Tetracuspid

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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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.