We recall the cycloid as the path traced out by a point on a circle rolling along a straight line. To find an inverse of the cycloid, the origin is an obvious point to take as centre of inversion.

Click the linked figure below, and then click the ‘Animate’ button to generate the inverse of the cycloid. Click the button again to stop the generation. You can also see the construction more slowly (and better!) by manually dragging the green point Q along the x-axis. Do you understand the construction? Notice that in the animation, the point Q is dragged well to the right of the window.

This is perhaps the least interesting of the inversion curves we consider here. It is clear that we could continue to obtain the inverse curve arching towards the origin by moving Q even further to the right. The inverse curve is just a poor distorted imitation of the cycloid itself.