Ellipse Rolling around a Circle

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


Varying the ellipse semimajor axis or eccentricity will change the circumference ratio between the central circle and the ellipse. A closed curve can be obtained after complete revolutions around the circle. By then the ellipse will have made revolutions around its axis.

Changing the pole offset will further create a 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 to the right of the central circle, we define two points:

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

2. The point , on the ellipse in the initial position, is at an arc length from the intersection with its semimajor axis

We also define two angles:

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

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

To roll the ellipse around the circle, two geometric transformations on points on the ellipse are needed. They are performed by the function :

1. a translation by the vector .

2. a rotation around through the angle .

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