Ellipse Rolling around a Circle

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

p+q

revolutions around its axis.

Changing the pole offset will further create a variety of curves. The bookmarks and snapshots give some examples.

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:

is the angle subtending an arc of length

on the circle

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

(x,y)

on the ellipse are needed. They are performed by the function

transfoEC(ϕ,{x,y},e,n)

1. a translation by the vector

E-C

2. a rotation around

through the angle

ϕ-τ

Erik Mahieu