NEPHROID

 Curve studied by Huygens, Tschirnhausen in 1679, Jacques Bernoulli in 1692, Daniel Bernoulli in 1725 and Proctor, who named it in 1878. From the Greek nephros "kidney".

 Complex parametrization: . Cartesian parametrization: . Cartesian equation: . Rational sextic. Polar equation: . Curvilinear abscissa: . Cartesian tangential angle: . Radius of curvature: . Intrinsic equation 1: . Intrinsic equation 2: . Pedal equation: . Length: 12 a; area: 3pa2.

The nephroid is an epicycloid with two cusps (circle with radius a/2 rolling outside a circle (C) with radius a).

It is also
- a pericycloid (circle with radius 3a/2 rolling on the circle (C), and containing it):

 Animation of the double generation

- the envelope of a diameter of a circle with radius a rolling without slipping on (C) externally.

- the envelope of a chord (PQ) of the circle with centre O and radius 2a (circumscribed circle of the nephroid), when P and Q travel along the circle in the same direction, the speed of one being three times the speed of the other (Cremona generation).
 Above, the points n and 3n modulo 30 are linked.

The nephroid is also:
- the envelope of the circles centred on a circle (here, (C)) and tangent to one of its diameters (here Ox). The nephroid is therefore the symmetric image of a diameter with respect to this circle.

- the caustic by reflexion of a circle (here, (C)) with a light source at infinity (here, the incident rays are parallel to Ox).

It is because of this property that the half-nephroid appears in your coffee cup, or here in an enamel plate.