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NEPHROID : Inversion
We first look at the central case.
Click the diagram below to activate the applet, and then click the ‘Animate’ button to generate the inverse of the nephroid with respect to the origin. Click the button again to stop the generation. Clicking the little red x at bottom right will clear the drawing from the applet window. Now drag the red point Q slowly along the segment. Can you describe the construction? Have we seen this inverse curve before?
The inverse here is an interesting lips-shaped curve – definitely not one we have seen before! What a lot of different curves there are! Let’s now look at the case where one of the cusps is taken as centre of inversion. Click the diagram below to activate the applet, and then click the ‘Animate’ button to generate the inverse of the nephroid with respect to the origin. Click the button again to stop the generation. Clicking the little red x at bottom right will clear the drawing from the applet window. Now drag the red point Q slowly along the segment. Can you describe the construction? Have we seen this inverse curve before?
The inverse here has some resemblance to a parabola. However, I can find no confirming support for this, and the mathematics is rather daunting! Working in the reverse direction, we discovered that the parabola has the cardioid and the cissoid of Diocles as inverses, but of course these used only two particular centres of inversion. Bibliography http://www.absoluteastronomy.com/topics/Nephroid
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We come now to the kidney shaped nephroid curve. There are two obvious points to take as centre of inversion: the central origin, and one of the cusps. 
