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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. 

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© Robert FERRÉOL 2017