Nephroid

In geometry, a nephroid (from Ancient Greek ὁ νεφρός (ho nephros) 'kidney-shaped') is a specific plane curve.

It is a type of epicycloid in which the smaller circle's radius differs from the larger one by a factor of one-half.

Although the term nephroid was used to describe other curves, it was applied to the curve in this article by Richard A. Proctor in 1878.

[1][2] A nephroid is If the small circle has radius

, the rolling angle of the small circle is

the starting point (see diagram) then one gets the parametric representation: The complex map

maps the unit circle to a nephroid[3] The proof of the parametric representation is easily done by using complex numbers and their representation as complex plane.

The movement of the small circle can be split into two rotations.

In the complex plane a rotation of a point

can be performed by the multiplication of point

of the nephroid is generated by the rotation of point

With one gets If the cusps are on the y-axis the parametric representation is and the implicit one: For the nephroid above the The proofs of these statements use suitable formulae on curves (arc length, area and radius of curvature) and the parametric representation above and their derivatives Let

The diameter may lie on the x-axis (see diagram).

The pencil of circles has equations: The envelope condition is One can easily check that the point of the nephroid

and hence a point of the envelope of the pencil of circles.

Similar to the generation of a cardioid as envelope of a pencil of lines the following procedure holds: The following consideration uses trigonometric formulae for

In order to keep the calculations simple, the proof is given for the nephroid with cusps on the y-axis.

Equation of the tangent: for the nephroid with parametric representation Herefrom one determines the normal vector

one gets the cusps of the nephroid, where there is no tangent.

to obtain Equation of the chord: to the circle with midpoint

: The equation of the chord containing the two points

the chord degenerates to a point.

is the parameter of the circle, whose chords are determined), for

Hence any chord from the circle above is tangent to the nephroid and The considerations made in the previous section give a proof for the fact, that the caustic of one half of a circle is a nephroid.

The circle may have the origin as midpoint (as in the previous section) and its radius is

The reflected ray has the normal vector (see diagram)

Hence the reflected ray is part of the line with equation which is tangent to the nephroid of the previous section at point The evolute of a curve is the locus of centers of curvature.

the suitably oriented unit normal.

For a nephroid one gets: The nephroid as shown in the picture has the parametric representation the unit normal vector pointing to the center of curvature and the radius of curvature

The inversion across the circle with midpoint