching: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And for the configuration with LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJiRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiPUYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUkjbW5HRiQ2JFElMC43NUYnRj5GPkYrRj4=: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number of normals passing through a given point:
root analysisthe discriminant of a quartic (general theory)Older books on the "theory of equations" have a lot to say about the roots of third and fourth degree polynomials. [We had a personal copy of "Theory of Equations" by Cyrus MacDuffee, Wiley, 1959, written in 1954.]Start with a generic quartic and divide through by its leading coefficient: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By changing the variable by a translation, we can eliminate the cubic term: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 is then possible to show that the following cubic has the same "discriminant" as the previous quartic, where the discriminant is a certain function of the roots of any polynomial which vanishes if there is a multiple root, and which can be expressed as a polynomial in the coefficients of the polynomial: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This discriminant is invariant under translation and takes a much simpler form for a cubic if the quadratic term is absent, 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 two nontrivial coefficients are traditionally called LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEicEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEicUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=: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 the discriminant is: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 is the discriminant of the original quartic when multiplied by LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JC1JKG1zdWJzdXBHRiQ2Jy1GLDYlUSJhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiQtSSNtbkdGJDYkUSI0RicvRjtRJ25vcm1hbEYnRkMtRkA2JFEiNkYnRkMvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnLyUvc3Vic2NyaXB0c2hpZnRHRkpGQ0YrRkM= to compensate for the first step in dividing through by the leading 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and it is zero when multiple roots exist. Here is a simple test of a quartic with obvious multiple roots: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 has a discriminant command which produces the same result once the factor of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiYUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUkjbW5HRiQ2JFEiNEYnL0Y2USdub3JtYWxGJ0Y+LyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGPg== is taken into account::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 when it is positive, there are 4 real distinct roots or 2 distinct complex conjugate pairs, and when it is negative, 2 real distinct roots and two distinct complex conjugate roots.discriminant sign 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 ROOTS:
all real: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one complex conjugate pair: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 complex conjugate pairs: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 when the discriminant is positive, the roots are all real and distinct or there are two distinct complex conjugate pairs. This is easy to show in general by considering the alternative formula for the discriminant as proportional to the product of the squares of the differences of all distinct root pairs.LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbW9HRiQ2LVEifkYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZDLUkjbWlHRiQ2JVEjRDRGJy8lJ2l0YWxpY0dRJXRydWVGJy9GMFEnaXRhbGljRictRiw2LVEiPUYnRi9GMkY1RjdGOUY7Rj1GPy9GQlEsMC4yNzc3Nzc4ZW1GJy9GRUZTLUklbXN1cEdGJDYlLUkobWZlbmNlZEdGJDYkLUYjNi8tRlk2JC1GIzYnLUklbXN1YkdGJDYlLUZHNiVRInhGJ0ZKRk0tRiM2JS1JI21uR0YkNiRRIjFGJ0YvLyUrZXhlY3V0YWJsZUdGNEYvLyUvc3Vic2NyaXB0c2hpZnRHUSIwRictRiw2LVEoJm1pbnVzO0YnRi9GMkY1RjdGOUY7Rj1GPy9GQlEsMC4yMjIyMjIyZW1GJy9GRUZgcC1GXG82JUZeby1GIzYlLUZkbzYkUSIyRidGL0Znb0YvRmlvRmdvRi9GLy1GLDYtUTEmSW52aXNpYmxlVGltZXM7RidGL0YyRjVGN0Y5RjtGPUY/RkFGRC1GWTYkLUYjNidGW29GXHAtRlxvNiVGXm8tRiM2JS1GZG82JFEiM0YnRi9GZ29GL0Zpb0Znb0YvRi9GaXAtRlk2JC1GIzYnRltvRlxwLUZcbzYlRl5vLUYjNiUtRmRvNiRRIjRGJ0YvRmdvRi9GaW9GZ29GL0YvRmlwLUZZNiQtRiM2J0ZicEZccEZgcUZnb0YvRi9GaXAtRlk2JC1GIzYnRmJwRlxwRltyRmdvRi9GL0ZpcC1GWTYkLUYjNidGYHFGXHBGW3JGZ29GL0YvRmdvRi9GL0ZmcC8lMXN1cGVyc2NyaXB0c2hpZnRHRltwRmdvRi8=The difference of complex conjugate root pairs is purely imaginary contributing a negative sign to the product, while the remaining root differences are either real or always occur in complex conjugate pairs contributing only positive signs to the product. Thus one complex conjugate root pair makes the discriminant negative but two make it positive, like the all real (distinct) root case.Let's keep the eccentricity combination from expanding in our manipulations by making the substitution 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 in the orthogonality condition: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The number of real roots of this quartic in LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiWEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= (the LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= coordinate of the point on the ellipse) depends on whether the point with the given coordinates LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkobWZlbmNlZEdGJDYmLUYjNiYtSSNtaUdGJDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiLEYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4zMzMzMzMzZW1GJy1GMTYlUSJ5RidGNEY3Rj5GPi8lJW9wZW5HUSJbRicvJSZjbG9zZUdRIl1GJ0Y+ lies inside or outside the evolute, so with some patience about the conditions for the exact solution of a quartic, one might be able to see exactly how the root merging occurs on the evolute. Looks like some effort is required.