Nephroids

Nephroids are a family of plane curves known for their kidney-like shape. The name derives from the Greek word nephros, meaning kidney. The most common nephroid is an epicycloid, formed when a circle of radius r rolls externally around a fixed circle of radius 2r, and a point on the rolling circle traces the curve. The nephroid has two cusps and is symmetric with respect to the line through the centers of the two circles.

A standard parametric form uses a positive scale parameter a (often taken as a = r): x = 3a cos t − a cos 3t, y = 3a sin t − a sin 3t, with t ranging from 0 to 2π. This parametrization describes the two-cusped epicycloid that defines the nephroid. Its shape can also be described by a Cartesian equation: (x^2 + y^2 − 4a^2)^3 = 108 a^4 y^2. The curve exhibits cusps at specific points and is of interest in both pure and applied contexts.

In optics, the nephroid appears as the catacaustic of a circle: when parallel light rays reflect inside a circular mirror, the envelope of the reflected rays forms a nephroid. This caustic interpretation links the nephroid to wavefronts and focal properties of circles. The nephroid also arises in studies of epicycloids and harmonic motion, and it serves as a classic example in discussions of special plane curves with cusps.

Historically, the nephroid has been studied in the context of geometric construction and optical caustics since the 17th century, with contributions from early geometers and analysts exploring its properties and applications.