Given is a hyperbola, defined by its foci points [math] F_1[/math] and [math]F_2 [/math], and point [math]C[/math]. We perform inversion with respect to the circle with center [math]O_c[/math] and radius r. Point D is a point on the circle. We can change the radius [math]r[/math] by dragging [math]D[/math]. Point [math]M[/math] is a random point on the hyperbola. Point [math]M'[/math] is the image of [math]M[/math] under inversion with respect to the above circle ([math] O_cM \cdot O_cM' = r^2[/math]).[br]As [math]M[/math] moves along the hyperbola, [math]M'[/math] will draw the locus of the inverse of the hyperbola. [br][b]If the center of the circle is in one of the foci, the inverse of the hyperbola is a Limaçon with an inner loop. [/b][br][list][br] [*] Move point [math]C[/math] or [math]F_2[/math] to change the hyperbola, and see the changes in the Limaçon. [br] [*]Drag point D to change the radius of the circle and see how this affects the Limaçon. [br] [*] Move the center of the circle to the center of the hyperbola. What is the inverse in this case?[br] [*] Continue to experiment by dragging the center [math]O_c[/math] of the circle to other locations.[br][/list]