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 hand we identify the coefficients of this Quartic for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JlEiYUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LlEiPUYnRjIvRjZRJ25vcm1hbEYnLyUmZmVuY2VHRjQvJSpzZXBhcmF0b3JHRjQvJSlzdHJldGNoeUdGNC8lKnN5bW1ldHJpY0dGNC8lKGxhcmdlb3BHRjQvJS5tb3ZhYmxlbGltaXRzR0Y0LyUnYWNjZW50R0Y0LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTi1JI21uR0YkNiVRIjFGJ0YyRjxGMkY8:LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2J0YrLUYjNictRiw2JVEjRDRGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYtUSM6PUYnL0Y6USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGRC8lKXN0cmV0Y2h5R0ZELyUqc3ltbWV0cmljR0ZELyUobGFyZ2VvcEdGRC8lLm1vdmFibGVsaW1pdHNHRkQvJSdhY2NlbnRHRkQvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZTLUkjbW5HRiQ2JFEiMUYnRkAvJStleGVjdXRhYmxlR0ZERkBGK0ZaRkAtRj02LVEiO0YnRkBGQi9GRkY4RkdGSUZLRk1GTy9GUlEmMC4wZW1GJ0ZURlpGQA==LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2J0YrLUYjNigtRiw2JVEjRDNGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYtUSM6PUYnL0Y6USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGRC8lKXN0cmV0Y2h5R0ZELyUqc3ltbWV0cmljR0ZELyUobGFyZ2VvcEdGRC8lLm1vdmFibGVsaW1pdHNHRkQvJSdhY2NlbnRHRkQvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZTLUYjNictRj02LVEqJnVtaW51czA7RidGQEZCRkVGR0ZJRktGTUZPL0ZSUSwwLjIyMjIyMjJlbUYnL0ZVRmZuLUYjNigtSSNtbkdGJDYkUSIyRidGQC1GPTYtUTEmSW52aXNpYmxlVGltZXM7RidGQEZCRkVGR0ZJRktGTUZPL0ZSUSYwLjBlbUYnL0ZVRmJvLUYjNiUtSSVtc3VwR0YkNiUtRiM2Jy1GLDYlUSRjb3NGJy9GN0ZERkAtRj02LVEwJkFwcGx5RnVuY3Rpb247RidGQEZCRkVGR0ZJRktGTUZPRmFvRmNvLUkobWZlbmNlZEdGJDYkLUYjNiUtRiw2JVEidEYnRjZGOS8lK2V4ZWN1dGFibGVHRkRGQEZARmpwRkAtRltvNiRRIjNGJ0ZALyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJ0ZqcEZARitGanBGQEYrRmpwRkBGK0ZqcEZARitGanBGQC1GPTYtUSI7RidGQEZCL0ZGRjhGR0ZJRktGTUZPRmFvRlRGanBGQA==LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2J0YrLUYjNigtRiw2JVEjRDJGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYtUSM6PUYnL0Y6USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGRC8lKXN0cmV0Y2h5R0ZELyUqc3ltbWV0cmljR0ZELyUobGFyZ2VvcEdGRC8lLm1vdmFibGVsaW1pdHNHRkQvJSdhY2NlbnRHRkQvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZTLUYjNipGKy1GIzYlLUklbXN1cEdGJDYlLUYjNictRiw2JVEkY29zRicvRjdGREZALUY9Ni1RMCZBcHBseUZ1bmN0aW9uO0YnRkBGQkZFRkdGSUZLRk1GTy9GUlEmMC4wZW1GJy9GVUZhby1JKG1mZW5jZWRHRiQ2JC1GIzYlLUYsNiVRInRGJ0Y2RjkvJStleGVjdXRhYmxlR0ZERkBGQEZbcEZALUkjbW5HRiQ2JFEiNkYnRkAvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnRltwRkAtRj02LVEiK0YnRkBGQkZFRkdGSUZLRk1GTy9GUlEsMC4yMjIyMjIyZW1GJy9GVUZocC1GIzYlLUZlbjYlLUYjNictRiw2JVEkc2luRidGXG9GQEZdb0Zjb0ZbcEZARl1wRmFwRltwRkAtRj02LVEoJm1pbnVzO0YnRkBGQkZFRkdGSUZLRk1GT0ZncEZpcC1GXnA2JFEiMUYnRkBGW3BGQEYrRltwRkBGK0ZbcEZALUY9Ni1RIjtGJ0ZARkIvRkZGOEZHRklGS0ZNRk9GYG9GVEZbcEZALUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2J0YrLUYjNigtRiw2JVEjRDFGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYtUSM6PUYnL0Y6USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGRC8lKXN0cmV0Y2h5R0ZELyUqc3ltbWV0cmljR0ZELyUobGFyZ2VvcEdGRC8lLm1vdmFibGVsaW1pdHNHRkQvJSdhY2NlbnRHRkQvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZTLUYjNigtSSNtbkdGJDYkUSIyRidGQC1GPTYtUTEmSW52aXNpYmxlVGltZXM7RidGQEZCRkVGR0ZJRktGTUZPL0ZSUSYwLjBlbUYnL0ZVRmpuLUYjNiUtSSVtc3VwR0YkNiUtRiM2Jy1GLDYlUSRjb3NGJy9GN0ZERkAtRj02LVEwJkFwcGx5RnVuY3Rpb247RidGQEZCRkVGR0ZJRktGTUZPRmluRltvLUkobWZlbmNlZEdGJDYkLUYjNiUtRiw2JVEidEYnRjZGOS8lK2V4ZWN1dGFibGVHRkRGQEZARmJwRkAtRlk2JFEiM0YnRkAvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnRmJwRkBGK0ZicEZARitGYnBGQEYrRmJwRkAtRj02LVEiO0YnRkBGQi9GRkY4RkdGSUZLRk1GT0ZpbkZURmJwRkA=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2J0YrLUYjNigtRiw2JVEjRDBGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYtUSM6PUYnL0Y6USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGRC8lKXN0cmV0Y2h5R0ZELyUqc3ltbWV0cmljR0ZELyUobGFyZ2VvcEdGRC8lLm1vdmFibGVsaW1pdHNHRkQvJSdhY2NlbnRHRkQvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZTLUYjNictRj02LVEqJnVtaW51czA7RidGQEZCRkVGR0ZJRktGTUZPL0ZSUSwwLjIyMjIyMjJlbUYnL0ZVRmZuLUYjNiUtSSVtc3VwR0YkNiUtRiM2Jy1GLDYlUSRjb3NGJy9GN0ZERkAtRj02LVEwJkFwcGx5RnVuY3Rpb247RidGQEZCRkVGR0ZJRktGTUZPL0ZSUSYwLjBlbUYnL0ZVRmdvLUkobWZlbmNlZEdGJDYkLUYjNiUtRiw2JVEidEYnRjZGOS8lK2V4ZWN1dGFibGVHRkRGQEZARmFwRkAtSSNtbkdGJDYkUSI2RidGQC8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRidGYXBGQEYrRmFwRkBGK0ZhcEZARitGYXBGQEYrRmFwRkA=Now we test the discriminant for the quartic in LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSRjb3NGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUkjbW9HRiQ2LVEwJkFwcGx5RnVuY3Rpb247RidGNy8lJmZlbmNlR0Y2LyUqc2VwYXJhdG9yR0Y2LyUpc3RyZXRjaHlHRjYvJSpzeW1tZXRyaWNHRjYvJShsYXJnZW9wR0Y2LyUubW92YWJsZWxpbWl0c0dGNi8lJ2FjY2VudEdGNi8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRk4tSShtZmVuY2VkR0YkNiQtRiM2JC1GLDYlUSJ0RicvRjVRJXRydWVGJy9GOFEnaXRhbGljRidGN0Y3RjdGK0Y3, using the formula from the subsection above.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This vanishing discriminant on the evolute means that there is at least one root of multiplicity greater than 1 there and so a maximum of 3 real distinct roots.Plot of our discriminantWe can plot the discriminant as a function of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= to see its behavior off the evolute. Since it depends on the parameters LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiYUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiYkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= only through the combinations 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, we can plot this as a function of the new position variables and get the whole story (incidentally, these are the variables which reduce the "stretched astroid" evolute curve to the standard astroid):LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2JS1GLDYlUShRdWFydGljRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvRjhRJ25vcm1hbEYnRitGOkY9LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2J0YrLUYjNigtRiw2JVErUXVhcnRpY191dkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIzo9RicvRjpRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZELyUpc3RyZXRjaHlHRkQvJSpzeW1tZXRyaWNHRkQvJShsYXJnZW9wR0ZELyUubW92YWJsZWxpbWl0c0dGRC8lJ2FjY2VudEdGRC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRlMtRiM2Jy1GLDYlUSVzdWJzRidGNkY5LUY9Ni1RMCZBcHBseUZ1bmN0aW9uO0YnRkBGQkZFRkdGSUZLRk1GTy9GUlEmMC4wZW1GJy9GVUZpbi1JKG1mZW5jZWRHRiQ2JC1GIzYrRistRiM2KC1GLDYlUSJ4RidGNkY5LUY9Ni1RIj1GJ0ZARkJGRUZHRklGS0ZNRk9GUUZULUYjNictSSVtc3VwR0YkNiUtRiw2JVEiY0YnRjZGOS1JI21uR0YkNiRRIjJGJ0ZALyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJy1GPTYtUTEmSW52aXNpYmxlVGltZXM7RidGQEZCRkVGR0ZJRktGTUZPRmhuRmpuLUYsNiVRInVGJ0Y2RjkvJStleGVjdXRhYmxlR0ZERkBGK0ZdcUZALUY9Ni1RIixGJ0ZARkIvRkZGOEZHRklGS0ZNRk9GaG4vRlVRLDAuMzMzMzMzM2VtRictRiM2Jy1GLDYlUSJ5RidGNkY5RmVvLUkmbWZyYWNHRiQ2KC1GIzYnRmpvRmdwLUYsNiVRInZGJ0Y2RjlGXXFGQC1GIzYlLUYsNiVRImJGJ0Y2RjlGXXFGQC8lLmxpbmV0aGlja25lc3NHUSIxRicvJStkZW5vbWFsaWduR1EnY2VudGVyRicvJSludW1hbGlnbkdGXHMvJSliZXZlbGxlZEdGREZdcUZARl9xLUYjNictRiw2JVEnZXhwYW5kRidGNkY5RmVuLUZcbzYkLUYjNiUtRltyNigtRiM2JS1GLDYlUShRdWFydGljRidGNkY5Rl1xRkAtRiM2J0YrLUYjNiUtRltwNiVGXXAtRmFwNiRRIjRGJ0ZARmRwRl1xRkBGK0ZdcUZARmdyRmpyRl1zRl9zRl1xRkBGQEZdcUZARitGXXFGQEZARl1xRkBGK0ZdcUZARitGXXFGQEYrRl1xRkA=Now we extract the coefficients and evaluate the 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This is also zero at the origin and along the axes, where because of symmetry, one has at least a pair of repeated real roots for the cosine, but four distinct values for the angle itself inside the evolute (check this statement).