11479

Elliptic Epitrochoid

This Demonstration traces the path of a point (known as the pole or generator) fixed to an ellipse that rolls without slipping around a stationary base ellipse.
If the circumference ratio between the ellipses is the rational number, a closed curve is obtained after complete revolutions of the rolling ellipse around the base. By then the rolling ellipse will have made revolutions around its center.
In this Demonstration, the circumference ratios are either integers () or of the form (). Consequently, a curve closes after one or two revolutions of the rolling ellipse around the base ellipse.
Moving the pole inside or outside the rolling ellipse makes the elliptic epitrochoid either curtate or prolate.
Changing the eccentricity of either ellipse creates a great variety of curves.

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DETAILS

With the rolling ellipse in its initial angular position , define two points:
1. The point on the base ellipse is at an arc length from its intersection with the positive axis.
2. The point on the rolling ellipse is at an arc length from the intersection with its semimajor axis.
Also define two angles:
3. is the angle that subtends an arc of length on the base ellipse.
4. is the angle between the tangent line on the rolling ellipse at and the axis.
Increasing rolls the ellipse around the base ellipse by means of two geometric transformations on points on the ellipse, performed by the Wolfram Language function transfoEC(ϕ,{x,y},e,n), which consists of a translation by the vector E C and a rotation around through the angle .
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