TETRACUSPID

 From the Greek tetra "four" and the Latin cuspis "tip". Name given by Bellavitis in 1854. Other name: quadricuspid (etymologically more logic...).

The tetracuspids are the closed curves with 4 cuspidal points.

Here are examples where the 4 cuspidal points are oriented towards the outside of the curve (protruding tetracuspid):

 The simplest is the astroid, as well as its affine deformations:

 The evolute of the ellipse with is a special case of scaled astroid.

Another example, for which the astroid is also a special case, is given by the

JOACHIMSTHAL TETRACUSPID

 Problem posed by Merlieux in 1842, solved in 1847 by Joachimsthal.

The Joachimsthal tetracuspid is the envelope of the line containing a segment of constant length a the ends of which move on two given secant lines.

 If the two secant lines are the lines forming an angle with Ox, and if , then we get: Cartesian parametrization: , with u and v linked by the relation .  Rational sextic. The points on the moving line describe ellipses.

Other protruding tetracuspids:
- The Maltese cross.
- some toroids
- some reptorias of ellipses

Here are examples where the 4 cuspidal points are oriented towards the inside (huddled tetracuspid):

 The epicycloid with four cusps: The family of curves parametrized by:  for odd values of n. Opposite, the case n = 5: It so happens that the case n = 3 can be obtained as the orthoptic of the cross curve: The Cartesian equation of the previous curves is where and, written this way, we can take any between 0 and 1. Opposite, the case = 1/2.