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We have to zoom in on the vertical scale to see what is happening since the graph is so flat:LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2J0YrLUYjNictRiw2JVEncGxvdDNkRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEwJkFwcGx5RnVuY3Rpb247RicvRjpRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZELyUpc3RyZXRjaHlHRkQvJSpzeW1tZXRyaWNHRkQvJShsYXJnZW9wR0ZELyUubW92YWJsZWxpbWl0c0dGRC8lJ2FjY2VudEdGRC8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRlMtSShtZmVuY2VkR0YkNiQtRiM2QUYrLUYjNiYtRiw2JVElc3Vic0YnRjZGOS1GVzYkLUYjNiotRiw2JVEiYUYnRjZGOS1GPTYtUSI9RidGQEZCRkVGR0ZJRktGTUZPL0ZSUSwwLjI3Nzc3NzhlbUYnL0ZVRmVvLUkjbW5HRiQ2JFEiMUYnRkAtRj02LVEiLEYnRkBGQi9GRkY4RkdGSUZLRk1GT0ZRL0ZVUSwwLjMzMzMzMzNlbUYnLUY9Ni1RIn5GJ0ZARkJGRUZHRklGS0ZNRk9GUUZULUYsNiVRKERpc2NfdXZGJ0Y2RjkvJStleGVjdXRhYmxlR0ZERkBGQEZncEZARitGW3AtRiw2JVEidUYnRjZGOUZhby1GIzYmLUY9Ni1RKiZ1bWludXMwO0YnRkBGQkZFRkdGSUZLRk1GTy9GUlEsMC4yMjIyMjIyZW1GJy9GVUZicUZnb0ZncEZALUY9Ni1RIy4uRidGQEZCRkVGR0ZJRktGTUZPRmFxRlRGZ29GW3AtRiw2JVEidkYnRjZGOUZhb0ZccUZkcUZnb0ZbcC1GLDYlUSV2aWV3RidGNkY5RmFvLUYjNiZGXnEtRmhvNiRRJjAuMDAxRidGQEZncEZARmRxLUZobzYkUSYwLjAwMkYnRkBGW3AtRiw2JVElYXhlc0YnRjZGOUZhby1GLDYlUSZib3hlZEYnRjZGOUZbcC1GLDYlUSZzdHlsZUYnRjZGOUZhby1GLDYlUShjb250b3VyRidGNkY5RmdwRkBGQEZncEZARitGZ3BGQEYrRmdwRkA=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2J0YrLUYjNictRiw2JVEncGxvdDNkRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEwJkFwcGx5RnVuY3Rpb247RicvRjpRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZELyUpc3RyZXRjaHlHRkQvJSpzeW1tZXRyaWNHRkQvJShsYXJnZW9wR0ZELyUubW92YWJsZWxpbWl0c0dGRC8lJ2FjY2VudEdGRC8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRlMtSShtZmVuY2VkR0YkNiQtRiM2O0YrLUYjNiYtRiw2JVElc3Vic0YnRjZGOS1GVzYkLUYjNiotRiw2JVEiYUYnRjZGOS1GPTYtUSI9RidGQEZCRkVGR0ZJRktGTUZPL0ZSUSwwLjI3Nzc3NzhlbUYnL0ZVRmVvLUkjbW5HRiQ2JFEiMUYnRkAtRj02LVEiLEYnRkBGQi9GRkY4RkdGSUZLRk1GT0ZRL0ZVUSwwLjMzMzMzMzNlbUYnLUY9Ni1RIn5GJ0ZARkJGRUZHRklGS0ZNRk9GUUZULUYsNiVRKERpc2NfdXZGJ0Y2RjkvJStleGVjdXRhYmxlR0ZERkBGQEZncEZARitGW3AtRiw2JVEidUYnRjZGOUZhby1GIzYoRistRiM2Ji1GPTYtUSomdW1pbnVzMDtGJ0ZARkJGRUZHRklGS0ZNRk8vRlJRLDAuMjIyMjIyMmVtRicvRlVGZHEtRmhvNiRRJDAuNUYnRkBGZ3BGQC1GPTYtUSMuLkYnRkBGQkZFRkdGSUZLRk1GT0ZjcUZURmZxRmdwRkBGW3AtRiw2JVEidkYnRjZGOUZhb0ZccUZbcC1GLDYlUSV2aWV3RidGNkY5RmFvLUYjNihGKy1GIzYmRmBxLUZobzYkUSYwLjAwMUYnRkBGZ3BGQEZpcS1GaG82JFEmMC4wMDJGJ0ZARmdwRkBGW3AtRiw2JVElYXhlc0YnRjZGOUZhby1GLDYlUSZib3hlZEYnRjZGOUZbcC1GLDYlUSZzdHlsZUYnRjZGOUZhby1GLDYlUShjb250b3VyRidGNkY5RmdwRkBGQEZncEZARitGZ3BGQEYrRmdwRkA=The discriminant is not a monotonic function along the radial direction, but has 4 local maxima inside the outer zero contour (and a local minimum with zero value at the origin), which seems to be the first closed contour enclosing all these contours around the local maxima. It is positive at all other points within the evolute, and negative off the axes outside the evolute, which correlates with the different possible subcases of multiple roots for a quartic.None of this yet prove our claims, since we must follow up with the analysis of the reality of the roots inside and outside the evolute, etc., but it does show that the evolute has to be the curve across which the number of normals through a given point changes.Discriminant in terms of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEidUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIixGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRiw2JVEidkYnRi9GMkY5 (never give up!)In these new variables the parametric equations for the evolute are: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and this confirms that the discriminant is zero on it: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The unparametrized equation for the evolute in the new variables 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which motivates the following variable change, where LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiVUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIixGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRiw2JVEiVkYnRi9GMkY5 are nonnegative variables needed to factor the discriminant: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 only depends on even powers of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEidUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIixGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRiw2JVEidkYnRi9GMkY5 so only nonnegative values of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiVUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIixGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRiw2JVEiVkYnRi9GMkY5 are relevant: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The only real solution setting the second factor to zero has 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: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referee sum of squares representationSince LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiVUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiVkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= must be nonnegative, there is not much advantage of the following sum of squares representation over the previous one for the only factor which has a zero: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The first factor of the discfactor (removing the factors which vanish on the axes) is zero exactly on the evolute, while the remaining factor is always positive except at this particular point where it has its absolute minimum, but which is irrelevant to our analysis since LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiVUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIixGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRiw2JVEiVkYnRi9GMkY5 are both negative there: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Thus we have succeeded in showing that the discriminant is zero only on the axes and on the evolute.Summarizing, the original discriminant has the following factorization in terms of the new variables: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 the first 4 factors have real roots, and only