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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 .