Ellipse Rolling inside a Circle

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


Varying the ellipse's semimajor axis or eccentricity changes the circumference ratio between the circle and the ellipse. A closed curve can be obtained after complete revolutions inside the circle. By then, the 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 ellipse in its initial position on the right inside the central circle, define two points:

1. The point on the circle is at an arclength from its intersection with the positive axis.

2. The point on the ellipse is at an arclength from the intersection with its semimajor axis.

Also define two angles:

3. is the angle subtending an arc of length on the circle.

4. is the angle between the tangent line on the ellipse at and the axis.

Increasing rolls the ellipse inside the circle by means of two geometric transformations on points on the ellipse, performed by the Mathematica function transfoEI(ϕ,{x,y},e,a), which consists of a translation by the vector and a rotation around through the angle .

In order for the ellipse to roll inside the central circle, the maximum radius of curvature of the ellipse, , must be smaller than the radius of the circle, equal to 1. This limits the eccentricity: .

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