the first 3 have nonnegative roots; the third factor is the equation of the evolute.the hyperbolas of ApolloniusAfter setting up all of the above routines, we then went to the history books and discovered the work of Apollonius.Apollonius did not have the tools of calculus or analytic geometry to explore this geometry, so he used ordinary geometry instead, which made his discussions very complicated. He studied many properties of the conic sections and in particular investigated the distance problem for each of them, namely how to construct the shortest and longest connecting line segments to the conic section from a given point in the plane, and how the distance between the given point and a point on the ellipse varied as one moved away from these special points on the ellipse where the distance is extremal. A construction with similar triangles leads to an equation of a hyperbola whose intersections with the ellipse are the points from which normals pass through the given point. Our construction reobtains his results analytically but not following his geometric approach, which is not easily digested. This hyperbola is a great help in visualizing what happens to the normals as one moves the given point around in the plane.Here is the hyperbola of Apollonius incorporated into the previous procedure to show the resulting geometry.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the extended procedure EllipserHyp(b,r,theta)LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzZgZWwtSSNtaUdGJDYlUSxFbGxpcHNlckh5cEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RKiZjb2xvbmVxO0YnL0YzUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGPS8lKXN0cmV0Y2h5R0Y9LyUqc3ltbWV0cmljR0Y9LyUobGFyZ2VvcEdGPS8lLm1vdmFibGVsaW1pdHNHRj0vJSdhY2NlbnRHRj0vJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZMLUY2Ni9RJXByb2NGJy8lJWJvbGRHRjEvRjNRJWJvbGRGJy8lK2ZvbnR3ZWlnaHRHRlVGO0Y+RkBGQkZERkZGSC9GS1EmMC4wZW1GJy9GTkZZLUkobWZlbmNlZEdGJDYkLUYjNiktRiw2JVEiYkYnRi9GMi1GNjYtUSIsRidGOUY7L0Y/RjFGQEZCRkRGRkZIRlgvRk5RLDAuMzMzMzMzM2VtRictRiw2JVEickYnRi9GMkZdby1GLDYlUSZ0aGV0YUYnL0YwRj1GOS8lK2V4ZWN1dGFibGVHRj1GOUY5LUY2Ni1RIn5GJ0Y5RjtGPkZARkJGREZGRkhGWEZaLUknbXNwYWNlR0YkNiYvJSdoZWlnaHRHUSYwLjBleEYnLyUmd2lkdGhHRlkvJSZkZXB0aEdGZHAvJSpsaW5lYnJlYWtHUShuZXdsaW5lRictRmBwNiZGYnBGZXBGZ3AvRmpwUSVhdXRvRictRjY2L1EmbG9jYWxGJ0ZSRlRGVkY7Rj5GQEZCRkRGRkZIRlhGWkZccC1GLDYlUSJhRidGL0YyRl1vLUYsNiVRJGVxMUYnRi9GMkZdby1GLDYlUSRlcTJGJ0YvRjJGXW8tRiw2JVEkZXEzRidGL0YyRl1vLUYsNiVRJENPU0YnRi9GMkZdby1GLDYlUSRTSU5GJ0YvRjJGXW8tRiw2JVElQ09TdEYnRi9GMkZdby1GLDYlUSVTSU50RidGL0YyRl1vLUYsNiVRI3R0RidGL0YyRl1vLUYsNiVRIlJGJ0YvRjJGXW8tRiw2JVEjQ0NGJ0YvRjJGXW8tRiw2JVEqQ3VydmVwbG90RidGL0YyRl1vLUYsNiVRJ0NDcGxvdEYnRi9GMkZdby1GLDYlUSRwcHRGJ0YvRjJGXW8tRiw2JVEjcHBGJ0YvRjJGXW8tRiw2JVEjcDFGJ0YvRjJGXW8tRiw2JVEjcDJGJ0YvRjJGXW8tRiw2JVEjcDNGJ0YvRjJGXW8tRiw2JVEjcDRGJ0YvRjJGXW8tRiw2JVEjcDVGJ0YvRjJGXW8tRiw2JVEjcGhGJ0YvRjJGXW8tRiw2JVElcHB0c0YnRi9GMkZdby1GLDYlUScmIzk0NTtGJ0Zpb0Y5Rl1vLUYsNiVRI3gwRidGL0YyRl1vLUYsNiVRI3kwRidGL0YyLUY2Ni1RIjtGJ0Y5RjtGYG9GQEZCRkRGRkZIRlhGTUZfcEZfcEZfcEZccEZjcUY1LUkjbW5HRiQ2JFEiMUYnRjktRjY2LVEiOkYnRjlGO0Y+RkBGQkZERkZGSEZKRk1GX3BGXHFGZnFGNS1GLDYlUSRzaW5GJ0Zpb0Y5LUZmbjYkLUYjNiUtRiw2JVEidEYnRi9GMkZqb0Y5RjlGXHAtRiw2JVEkY29zRidGaW9GOUZbd0ZccC1GZm42JC1GIzYnLUklbXN1cEdGJDYlRmpuLUYjNiUtRmJ2NiRRIjJGJ0Y5RmpvRjkvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnLUY2Ni1RKCZtaW51cztGJ0Y5RjtGPkZARkJGREZGRkgvRktRLDAuMjIyMjIyMmVtRicvRk5GaHhGYXZGam9GOUY5LUY2Ni1RIitGJ0Y5RjtGPkZARkJGREZGRkhGZ3hGaXhGXHBGaHZGW3dGXHBGY29GXHBGYnctRmZuNiQtRiM2JS1GLDYlUScmIzk1MjtGJ0Zpb0Y5RmpvRjlGOUZkeEZqbkZccEZjb0ZccEZid0Zbd0ZccEZodkZdeUZldkZfcEZccEZickY1LUYsNiVRJnNvbHZlRidGL0YyLUZmbjYkLUYjNihGZnFGXW9GaHZGW3dGam9GOUY5RmV2Rl9wRlxwRl9yRjVGZHktRmZuNiQtRiM2KEZmcUZdb0Zid0Zbd0Zqb0Y5RjlGXnZGX3BGXHAtRjY2L1EjaWZGJ0ZSRlRGVkY7Rj5GQEZCRkRGRkZIRlhGWkZccEZfci1GNjYtUSI9RidGOUY7Rj5GQEZCRkRGRkZIRkpGTS1GYnY2JEZjeEY5RlxwLUY2Ni9RJXRoZW5GJ0ZSRlRGVkY7Rj5GQEZCRkRGRkZIRlhGWkZfcEZccUZccEZccEZpcUY1LUZmbjYkLUYjNilGYXYtRjY2LVEqJnVtaW51czA7RidGOUY7Rj5GQEZCRkRGRkZIRmd4Rml4RmJ3LUZqdzYlRlt3Rlx4RmF4LUYsNiNRIUYnRmpvRjlGOUZccC1GLDYlUSZkZW5vbUYnRi9GMi1Ganc2JS1GZm42JC1GIzYlRmJyRmpvRjlGOUZceEZheEZiei1GLDYlUSZudW1lckYnRi9GMkZpW2xGXnZGX3BGXHFGXHBGXHBGZXJGNS1GZm42Ji1GIzYmRmR5LUZmbjYkLUYjNihGaXFGXW9GYndGW3dGam9GOUY5RmpvRjlGOS8lJW9wZW5HUSJbRicvJSZjbG9zZUdRIl1GJ0ZedkZfcEZccUZccEZccEZockY1LUYsNiVRJG1hcEYnRi9GMi1GZm42JC1GIzYqLUYsNiVRInpGJ0YvRjItRjY2LVEoJnNyYXJyO0YnRjlGO0Y+RkBGQkZERkZGSEZKRk0tRiw2JVElc3Vic0YnRi9GMi1GZm42JC1GIzYqRmJ3Rlt3RmJ6RmddbEZdb0ZickZqb0Y5RjlGXW9GZXJGam9GOUY5Rl52Rl9wRlxxRlxwRlxwRltzRjUtRiw2JVEkemlwRidGL0YyLUZmbjYkLUYjNiwtRmZuNiQtRiM2Jy1GLDYlUSJ4RidGL0YyRl1vLUYsNiVRInlGJ0YvRjJGam9GOUY5RmpdbC1GLDYlUSdhcmN0YW5GJ0Zpb0Y5LUZmbjYkLUYjNidGYl9sRl1vRl9fbEZqb0Y5RjlGXW9GZXJGXW9GaHJGam9GOUY5Rl52Rl9wRlxxRlxwRlxwRltzRjUtRmZuNiYtRiM2Ky1GLDYlUSNvcEYnRi9GMi1GZm42JC1GIzYlRltzRmpvRjlGOUZdby1JJm1mcmFjR0YkNigtRiw2JVEnJiM5NjA7RidGaW9GOUZceC8lLmxpbmV0aGlja25lc3NHRmR2LyUrZGVub21hbGlnbkdRJ2NlbnRlckYnLyUpbnVtYWxpZ25HRmFhbC8lKWJldmVsbGVkR0Y9Rl1vRl5bbEZnYGxGam9GOUY5RmpcbEZdXWxGZXZGX3BGXHAtRjY2L1ElZWxpZkYnRlJGVEZWRjtGPkZARkJGREZGRkhGWEZaRlxwRmJyRmJ6RmV6RlxwRmd6Rl9wRlxxRlxwRlxwRlxyRjUtRmZuNiQtRiM2KUZhdkZeW2xGaHZGYVtsRmNbbEZqb0Y5RjlGXHBGZltsLUZqdzYlLUZmbjYkLUYjNiVGX3JGam9GOUY5Rlx4RmF4RmJ6Rl9cbEZdYmxGXnZGX3BGXHFGXHBGXHBGaHJGNS1GZm42Ji1GIzYmRmR5LUZmbjYkLUYjNihGXHJGXW9GaHZGW3dGam9GOUY5RmpvRjlGOUZqXGxGXV1sRl52Rl9wRlxxRlxwRlxwRmVyRjVGYF1sLUZmbjYkLUYjNipGZ11sRmpdbEZdXmwtRmZuNiQtRiM2KkZodkZbd0ZiekZnXWxGXW9GX3JGam9GOUY5Rl1vRmhyRmpvRjlGOUZedkZfcEZccUZccEZccEZbc0Y1RmRebEZnXmxGXnZGX3BGXHFGXHBGXHBGW3NGNS1GZm42Ji1GIzYqRmBgbEZjYGxGXW9GZXpGXW8tRiw2JVEjUGlGJ0Zpb0Y5RmpvRjlGOUZqXGxGXV1sRmV2Rl9wRlxxLUY2Ni9RJWVsc2VGJ0ZSRlRGVkY7Rj5GQEZCRkRGRkZIRlhGWkZfcEZccUZccEZccEZpcUY1Rmp6RlxwRmZbbEZpW2xGYnpGX1xsRmlbbEZedkZfcEZccUZccEZccEZlckY1RmJcbEZedkZfcEZccUZccEZccEZockY1RmBdbEZjXWxGXnZGX3BGXHFGXHBGXHBGW3NGNUZkXmxGZ15sRl52Rl9wRlxwLUY2Ni9RJGVuZEYnRlJGVEZWRjtGPkZARkJGREZGRkhGWEZaRlxwRl96Rl52RlxwRl9wRlxxRl5zRjVGX3ctRjY2LUZcXmxGOUY7Rj5GQEZCRkRGRkZIRlhGWi1GZm42Ji1GIzYvRmNxRlxwRmJ3Rlt3Rl1vRmpuRlxwRmh2Rlt3Rl1vRmV6RmpvRjlGOS9GW11sUScmbGFuZztGJy9GXl1sUScmcmFuZztGJ0ZldkZccEZec0Zbd0ZedkZfcEZccUZhc0Y1LUYsNiVRKHVuYXBwbHlGJ0YvRjItRmZuNiQtRiM2Jy1GZm42JC1GIzYrRl5zRlt3Rmp4LUZoYGw2KEZhdi1GIzYnLUYsNiVRKkN1cnZhdHVyZUYnRi9GMi1GZm42JC1GIzYlRl5zRmpvRjlGOUZbd0Zqb0Y5Rl1hbEZfYWxGYmFsRmRhbC1GNjYtUScmc2RvdDtGJ0Y5RjtGPkZARkJGREZGRkhGWEZaLUYsNiVRKVROQkZyYW1lRidGL0YyLUZmbjYkLUYjNi5GXnNGW3dGXW9GX3dGXW8tRjY2LVEiJ0YnRjlGO0Y+RkBGQkZERkZGSC9GS1EsMC4xMTExMTExZW1GJ0ZaLUYsNiVRJ291dHB1dEYnRi9GMkZqZmxGYnotRmZuNiYtRiM2J0ZqZmwtRiw2JVEiTkYnRi9GMkZqZmxGam9GOUY5RmpcbEZdXWxGam9GOUY5RmpvRjlGOUZdb0Zfd0Zqb0Y5RjlGZXZGXHFGX3BGX3BGX3BGXHBGZHNGNS1GLDYlUSVwbG90RidGL0YyLUZmbjYkLUYjNjEtRmZuNiYtRiM2M0Zec0Zbdy1GZm42Ji1GIzYlRmF2RmpvRjlGOUZqXGxGXV1sRl1vRl5zRlt3LUZmbjYmRlx4RjlGalxsRl1dbEZdb0Zfd0ZiekZlei1GNjYtUSMuLkYnRjlGO0Y+RkBGQkZERkZGSEZneEZaRl54RlxwLUYsNiVRJSZwaTtGJ0Zpb0Y5RmpvRjlGOUZqXGxGXV1sRl1vLUYsNiVRKm51bXBvaW50c0YnRi9GMkZiei1GYnY2JFEkMTkzRidGOUZdby1GLDYlUShzY2FsaW5nRidGL0YyRmJ6LUYsNiVRLGNvbnN0cmFpbmVkRidGL0YyRl1vLUYsNiVRJmNvbG9yRidGL0YyRmJ6LUYsNiVRJWJsdWVGJ0YvRjJGam9GOUY5RmV2Rl9wRlxwRmdzRjVGaWdsLUZmbjYkLUYjNjEtRmZuNiYtRiM2M0Zhc0Zbd0ZkaGxGXW9GYXNGW3dGaGhsRl1vRl93RmJ6RmV6RmpobEZeeEZccEZdaWxGam9GOUY5RmpcbEZdXWxGXW9GYGlsRmJ6RmNpbEZdb0ZmaWxGYnpGaWlsRl1vRlxqbEZiei1GLDYlUSZibGFja0YnRi9GMkZqb0Y5RjlGZXZGX3BGXHBGaWdsLUZmbjYkLUYjNjEtRmZuNiYtRiM2JS1GZm42Ji1GIzYtRmNvRlxwRmJ3Rl15Rl1vRmNvRlxwRmh2Rl15RmpvRjlGOUZqXGxGXV1sRmpvRjlGOUZqXGxGXV1sRl1vLUYsNiVRJnN0eWxlRidGL0YyRmJ6LUYsNiVRJnBvaW50RidGL0YyRl1vLUYsNiVRJ3N5bWJvbEYnRi9GMkZiei1GLDYlUSdDSVJDTEVGJ0YvRjJGXW9GXGpsRmJ6Rl9qbEZqb0Y5RjlGZXZGXHBGanNGNS1GLDYlUSIlRidGL0YyRmV2Rl9wRlxxRmlnbC1GZm42JC1GIzYpLUZmbjYmLUYjNictRmZuNiYtRiM2J0ZlekZdb0ZlekZqb0Y5RjlGalxsRl1dbEZdb0ZlW21Gam9GOUY5RmpcbEZdXWxGXW9GXGpsRmJ6LUYsNiVRJWdyZXlGJ0YvRjJGam9GOUY5RmV2RlxwRl10RjVGZVxtRmV2Rl9wRlxxRmlnbC1GZm42JC1GIzYpLUZmbjYmLUYjNidGZVttRl1vLUZmbjYmLUYjNitGYnctRmZuNiQtRiM2JkZbc0ZkaGxGam9GOUY5Rl1vRmpuRlxwLUYsNiVGanZGL0YyRmNebUZqb0Y5RjlGalxsRl1dbEZqb0Y5RjlGalxsRl1dbEZdb0ZcamxGYnotRiw2JVEkcmVkRidGL0YyRmpvRjlGOUZldkZccEZgdEY1RmVcbUZldkZfcEZccUZpZ2wtRmZuNiQtRiM2KS1GZm42Ji1GIzYnRmVbbUZdby1GZm42Ji1GIzYrRmJ3LUZmbjYkLUYjNiZGW3NGaGhsRmpvRjlGOUZdb0ZqbkZccEZnXm1GaF9tRmpvRjlGOUZqXGxGXV1sRmpvRjlGOUZqXGxGXV1sRl1vRlxqbEZiekZpXm1Gam9GOUY5RmV2RlxwRmN0RjVGZVxtRmV2Rl9wRlxxRmlnbC1GZm42JC1GIzYpLUZmbjYmLUYjNidGZVttRl1vLUZmbjYmLUYjNitGYnctRmZuNiQtRiM2JkZbcy1GZm42Ji1GIzYlLUZidjYkUSIzRidGOUZqb0Y5RjlGalxsRl1dbEZqb0Y5RjlGXW9Gam5GXHBGZ15tRmhgbUZqb0Y5RjlGalxsRl1dbEZqb0Y5RjlGalxsRl1dbEZdb0ZcamxGYnpGaV5tRmpvRjlGOUZldkZccEZmdEY1RmVcbUZldkZfcEZccUZpZ2wtRmZuNiQtRiM2KS1GZm42Ji1GIzYnRmVbbUZdby1GZm42Ji1GIzYrRmJ3LUZmbjYkLUYjNiZGW3MtRmZuNiYtRiM2JS1GYnY2JFEiNEYnRjlGam9GOUY5RmpcbEZdXWxGam9GOUY5Rl1vRmpuRlxwRmdebUZfYm1Gam9GOUY5RmpcbEZdXWxGam9GOUY5RmpcbEZdXWxGXW9GXGpsRmJ6RmlebUZqb0Y5RjlGZXZGXHBGaXRGNUZlXG1GZXZGX3BGXHBGaHVGNUZjb0ZccEZid0ZdeUZldkZccEZbdkY1RmNvRlxwRmh2Rl15RmV2Rl9wRlxwLUYsNiVRLWltcGxpY2l0cGxvdEYnRi9GMi1GZm42JC1GIzZHRl9fbEZccEZiX2xGXHAtRmZuNiQtRiM2J0ZhdkZkeEZpd0Zqb0Y5RjlGZHhGaHVGXHBGYl9sRmp4Rlt2RlxwRl9fbEZccEZpd0ZiekZlekZdb0ZfX2xGYnpGXltsRmF2RmpobEZhdkZdb0ZiX2xGYnpGXltsRmpuRmpobEZqbkZdb0ZcamxGYnotRiw2JVEmZ3JlZW5GJ0YvRjJGam9GOUY5RmV2RlxwRlx1RjVGZVxtRmV2Rl9wRlxxRl9wRlxwLUYsNiVRKGRpc3BsYXlGJ0YvRjItRmZuNiQtRiM2NUZkc0Zdb0Znc0Zdb0Zqc0Zdb0ZddEZdb0ZgdEZdb0ZjdEZdb0ZmdEZdb0ZpdEZdb0ZcdUZqb0Y5RjlGXnZGXHBGX3BGXHFGXWRsRlxwRk9GZXZGX3BGXHFGX3BGXHFGY1tsRmpvRjk=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2Jy1GLDYlUSxFbGxpcHNlckh5cEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RMCZBcHBseUZ1bmN0aW9uO0YnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZRLUkobWZlbmNlZEdGJDYkLUYjNiotSSNtbkdGJDYkUSUwLjc1RidGPi1GOzYtUSIsRidGPkZAL0ZERjZGRUZHRklGS0ZNRk8vRlNRLDAuMzMzMzMzM2VtRictRlo2JFEkMC4yRidGPkZnbi1GIzYmLUY7Ni1RKiZ1bWludXMwO0YnRj5GQEZDRkVGR0ZJRktGTS9GUFEsMC4yMjIyMjIyZW1GJy9GU0Zmby1JJm1mcmFjR0YkNigtRiM2JS1GLDYlUSUmcGk7RicvRjVGQkY+LyUrZXhlY3V0YWJsZUdGQkY+LUYjNiUtRlo2JFEiNEYnRj5GYXBGPi8lLmxpbmV0aGlja25lc3NHUSIxRicvJStkZW5vbWFsaWduR1EnY2VudGVyRicvJSludW1hbGlnbkdGXXEvJSliZXZlbGxlZEdGQkZhcEY+RitGYXBGPkY+RmFwRj5GK0ZhcEY+Then a no-tickmark black and white version with explicit circle points added in for all the important points in the diagram, and a piece of the tangent line at each point on the ellipse from which a normal passes through the given point.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LUkmbXRleHRHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2KVFicCNwbG90c2V0dXAocHMscGxvdG9wdGlvbnM9YG5vYm9yZGVyLHBvcnRyYWl0YCxwbG90b3V0cHV0PWBjOi9sb2NhbC9maWdfb2ZmY2VudGVyLmVwc2ApO0YnLyUnZmFtaWx5R1EwVGltZXN+TmV3flJvbWFuRicvJSVib2xkR1EmZmFsc2VGJy8lK2ZvcmVncm91bmRHUShbMCwwLDBdRicvJStleGVjdXRhYmxlR0YxLyUwZm9udF9zdHlsZV9uYW1lR1EpMkR+SW5wdXRGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRic=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2J0YrLUYjNictRiw2JVEzRWxsaXBzZXJIeXBOb1RpY2tzRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEwJkFwcGx5RnVuY3Rpb247RicvRjpRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZELyUpc3RyZXRjaHlHRkQvJSpzeW1tZXRyaWNHRkQvJShsYXJnZW9wR0ZELyUubW92YWJsZWxpbWl0c0dGRC8lJ2FjY2VudEdGRC8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRlMtSShtZmVuY2VkR0YkNiQtRiM2Ki1JI21uR0YkNiRRJTAuNzVGJ0ZALUY9Ni1RIixGJ0ZARkIvRkZGOEZHRklGS0ZNRk9GUS9GVVEsMC4zMzMzMzMzZW1GJy1GZm42JFElMC4wOEYnRkBGaW4tRiM2Ji1GPTYtUSomdW1pbnVzMDtGJ0ZARkJGRUZHRklGS0ZNRk8vRlJRLDAuMjIyMjIyMmVtRicvRlVGaG8tSSZtZnJhY0dGJDYoLUYjNiUtRiw2JVEnJiM5NjA7RicvRjdGREZALyUrZXhlY3V0YWJsZUdGREZALUYjNiUtRmZuNiRRIjRGJ0ZARmNwRkAvJS5saW5ldGhpY2tuZXNzR1EiMUYnLyUrZGVub21hbGlnbkdRJ2NlbnRlckYnLyUpbnVtYWxpZ25HRl9xLyUpYmV2ZWxsZWRHRkRGY3BGQEYrRmNwRkBGQEZjcEZARitGY3BGQEYrRmNwRkA=LUkmbXRleHRHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2KVE0I3Bsb3RzZXR1cChpbmxpbmUpOkYnLyUnZmFtaWx5R1EwVGltZXN+TmV3flJvbWFuRicvJSVib2xkR1EmZmFsc2VGJy8lK2ZvcmVncm91bmRHUShbMCwwLDBdRicvJStleGVjdXRhYmxlR0YxLyUwZm9udF9zdHlsZV9uYW1lR1EpMkR+SW5wdXRGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRic=Just approaching the evolute from inside: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On the evolute for 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: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LUkmbXRleHRHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2KVE0I3Bsb3RzZXR1cChpbmxpbmUpOkYnLyUnZmFtaWx5R1EwVGltZXN+TmV3flJvbWFuRicvJSVib2xkR1EmZmFsc2VGJy8lK2ZvcmVncm91bmRHUShbMCwwLDBdRicvJStleGVjdXRhYmxlR0YxLyUwZm9udF9zdHlsZV9uYW1lR1EpMkR+SW5wdXRGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRic=Almost on the major axis: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 the major axis:LUkmbXRleHRHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2KVFncCNwbG90c2V0dXAocHMscGxvdG9wdGlvbnM9YG5vYm9yZGVyLHBvcnRyYWl0YCxwbG90b3V0cHV0PWBjOi9sb2NhbC9maWdfZXZvbHV0ZXZlcnRleDIuZXBzYCk7RicvJSdmYW1pbHlHUTBUaW1lc35OZXd+Um9tYW5GJy8lJWJvbGRHUSZmYWxzZUYnLyUrZm9yZWdyb3VuZEdRKFswLDAsMF1GJy8lK2V4ZWN1dGFibGVHRjEvJTBmb250X3N0eWxlX25hbWVHUSkyRH5JbnB1dEYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJw==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 this particular configuration, the major axis is along the LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=-axis, which is important in this derivation.
