Using a Bezier Curve to Model a Hershey KissKlaus Volpert 28-sep-2014
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promised to show you how do draw a rendition of a Hershey Kiss in Maple. For that I will first model the outline of the shape in the LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkobWZlbmNlZEdGJDYkLUYjNiYtSSNtaUdGJDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiLEYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4zMzMzMzMzZW1GJy1GMTYlUSJ5RidGNEY3Rj5GPi1GMTYjUSFGJ0Y+ plane using cubic Bezier curves (described in this Wiki article), and then revolve it around the LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=-axis to create the solid region approximating the shape of this well known chocolate.finding the boundary curve and plotting the solidFirst the outline: I experimented a bit with the Java applet here, to obtain a reasonable outline.I estimated the coordinates of the 4 control points and now use them here in Maple:LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEocmVzdGFydEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIjpGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTA==QyY+SSJ4RzYiLCoqJkkjeDBHRiUiIiIsJkYpRilJInRHRiUhIiIiIiRGKSooSSN4MUdGJUYpRioiIiNGK0YpRi0qKEkjeDJHRiVGKUYqRilGK0YwRi0qJkkjeDNHRiVGKUYrRi1GKUYsPkkieUdGJSwqKiZJI3kwR0YlRilGKkYtRikqKEkjeTFHRiVGKUYqRjBGK0YpRi0qKEkjeTJHRiVGKUYqRilGK0YwRi0qJkkjeTNHRiVGKUYrRi1GKUYsprint(); # input placeholderThe coordinates of the four points I use 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%;The curve leaves the first point (at the top tip of the Kiss) along the direction towards the second point, and arrives at the final point on the horizontal axis from the direction of the third point. The two intermediate points are called control points.Now I revolve this curve around the y-axis: (for the color and style choices I right-clicked on the picture, and used the menu)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%;Pretty no? We could experiment with a more chocolate color?
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%;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%;Looks like Maple has no preset color for dark chocolate.volume of a Kiss?What about the volume of this shape, given by the formula LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYqLUkobXN1YnN1cEdGJDYnLUkjbW9HRiQ2LVErJkludGVncmFsO0YnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmdW5zZXRGJy8lKnNlcGFyYXRvckdGNy8lKXN0cmV0Y2h5R1EldHJ1ZUYnLyUqc3ltbWV0cmljR0Y3LyUobGFyZ2VvcEdGPC8lLm1vdmFibGVsaW1pdHNHRjcvJSdhY2NlbnRHRjcvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZHLUYjNiQtSSNtaUdGJDYjUSdgeV9fMWBGJ0YyLUYjNiQtRk02I1EnYHlfXzJgRidGMi8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRicvJS9zdWJzY3JpcHRzaGlmdEdGVy1GTTYlUScmIzk2MDtGJy8lJ2l0YWxpY0dRJmZhbHNlRidGMi1GLzYtUSJ+RidGMi9GNkZpbi9GOUZpbi9GO0Zpbi9GPkZpbi9GQEZpbi9GQkZpbi9GREZpbkZFRkgtSSVtc3VwR0YkNiUtRk02JVEieEYnL0ZobkY8L0YzUSdpdGFsaWNGJy1GIzYmLUkjbW5HRiQ2JVEiMkYnLyU2c2VsZWN0aW9uLXBsYWNlaG9sZGVyR0Y8RjJGam9GY3BGW3BGVS1GLzYuRlxvRmNwRjJGXW9GXm9GX29GYG9GYW9GYm9GY29GRUZILUYvNi1RMCZEaWZmZXJlbnRpYWxEO0YnRjJGNUY4L0Y7RjdGPS9GQEY3RkFGQ0ZFRkgtRk02JVEieUYnRmpvRltwRjI= for a curve revolved about the LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=-axis? We only need use the chain rule for this parametrized curve to make the replacement 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. But since we trace out the curve from top to bottom as we increase LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= (which one learns by glancing at the plot of this derivative), we need to reverse the sign of the derivative of the decreasing function LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRInRGJ0YvRjIvRjNRJ25vcm1hbEYnRj0tSSNtb0dGJDYtUSIuRidGPS8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGRS8lKXN0cmV0Y2h5R0ZFLyUqc3ltbWV0cmljR0ZFLyUobGFyZ2VvcEdGRS8lLm1vdmFibGVsaW1pdHNHRkUvJSdhY2NlbnRHRkUvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZURj0=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(); # input placeholderLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVElcGxvdEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYpLUY2NiYtRiM2Jy1GLDYlUSJ5RidGL0YyLUkjbW9HRiQ2LVEiLEYnL0YzUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGMS8lKXN0cmV0Y2h5R0ZJLyUqc3ltbWV0cmljR0ZJLyUobGFyZ2VvcEdGSS8lLm1vdmFibGVsaW1pdHNHRkkvJSdhY2NlbnRHRkkvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4zMzMzMzMzZW1GJy1JJm1mcmFjR0YkNigtRkI2LVEwJkRpZmZlcmVudGlhbEQ7RidGRS9GSFEmdW5zZXRGJy9GS0Zdby9GTUZdby9GT0Zdby9GUUZdby9GU0Zdby9GVUZdb0ZWL0ZaRlgtRiM2JUZpbi1GQjYtUSJ+RidGRUZHL0ZLRklGTEZORlBGUkZURlZGZG8tRiw2JVEidEYnRi9GMi8lLmxpbmV0aGlja25lc3NHUSIxRicvJStkZW5vbWFsaWduR1EnY2VudGVyRicvJSludW1hbGlnbkdGY3AvJSliZXZlbGxlZEdGSUZnb0Y+RkUvJSVvcGVuR1EiW0YnLyUmY2xvc2VHUSJdRidGQUZbcC1GQjYtUSI9RidGRUZHRmpvRkxGTkZQRlJGVC9GV1EsMC4yNzc3Nzc4ZW1GJy9GWkZicS1JI21uR0YkNiRRIjBGJ0ZFLUZCNi1RIy4uRidGRUZHRmpvRkxGTkZQRlJGVC9GV1EsMC4yMjIyMjIyZW1GJ0Zkby1GZXE2JEZgcEZFRkU=%;So we evaluate the above volume integral.PkkkVm9sRzYiLCQtSSRpbnRHRiQ2JCooSSNQaUclKnByb3RlY3RlZEciIiJJInhHRiQiIiMtSSVkaWZmR0YrNiRJInlHRiRJInRHRiRGLC9GMzsiIiFGLCEiIg==print(); # input placeholderJSFHcomparison with a cone of the same base and heightThe formula for the volume of a cone of base LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiYkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= and height LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiaEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= is 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Qyc+SShWX19jb25lRzYiKiYtSSIqRyUqcHJvdGVjdGVkRzYkLCRJI1BpR0YpIyIiIiIiJCIkKyJGLi1JIl5HRik2JCIjZyIiI0YuRi4qJkkkVm9sR0YlRi5JIiVHRiUhIiJGLi1JJmV2YWxmR0YpNiNGOA==The Kiss has a slightly bigger volume.Compare their profiles:QyUtSSVwbG90RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiQ3JDckSSN4MEdGKEkjeTBHRig3JEkjeDNHRihJI3kzR0YoL0kmY29sb3JHRihJJWJsdWVHRighIiItSShkaXNwbGF5R0YoNiRJJmN1cnZlR0YoSSIlR0YoClearly the Kiss bulges outward where it counts at large radii from the symmetry axis, and dips inward where there is little contribution to the total volume.Of course these particular dimensions chosen to make the nice curve are not relevant to an actual Hershey kiss. We can derive the formula for a unit base circular radius and then from that formula scale up to a real Kiss by measuring half the base diameter. Comparing it to a spherical volume 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 with the same unit radius is helpful to make sure our numbers make sense. First we have to scale down the volume by 3 factors of 60 to reduce all the dimensions uniformly so that the new base radius changes from 60 to 1. 