# The Inverse of an Ellipse

- Author:
- Irina Boyadzhiev

- Topic:
- Ellipse

Given is an ellipse, defined by its foci points and , and point . We perform inversion with respect to the circle with center and radius r. Point D is a point on the circle. We can change the radius by dragging . Point is a random point on the ellipse. Point is the image of under inversion with respect to the above circle ( ).
As moves along the ellipse, will draw the locus of the inverse of the ellipse.

**If the center of the circle is in one of the foci, the inverse of the ellipse is a Limaçon with no loop ( a dimpled Limaçon).**- Move point
or to change the ellipse, and see the changes in the Limaçon. - Drag point D to change the radius of the circle and see how this affects the Limaçon.
- Continue to experiment by dragging the center
of the circle to other locations.