The rightmost point P on the ellipse in the first quadrant from which a normal passes through the given point O in the fourth quadrant is a configuration for which the similar triangle construction diagram is particularly simple. Dropping a perpendicular to the LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=-axis from O defines the point M, while doing the same from P defines the point N, while let G be the point where the normal line crosses the LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=-axis, and finally let T be the intersection of the tangent line to the ellipse at P with the LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=-axis.setting up the diagramThe given point O(LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUkjbW5HRiQ2JFEiMEYnL0Y2USdub3JtYWxGJ0Y+LyUvc3Vic2NyaXB0c2hpZnRHRj0tSSNtb0dGJDYtUSIsRidGPi8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0ZILyUqc3ltbWV0cmljR0ZILyUobGFyZ2VvcEdGSC8lLm1vdmFibGVsaW1pdHNHRkgvJSdhY2NlbnRHRkgvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4zMzMzMzMzZW1GJy1GLDYlLUYvNiVRInlGJ0YyRjVGOEZARj4=) has polar coordinates (LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEickYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIixGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRiw2JVEoJnRoZXRhO0YnL0YwRj1GOUY5):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We find the feet of the normals passing through O: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Let P(LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIixGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRiw2JVEieUYnRi9GMkY5) be correspond to the third solution in the first quadrant: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 we plot the similar triangles with the ellipse in the right half plane, using calculus to evaluate the LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=-intercept LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkmbWZyYWNHRiQ2KC1GIzYkLUklbXN1cEdGJDYlLUkjbWlHRiQ2JVEiYUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21uR0YkNiRRIjJGJy9GO1Enbm9ybWFsRicvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnRkEtRiM2JC1GNDYlUSJ4RidGN0Y6RkEvJS5saW5ldGhpY2tuZXNzR1EiMUYnLyUrZGVub21hbGlnbkdRJ2NlbnRlckYnLyUpbnVtYWxpZ25HRlAvJSliZXZlbGxlZEdRJmZhbHNlRidGQQ== of the tangent line at P , and using a relation derived below from similar triangles to locate the LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=-intercept of the normal line, and then adding in the point letter symbols (it is insane that the graphics text placement is still so primitive!):LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYrLUkjbWlHRiQ2JVEjeDBGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYtUSIsRicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0YxLyUpc3RyZXRjaHlHRj0vJSpzeW1tZXRyaWNHRj0vJShsYXJnZW9wR0Y9LyUubW92YWJsZWxpbWl0c0dGPS8lJ2FjY2VudEdGPS8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHUSwwLjMzMzMzMzNlbUYnLUYsNiVRI3kwRidGL0YyRjUtRiw2JVEieEYnRi9GMkY1LUYsNiVRInlGJ0YvRjIvJStleGVjdXRhYmxlR0Y9Rjk=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 can be improved with the math expression editing directly on the plot available in Maple 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triangle 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LUkmbXRleHRHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2KVFhcCNwbG90c2V0dXAocHMscGxvdG9wdGlvbnM9YG5vYm9yZGVyLHBvcnRyYWl0YCxwbG90b3V0cHV0PWBjOi9sb2NhbC9maWdfdHJpYW5nbGUuZXBzYCk7RicvJSdmYW1pbHlHUTBUaW1lc35OZXd+Um9tYW5GJy8lJWJvbGRHUSZmYWxzZUYnLyUrZm9yZWdyb3VuZEdRKFswLDAsMF1GJy8lK2V4ZWN1dGFibGVHRjEvJTBmb250X3N0eWxlX25hbWVHUSkyRH5JbnB1dEYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJw==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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2J0YrLUYjNictRiw2JVE3VHJpYW5nbGVfRmlndXJlX05ld2VzdEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIzo9RicvRjpRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZELyUpc3RyZXRjaHlHRkQvJSpzeW1tZXRyaWNHRkQvJShsYXJnZW9wR0ZELyUubW92YWJsZWxpbWl0c0dGRC8lJ2FjY2VudEdGRC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRlMtRiw2JVEiJUYnRjZGOS8lK2V4ZWN1dGFibGVHRkRGQEYrRllGQEYrRllGQA==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LUkmbXRleHRHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2KVFlcCNwbG90c2V0dXAocHMscGxvdG9wdGlvbnM9YG5vYm9yZGVyLHBvcnRyYWl0YCxwbG90b3V0cHV0PWBjOi9sb2NhbC9maWdfdHJpYW5nbGVfb2xkLmVwc2ApO0YnLyUnZmFtaWx5R1EwVGltZXN+TmV3flJvbWFuRicvJSVib2xkR1EmZmFsc2VGJy8lK2ZvcmVncm91bmRHUShbMCwwLDBdRicvJStleGVjdXRhYmxlR0YxLyUwZm9udF9zdHlsZV9uYW1lR1EpMkR+SW5wdXRGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRic=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2JS1GLDYlUTBUcmlhbmdsZV9GaWd1cmVGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvJStleGVjdXRhYmxlR1EmZmFsc2VGJy9GOFEnbm9ybWFsRidGK0Y6Rj0=LUkmbXRleHRHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2KVE0I3Bsb3RzZXR1cChpbmxpbmUpOkYnLyUnZmFtaWx5R1EwVGltZXN+TmV3flJvbWFuRicvJSVib2xkR1EmZmFsc2VGJy8lK2ZvcmVncm91bmRHUShbMCwwLDBdRicvJStleGVjdXRhYmxlR0YxLyUwZm9udF9zdHlsZV9uYW1lR1EpMkR+SW5wdXRGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRic=Letting LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEic0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= be the lengths of MG and GN, it follows 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Triangles GMO, GNP and PNT are all similar right triangles, leading to two equalities: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where LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkmbWZyYWNHRiQ2KC1GIzYkLUklbXN1cEdGJDYlLUkjbWlHRiQ2JVEiYUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21uR0YkNiRRIjJGJy9GO1Enbm9ybWFsRicvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnRkEtRiM2JC1GNDYlUSJ4RidGN0Y6RkEvJS5saW5ldGhpY2tuZXNzR1EiMUYnLyUrZGVub21hbGlnbkdRJ2NlbnRlckYnLyUpbnVtYWxpZ25HRlAvJSliZXZlbGxlZEdRJmZhbHNlRidGQQ== is the LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=-intercept of the tangent line (LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=-coordinate of the point T). This follows from using calculus to obtain the equation of the tangent line to the ellipse at P, which can be written using (LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEidUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIixGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRiw2JVEidkYnRi9GMkY5) as the coordinates since (LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIixGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRiw2JVEieUYnRi9GMkY5) are the coordinates of the point of tangency, from which setting LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJ2RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiPUYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUkjbW5HRiQ2JFEiMEYnRj5GPkYrRj4= and solving for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEidUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= yields the LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=-intercept:LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2J0YrLUYjNihGKy1GIzYnLUkmbWZyYWNHRiQ2KC1GIzYnLUYsNiVRInVGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYtUTEmSW52aXNpYmxlVGltZXM7RicvRkFRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZLLyUpc3RyZXRjaHlHRksvJSpzeW1tZXRyaWNHRksvJShsYXJnZW9wR0ZLLyUubW92YWJsZWxpbWl0c0dGSy8lJ2FjY2VudEdGSy8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRlotRiw2JVEieEYnRj1GQC8lK2V4ZWN1dGFibGVHRktGRy1GIzYnRistRiM2JS1JJW1zdXBHRiQ2JS1GLDYlUSJhRidGPUZALUkjbW5HRiQ2JFEiMkYnRkcvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnRmpuRkdGK0ZqbkZHLyUubGluZXRoaWNrbmVzc0dRIjFGJy8lK2Rlbm9tYWxpZ25HUSdjZW50ZXJGJy8lKW51bWFsaWduR0ZicC8lKWJldmVsbGVkR0ZLLUZENi1RIitGJ0ZHRklGTEZORlBGUkZURlYvRllRLDAuMjIyMjIyMmVtRicvRmZuRltxLUY2NigtRiM2Jy1GLDYlUSJ2RidGPUZARkMtRiw2JVEieUYnRj1GQEZqbkZHLUYjNidGKy1GIzYlLUZhbzYlLUYsNiVRImJGJ0Y9RkBGZm9Gam9Gam5GR0YrRmpuRkdGXXBGYHBGY3BGZXBGam5GRy1GRDYtUSI9RidGR0ZJRkxGTkZQRlJGVEZWL0ZZUSwwLjI3Nzc3NzhlbUYnL0ZmbkZkci1GZ282JEZfcEZHRmpuRkdGK0ZqbkZHLUZENi1RIjtGJ0ZHRkkvRk1GP0ZORlBGUkZURlZGWEZlckZqbkZHLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2J0YrLUYjNictRiw2JVEmc29sdmVGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYtUTAmQXBwbHlGdW5jdGlvbjtGJy9GOlEnbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkQvJSlzdHJldGNoeUdGRC8lKnN5bW1ldHJpY0dGRC8lKGxhcmdlb3BHRkQvJS5tb3ZhYmxlbGltaXRzR0ZELyUnYWNjZW50R0ZELyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGUy1JKG1mZW5jZWRHRiQ2JC1GIzYoRistRiM2Jy1GLDYlUSVzdWJzRidGNkY5RjwtRlc2JC1GIzYoRistRiM2Jy1GLDYlUSJ2RidGNkY5LUY9Ni1RIj1GJ0ZARkJGRUZHRklGS0ZNRk8vRlJRLDAuMjc3Nzc3OGVtRicvRlVGZ28tSSNtbkdGJDYkUSIwRidGQC8lK2V4ZWN1dGFibGVHRkRGQC1GPTYtUSIsRidGQEZCL0ZGRjhGR0ZJRktGTUZPRlEvRlVRLDAuMzMzMzMzM2VtRictRiw2JVEiJUYnRjZGOUZdcEZARkBGXXBGQEZfcC1GLDYlUSJ1RidGNkY5Rl1wRkBGQEZdcEZARitGXXBGQEYrRl1wRkA=From these three equations, one can eliminate LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEic0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=, leaving only one equation left: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This last equation just equates the negative reciprocal of the normal line slope to the tangent line slope (the orthogonality condition), so we could have avoided using the similar triangle relationships to get this. Finally, one can simplify this last equation using the equation of the ellipse: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This is the equation of a hyperbola whose asymptotes are a pair of horizontal and vertical lines (since there are no squared terms in the equation) and which passes through both the origin and the given point (LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEjeDBGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYtUSIsRicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0YxLyUpc3RyZXRjaHlHRj0vJSpzeW1tZXRyaWNHRj0vJShsYXJnZW9wR0Y9LyUubW92YWJsZWxpbWl0c0dGPS8lJ2FjY2VudEdGPS8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHUSwwLjMzMzMzMzNlbUYnLUYsNiVRI3kwRidGL0YyRjk=). By considering translations of the hyperbola 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 to 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, and requiring that 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 to eliminate the constant term shows that the center of the hyperbola has coordinates: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 (p.166) refers only to similar triangles GMO and GNP and the corresponding eq1 of similar sides in re-interpreting the work Apollonius using analytic geometry. Somehow Apollonius obtained the result that the normals which pass through O come from the points P on the ellipse which are the intersections with this hyperbola, although he did not write down any equations for it.Note that the LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=-coordinate of the point G, which is the intercept of the normal line, is just: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 we used in placing the letter G in the above graphic.To recap, we needed only 3 inputs to derive the equation of the hyperbola of Apollonius: the equation of the ellipse itself, the intercept LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkmbWZyYWNHRiQ2KC1GIzYkLUklbXN1cEdGJDYlLUkjbWlHRiQ2JVEiYUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21uR0YkNiRRIjJGJy9GO1Enbm9ybWFsRicvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnRkEtRiM2JC1GNDYlUSJ4RidGN0Y6RkEvJS5saW5ldGhpY2tuZXNzR1EiMUYnLyUrZGVub21hbGlnbkdRJ2NlbnRlckYnLyUpbnVtYWxpZ25HRlAvJSliZXZlbGxlZEdRJmZhbHNlRidGQQ== with its major axis of the tangent line from the point (LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=,LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=) on the ellipse, and the orthogonality condition between the tangent line segment and the normal line from the given point. LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2JS1GLDYlUStFcW5FbGxpcHNlRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvRjhRJ25vcm1hbEYnLUkjbW9HRiQ2LVEiO0YnRj0vJSZmZW5jZUdGPC8lKnNlcGFyYXRvckdGNi8lKXN0cmV0Y2h5R0Y8LyUqc3ltbWV0cmljR0Y8LyUobGFyZ2VvcEdGPC8lLm1vdmFibGVsaW1pdHNHRjwvJSdhY2NlbnRHRjwvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4yNzc3Nzc4ZW1GJ0Y6Rj0=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 of the ellipse and hyperbola:
