Ellipse Rolling around Another Ellipse

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This Demonstration draws the roulette of a generator point on an ellipse that rolls without slipping around a base ellipse.


Varying the semimajor axis or eccentricity of the rolling ellipse changes the circumference ratio between the base ellipse and the rolling ellipse. A closed curve can be obtained after complete revolutions around the base. By then the rolling ellipse will have made revolutions around its center.

Change the pole offset to create an even greater variety of curves. The bookmarks and snapshots give some examples.


Contributed by: Erik Mahieu (July 2014)
Open content licensed under CC BY-NC-SA



With the rolling ellipse in its initial position to the right of the base ellipse, 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 subtending 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) that consists of a translation by the vector and a rotation around through the angle .

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