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(); # input placeholderWe look at the coefficient of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnJiM5NjA7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yy in the unit sphere volume and compare the scaled down Kiss volume coefficient of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnJiM5NjA7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yy to 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print(); # input placeholderDoes this make sense? Well, we can compute the average radius of a Kiss from its center of mass or centroid, since it is a uniformly dense solid (presumably). The centroid will be on the LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=-axis somewhere lower than the midpoint of the height since most of the Kiss is lower than the halfway point. The average height is the centroid value ofLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= along the axis. We just have to evaluate the differential of volume for a volume of revolution about the LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= axis, in turn with a boundary parametrized by LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=.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print(); # input 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 , 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So explicitly we integrate from the vertical axis out to its bounding curve LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRInRGJ0YvRjIvRjNRJ25vcm1hbEYnRj1GPQ== and then integrate vertically.LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=JSFHLet's first look at how we can do this for a unit sphere before tackling the Kiss. One revolves the right semicircle around the vertical 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(); # input placeholderThe horizontal integral out to the perimeter curve is by the method of "discs", followed by the vertical integral, first applied to find the volume of a unit sphere, gives the correct result in this parametrized curve boundary approach:
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print(); # input placeholderNow we repeat to get the centroid vertical coordinate by integrating LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=, which by symmetry must be 0: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(); # input placeholderSeems to work. Now let's use it for the Kiss case, scaled back to a unit base radius.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(); # input placeholderThe differential of area in the circular cross-sections is 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 where in this context LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEickYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= is the variable LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=, but we use a dummy variable of integration, and integrate out to the boundary curve as a function of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=, then integrate with respect to LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEjZHlGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvRjNRJ25vcm1hbEYn re-expressed in terms of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEjZHRGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvRjNRJ25vcm1hbEYn by the chain rule, and to reduce the volume integral to the unit base radius case, we further divide by 3 powers of 60.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print(); # input placeholderand the y integral where we have to scale down the extra factor of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= in the integrand by one more factor of 60
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(); # input placeholder6Since we already have the original curve, we scale back up to plot the centroid on the axis for the volume of revolution.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%;Seems reasonable. Notice that if one imagines radial line segments from the centroid to the boundary sweeping from the downward vertical direction around the boundary to the upward vertical direction, 60 is the maximum distance from the centroid except for a smal cone around the upward cirection, so if we compare this to a sphere of radius 60 centered there, it is clear that the average radius is a fraction of 60.
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%;If we actually wanted to compute the average radius of the boundary of the Kiss solid, with respect to its centroid we would have to break the volume integrals into one with the base as boundary, and one with the lateral Kiss face as the boundary. This would be a challenging problem, but doable, just as we did above. LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW9HRiQ2LVEiLEYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdRJXRydWVGJy8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4zMzMzMzMzZW1GJ0Yv