nonparametric approach (hindsight)Once we have found that the feet of the normals from the ellipse to a given point in the plane are the intersection points of the ellipse with the hyperbola of Apollonius associated with the given point and the ellipse, these feet are the solution of the pair of quadratic equations in two variables: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When LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSN5MEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RJSZuZTtGJy9GOFEnbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkIvJSlzdHJldGNoeUdGQi8lKnN5bW1ldHJpY0dGQi8lKGxhcmdlb3BHRkIvJS5tb3ZhYmxlbGltaXRzR0ZCLyUnYWNjZW50R0ZCLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGUS1JI21uR0YkNiRRIjBGJ0Y+Rj5GK0Y+, one can eliminate LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= to get a fourth order equation for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=: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In the parametric approach with 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, eliminating the sine to get a fourth order equation for the cosine is equivalent to eliminating LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= to get this fourth order equation for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=, differing only by the substitution 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. 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Since the evolute 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 is intimately related to the number of roots of this polynomial, simplifying the evolute by a change of variables will simplify the analysis of the roots of the polynomial. This suggests the first change of variables mapping the evolute onto the standard astroid: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or equivalently to the cosine and sine variables in the parametric approach: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and the second change of variables converts the evolute equation into a linear condition: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These enable one to factor the discriminant of the fourth order polynomial in the above section analyzing the number of its roots.properties of the hyperbolaHere we do the reverse: translate the given hyperbola back to the origin, where the coefficients of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= must vanish for it to have the form LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2J0YrLUYjNigtRiw2JVEiQUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RMSZJbnZpc2libGVUaW1lcztGJy9GOlEnbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkQvJSlzdHJldGNoeUdGRC8lKnN5bW1ldHJpY0dGRC8lKGxhcmdlb3BHRkQvJS5tb3ZhYmxlbGltaXRzR0ZELyUnYWNjZW50R0ZELyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGUy1GLDYlUSJ4RidGNkY5RjwtRiw2JVEieUYnRjZGOUZALUY9Ni1RIj1GJ0ZARkJGRUZHRklGS0ZNRk8vRlJRLDAuMjc3Nzc3OGVtRicvRlVGam5GM0ZARitGQA==:LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2JS1GLDYlUS1FcW5IeXBlcmJvbGFGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvJStleGVjdXRhYmxlR1EmZmFsc2VGJy9GOFEnbm9ybWFsRictSSNtb0dGJDYtUSI7RidGPS8lJmZlbmNlR0Y8LyUqc2VwYXJhdG9yR0Y2LyUpc3RyZXRjaHlHRjwvJSpzeW1tZXRyaWNHRjwvJShsYXJnZW9wR0Y8LyUubW92YWJsZWxpbWl0c0dGPC8lJ2FjY2VudEdGPC8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHUSwwLjI3Nzc3NzhlbUYnRjpGPQ==LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYpLUkjbWlHRiQ2JVE3RXFuSHlwZXJib2xhVHJhbnNsYXRlZEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIzo9RicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y9LyUpc3RyZXRjaHlHRj0vJSpzeW1tZXRyaWNHRj0vJShsYXJnZW9wR0Y9LyUubW92YWJsZWxpbWl0c0dGPS8lJ2FjY2VudEdGPS8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkwtRiw2JVElc3Vic0YnRi9GMi1JKG1mZW5jZWRHRiQ2JC1GIzYqLUYsNiNRIUYnLUYjNigtRiw2JVEieEYnRi9GMi1GNjYtUSI9RidGOUY7Rj5GQEZCRkRGRkZIRkpGTS1GIzYnLUYsNiVRIlhGJ0YvRjItRjY2LVEiK0YnRjlGO0Y+RkBGQkZERkZGSC9GS1EsMC4yMjIyMjIyZW1GJy9GTkZlby1GLDYlUSN4MUYnRi9GMi8lK2V4ZWN1dGFibGVHRj1GOUZXRmpvRjktRjY2LVEiLEYnRjlGOy9GP0YxRkBGQkZERkZGSC9GS1EmMC4wZW1GJy9GTlEsMC4zMzMzMzMzZW1GJy1GIzYoLUYsNiVRInlGJ0YvRjJGaW4tRiM2Jy1GLDYlUSJZRidGL0YyRmFvLUYsNiVRI3kxRidGL0YyRmpvRjlGV0Zqb0Y5RlxwLUYsNiVRLUVxbkh5cGVyYm9sYUYnRi9GMkZqb0Y5RjktRjY2LVEiOkYnRjlGO0Y+RkBGQkZERkZGSEZKRk1Gam9GOQ==LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEnZXhwYW5kRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkobWZlbmNlZEdGJDYkLUYjNiUtRiw2JVEiJUYnRi9GMi8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnL0YzUSdub3JtYWxGJ0ZALUkjbW9HRiQ2LVEiOkYnRkAvJSZmZW5jZUdGPy8lKnNlcGFyYXRvckdGPy8lKXN0cmV0Y2h5R0Y/LyUqc3ltbWV0cmljR0Y/LyUobGFyZ2VvcEdGPy8lLm1vdmFibGVsaW1pdHNHRj8vJSdhY2NlbnRHRj8vJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZWRj1GQA==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Note that CH stands for the coordinates of the center of the hyperbola.The translated hyperbola is just: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the extended procedure EllipserHypCenter(b,r,theta) and its 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 we animate in the radial parameter of the chosen point O for fixed angle and slow up the action as the point approaches the evolute: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 exercise for a student would be to understand what the three formulas for the stopping radii in the above procedure are or how the slowup of the action was achieved.final frame 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LUkmbXRleHRHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2KVE0I3Bsb3RzZXR1cChpbmxpbmUpOkYnLyUnZmFtaWx5R1EwVGltZXN+TmV3flJvbWFuRicvJSVib2xkR1EmZmFsc2VGJy8lK2ZvcmVncm91bmRHUShbMCwwLDBdRicvJStleGVjdXRhYmxlR0YxLyUwZm9udF9zdHlsZV9uYW1lR1EpMkR+SW5wdXRGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRic=With the animation one sees how the situation develops as the given point (x0,y0) = (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) moves out on a ray from the origin at a fixed angle. The center of the hyperbola moves out on the other side of the (major) LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=-axis at a smaller angle, while the vertices of the hyperbola move out from its center, all at distances proportional to LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEickYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=. Inside the evolute the branch of the hyperbola not passing through the given point moves out from the center of the ellipse still intersecting the ellipse in two points, but as the given point passes through the evolute, the second branch becomes tangent to the ellipse and then moves out of the ellipse, reducing the number of normals to 3 on the evolute and 2 outside the evolute, provided one is not moving along one of the axes of the ellipse. In that case the vertices of the ellipse on that axis remain as two intersection points, while the remaining pair degenerates to one of these semi-axis vertices as the given point reaches the cusp of the evolute, so 3 become 1, the total dropping from 4 to 2 directly.bonus: evolute as the envelope of normals
and the locus of singularities of the family of parallels
(graphical)In the following procedure we need to specify how many normals to plot in addition to the shape parameter. We terminate each normal at a fixed distance inside and outside the ellipse, so the tips are "parallel curves" in the differential geometry language. One can adjust these distances to achieve different goals by editing the procedure. This shows very nicely the fact that the envelope of all the normals is the evolute.EllipseEvoluteNormals(b,n)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We make the normal vector a Maple Vector function and then do the same for the normal line as a function of two variables. We fought with this for hours to get the functions to work correctly in upgrading this from Maple 9.5 classic to Maple11.02 Standard 2d math 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We can easily plot directly the parallel curves by replacing the radius of curvature by an adjustable arclength parameter LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEic0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= parametrizing this family, to see that the "ingoinging wavefronts" built from rays which emanate from the "initial elliptical wavefront" (where they are along the normal direction) develop singularities along the evolute.EllipseEvoluteParallels(b,s)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An animation might do this one better so we jiggle our procedure in a different way.EllipseEvoluteParallelsMovie(b,n,sfinal)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In each one of these procedures we have not tried to optimize the code but just stumble from one to the next by making small changes the way an unsophistocated user might try to use Maple to solve a new problem for which that user has no special Maple expertise, which is indeed our own situation. However, during the upgrade from Maple Classic to Maple Standard with Vector Calculus, we had to improve the sloppy coding somewhat just to make things work.bonus: evolute as the curve of limiting
intersections of successive normals (symbolic)The idea that the evolute is the curve of limiting intersections of successive normals of the original curve is a way of actually calculating the ellipse as the envelope of its normals. One can write down the equation of the normal line to the parametrized ellipse, and then its neighbor at a nearby point on the ellipse, find their intersection, and then take the limit as the nearby point approaches the original one.
