Nephroid

The nephroid (from the Greek ὁ νεφρός ho nephros) is a plane curve whose name means kidney-shaped (compare nephrology.) Although the term nephroid was used to describe other curves, it was applied to the curve in this article by Proctor in 1878.

A nephroid is an algebraic curve of degree 6. It can be generated by a rolling circle with radius ${\displaystyle a}$ on the outside of a fixed circle with radius ${\displaystyle 2a}$. Hence a nephroid is an epicycloid.

If the small circle has radius ${\displaystyle a}$, the fixed circle has midpoint ${\displaystyle (0,0)}$ and radius ${\displaystyle 2a}$, the rolling angle of the small circle is ${\displaystyle 2\varphi }$ and point ${\displaystyle (2a,0)}$ the starting point (s. picture) then one gets the

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