ellipse calculation of osculating circlecreated 1994, updated 2018Stewart Calculus 8e. 13.3.51.
See the worksheet osculate-ellipse.mw for visualizing the ellipse osculating circles with semiaxes 3 and 2 instead of 4 and 1, easily edited and re-executed.deriving the formulasThis shows how to derive the geometric information about a parametrized curve, including the unit tangent, unit normal, center of the osculating circle (the curve of centers is called the "evolute" of the original curve), and the parametrization of the osculating circle itself. The end formulas can then be used to both plot and animate the osculating circle above. The ellipse 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, represented by a deformed circle parametrization (semiaxes 2 and 4 respectively is the example used here.[Recall that "osculate" comes from the latin verb "to kiss" which is what the circle of best fit to the curvature does to the curve at the point of tangency.]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 introduce the unit tangent: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Over the complex numbers square roots are a problem, so we assume our parameter is real. The unit normal is then the direction of the rate of change of the unit tangent:LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEnYXNzdW1lRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkobWZlbmNlZEdGJDYkLUYjNictRiw2JVEidEYnRi9GMi1JI21vR0YkNi1RIixGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGRS8lKnN5bW1ldHJpY0dGRS8lKGxhcmdlb3BHRkUvJS5tb3ZhYmxlbGltaXRzR0ZFLyUnYWNjZW50R0ZFLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRiw2JVElcmVhbEYnRi9GMi8lK2V4ZWN1dGFibGVHRkVGQUZBRmVuRkE=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The curvature is the length of the acceleration vector divided by the length of the tangent by the chain rule:
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The radius of curvature is the reciprocal of the curvature: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 has integer values at the axis intercepts LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkobWZlbmNlZEdGJDYmLUYjNigtSSNtb0dGJDYtUSUmcG07RicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y5LyUpc3RyZXRjaHlHRjkvJSpzeW1tZXRyaWNHRjkvJShsYXJnZW9wR0Y5LyUubW92YWJsZWxpbWl0c0dGOS8lJ2FjY2VudEdGOS8lJ2xzcGFjZUdRLDAuMjIyMjIyMmVtRicvJSdyc3BhY2VHRkgtSSNtbkdGJDYkUSI4RidGNC1GMTYtUSIsRidGNEY3L0Y7USV0cnVlRidGPEY+RkBGQkZEL0ZHUSYwLjBlbUYnL0ZKUSwwLjMzMzMzMzNlbUYnLUZMNiRRIjBGJ0Y0LyUrZXhlY3V0YWJsZUdGOUY0RjQvJSVvcGVuR1EiW0YnLyUmY2xvc2VHUSJdRidGTy1GLDYmLUYjNihGWEZPRjAtRkw2JFEiMUYnRjRGZW5GNEY0RmduRmpuRmVuRjQ=:LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYpLUkjbWlHRiQ2JVEnJiM5NjE7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JKG1mZW5jZWRHRiQ2JC1GIzYlLUkjbW5HRiQ2JFEiMEYnRjIvJStleGVjdXRhYmxlR0YxRjJGMi1JI21vR0YkNi1RIjtGJ0YyLyUmZmVuY2VHRjEvJSpzZXBhcmF0b3JHUSV0cnVlRicvJSlzdHJldGNoeUdGMS8lKnN5bW1ldHJpY0dGMS8lKGxhcmdlb3BHRjEvJS5tb3ZhYmxlbGltaXRzR0YxLyUnYWNjZW50R0YxLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMjc3Nzc3OGVtRidGKy1GNjYkLUYjNiUtSSZtZnJhY0dGJDYoLUYsNiVRJSZwaTtGJ0YvRjItRiM2JS1GOzYkUSIyRidGMi9GMEZIL0YzUSdpdGFsaWNGJy8lLmxpbmV0aGlja25lc3NHUSIxRicvJStkZW5vbWFsaWduR1EnY2VudGVyRicvJSludW1hbGlnbkdGam8vJSliZXZlbGxlZEdGMUY+RjJGMkY+RjI= The position vector of the center of the osculating circle is a distance LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2I1EhRictRiM2KC1GLDYmUSYmcmhvO0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lK2V4ZWN1dGFibGVHRjYvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUkjbW9HRiQ2PlEwJkFwcGx5RnVuY3Rpb247RicvJSdmYW1pbHlHUTBUaW1lc35OZXd+Um9tYW5GJy8lJXNpemVHUSMxMkYnLyUlYm9sZEdGNkY0LyUqdW5kZXJsaW5lR0Y2LyUqc3Vic2NyaXB0R0Y2LyUsc3VwZXJzY3JpcHRHRjYvJStmb3JlZ3JvdW5kR1EoWzAsMCwwXUYnLyUrYmFja2dyb3VuZEdRLlsyNTUsMjU1LDI1NV1GJy8lJ29wYXF1ZUdGNkY3LyUpcmVhZG9ubHlHRjYvJSljb21wb3NlZEdGNi8lKmNvbnZlcnRlZEdGNi8lK2ltc2VsZWN0ZWRHRjYvJSxwbGFjZWhvbGRlckdGNi8lNnNlbGVjdGlvbi1wbGFjZWhvbGRlckdGNkY5LyUmZmVuY2VHRjYvJSpzZXBhcmF0b3JHRjYvJSlzdHJldGNoeUdGNi8lKnN5bW1ldHJpY0dGNi8lKGxhcmdlb3BHRjYvJS5tb3ZhYmxlbGltaXRzR0Y2LyUnYWNjZW50R0Y2LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGXHAtSShtZmVuY2VkR0YkNiUtRiM2Ji1GLDYmUSJ0RicvRjVRJXRydWVGJ0Y3L0Y6USdpdGFsaWNGJ0Y3LyUwZm9udF9zdHlsZV9uYW1lR1EnTm9ybWFsRidGOUY3RjlGN0ZbcUY5RitGN0ZbcUY5 along the unit normal 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 from the point 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 on the curve, so by vector addition: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Finally the osculating circle itself can be parametrized by first parametrizing the difference vector (below called RAD for radius vector) from the center to the circle itself as a function of the angle LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEoJnRoZXRhO0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRidGMg== from the polar axis (chosen to be 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 in the counterclockwise direction towards LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiVEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRInRGJ0YvRjIvRjNRJ25vcm1hbEYnRj1GPQ== (thinking of the vectors LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbW9HRiQ2LVEqJnVtaW51czA7RicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRLDAuMjIyMjIyMmVtRicvJSdyc3BhY2VHRkMtSSNtaUdGJDYlUSJORicvJSdpdGFsaWNHUSV0cnVlRicvRjBRJ2l0YWxpY0YnLUkobWZlbmNlZEdGJDYkLUYjNiQtRkc2JVEidEYnRkpGTUYvRi9GLw== and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiVEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRInRGJ0YvRjIvRjNRJ25vcm1hbEYnRj0tSSNtb0dGJDYtUSJ+RidGPS8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGRS8lKXN0cmV0Y2h5R0ZFLyUqc3ltbWV0cmljR0ZFLyUobGFyZ2VvcEdGRS8lLm1vdmFibGVsaW1pdHNHRkUvJSdhY2NlbnRHRkUvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZURj0=with their initial pointsat the center of the osculating circle):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The osculating circle is obtained by vector addition of this to the position vector of the center: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 normal line segment from the point of contact to the center of the osculating circle is the following where 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: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JSFH