The Inverse of an 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.