First we need the normal to the ellipse in order to write an equation for the normal line. This can be done by hand but Maple can easily give us what we need and it is obviously orthogonal to the tangent, so we could have written this down easily by inspection: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In the plane, all we need to do is rotate the tangent by 90 degrees: 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 to get a normal. We don't need the unit normal to write the equation of the normal line, so we will drop the normalizing denominator.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 fact we only need the slope to write this equation: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We repeat at the nearby point corresponding to LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJ0RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiK0YnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjIyMjIyMjJlbUYnLyUncnNwYWNlR0ZRLUYsNiVRKCZkZWx0YTtGJy9GNUZCRj5GPkYrRj4=:LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2J0YrLUYjNigtRiw2JVEjTDJGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYtUSM6PUYnL0Y6USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGRC8lKXN0cmV0Y2h5R0ZELyUqc3ltbWV0cmljR0ZELyUobGFyZ2VvcEdGRC8lLm1vdmFibGVsaW1pdHNHRkQvJSdhY2NlbnRHRkQvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZTLUYjNictSSZtZnJhY0dGJDYoLUYjNidGKy1GIzYnLUYsNiVRInlGJ0Y2RjktRj02LVEoJm1pbnVzO0YnRkBGQkZFRkdGSUZLRk1GTy9GUlEsMC4yMjIyMjIyZW1GJy9GVUZgby1JJW1zdWJHRiQ2JS1GIzYnLUYsNiVRIlJGJ0Y2RjktRj02LVEwJkFwcGx5RnVuY3Rpb247RidGQEZCRkVGR0ZJRktGTUZPL0ZSUSYwLjBlbUYnL0ZVRl5wLUkobWZlbmNlZEdGJDYkLUYjNidGKy1GIzYnLUYsNiVRInRGJ0Y2RjktRj02LVEiK0YnRkBGQkZFRkdGSUZLRk1GT0Zfb0Zhby1GLDYlUScmIzk0ODtGJy9GN0ZERkAvJStleGVjdXRhYmxlR0ZERkBGK0ZhcUZARkBGYXFGQC1GIzYlLUkjbW5HRiQ2JFEiMkYnRkBGYXFGQC8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnRmFxRkBGK0ZhcUZALUYjNidGKy1GIzYnLUYsNiVRInhGJ0Y2RjlGXG8tRmNvNiVGZW8tRiM2JS1GZnE2JFEiMUYnRkBGYXFGQEZpcUZhcUZARitGYXFGQC8lLmxpbmV0aGlja25lc3NHRmlyLyUrZGVub21hbGlnbkdRJ2NlbnRlckYnLyUpbnVtYWxpZ25HRl5zLyUpYmV2ZWxsZWRHRkQtRj02LVEiPUYnRkBGQkZFRkdGSUZLRk1GT0ZRRlQtRlk2KC1GIzYoLUYsNiVRImFGJ0Y2RjktRj02LVExJkludmlzaWJsZVRpbWVzO0YnRkBGQkZFRkdGSUZLRk1GT0ZdcEZfcC1GIzYnLUYsNiVRJHNpbkYnRmBxRkBGam9GYHBGYXFGQEYrRmFxRkAtRiM2J0YrLUYjNigtRiw2JVEiYkYnRjZGOUZddC1GIzYnLUYsNiVRJGNvc0YnRmBxRkBGam9GYHBGYXFGQEYrRmFxRkBGK0ZhcUZARmpyRlxzRl9zRmFzRmFxRkBGK0ZhcUZARitGYXFGQEYrRmFxRkA=First we find the coordinates (LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIixGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRiw2JVEieUYnRi9GMkY5) of the intersection of these two adjacent normals to the 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and reorder them into the corresponding position vector and then take the limit of this intersection point as LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEoJmRlbHRhO0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRidGMg== tends to 0. The second component requires some extra simplification since the cosine terms must be converted to sine terms while simplify converts as much to cosines as possible: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So the final result is the position vector of the parametrized evolute given above: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One can easily take the limit by inspection of the simplified equations above, but the initial simplification step by Maple avoided tedious algebra.references and web linksThe evolute of the ellipse was calculated long before calculus existed:
http://geometryalgorithms.com/history.htmby Apollonius of Perga (262-190 BC) who wrote 8 volumes on various properties of the conic sections, with volume 5 devoted entirely to the distance problem.
http://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/Apollonius.html
See also Dan Margolit's "A History of Curvature":
http://www3.villanova.edu/maple/misc/history_of_curvature/index.htmand WikiPedia:http://en.wikipedia.org/wiki/Apollonius_of_Perga
The MacTutor History of Mathematics Archives, Famous Curves Index, has some interesting things to say about the ellipse and its evolute:
http://www-gap.dcs.st-and.ac.uk/~history/Curves/Ellipse.htmlincluding Java applets where one can adjust the semiaxes and see the evolute or the involutes etc. (and stating the result that 4 normals intersect the point inside the evolute but only 2 outside).For a Java applet where one can adjust the semiaxes of the ellipse with sliders and see the evolute move (German text):
http://www.uni-flensburg.de/mathe/zero/fgalerie/kurven/evolute_einer_ellipse.htmlSome older postscript notes on differential geometry by Bal\303\241zs Csik\303\263s [see unit 3, page 4]
http://www.cs.elte.hu/geometry/csikos/dif/dif.html
state that Hyugens (teacher of Liebnitz who together with Newton was one of the fathers of modern calculus) introduced the evolute of an ellipse when studying wavefronts, giving the 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The Mathematica Math World site does this one better by also giving the unparametrized equations of the evolute and stating the result that 4 normals intersect the point inside the evolute but only 2 outside:
http://mathworld.wolfram.com/EllipseEvolute.htmlwhich is apparently called an stretched astroid:
http://mathworld.wolfram.com/Astroid.html
By web searching on this topic, one digs up an incredible wealth of stuff... but life is too short.However, finding time to go to the library, browsing the differential geometry section turned up a recent book
Elementary Geometry of Differentiable Curves: An Undergraduate Introduction
by C. G. Gibson (Cambridge University Press, 2001)with some more detailed explanation of some of the claims we'd seen web browsing, explaining the evolute as the envelope of the normals of the ellipse and the locus of the singular points in the family of parallel curves generated by an ellipse, constructed as the loci of points equidistant along the normal lines at each point of the ellipse, relevant from the wave theory of optics which motivated Huygens to first examine the ellipse in this context. [The author used MAPLE for many of the illustrations.] This inspired us to add the related section above on the envelope of the normals and the family of parallels.More library work produced the remarkable book:
Apollonius of Perga: Treatise on Conic Sections, edited in modern notation with introductions including an essay on the earlier history of the subject, by T.L. Heath, Heffer & Sons, Cambridge, 1896. [Amazon]
Apparently the notation and writing style of Apollonius made his books nearly impenetrable, so his work was largely overlooked, but this mathematician took it upon himself to modernize the document so it could be appreciated. The work of Apollonius is a perfect example of the lengths one has to go when one does not have the appropriate tools to attack a given problem. He studied the geometry of the max/min problem for all the conics, and apparently (?) derived all their evolutes in this context, using very complicated geometrical reasoning.Then there is a recent two volume set which is a retranslation from the Arabic translation of the lost Greek original volumes of Apollonius, with added commentary:
Apollonius: Conics -- Books V to VII, by G.J. Toomer, Springer-Verlag, New York, 1990.Some of the original works of Apollonious (books I-IV) are available "in translation" for a reasonable price by Green Lion Press Books [I-III, IV]
A brief introduction to Apollonius may be found in:
A Dictionary of Scientific Biography, by C. Gillispie, Scribners, New York, 1970, "Apollonius of Perga," by G.J. Toomer, pp. 179-193.
Finally there is also:
A History of Greek Mathematics, by Thomas Heath, Clarendon Press, Oxford, 1921, "Conic Sections. Apollonius of Perga", Volume 2, pp.110-196; reprinted by Oxford University Press, 1960, 1965.Here is the worksheet end time (restarts prevent taking the difference with the start time):LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYpLUkjbWlHRiQ2JVEoRW5kVGltZUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RKiZjb2xvbmVxO0YnL0YzUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGPS8lKXN0cmV0Y2h5R0Y9LyUqc3ltbWV0cmljR0Y9LyUobGFyZ2VvcEdGPS8lLm1vdmFibGVsaW1pdHNHRj0vJSdhY2NlbnRHRj0vJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZMLUYsNiVRJXRpbWVGJ0YvRjItSShtZmVuY2VkR0YkNiQtRiM2JS1GLDYjUSFGJy8lK2V4ZWN1dGFibGVHRj1GOUY5RldGWkY5JSFH