A hypocycloid curve is a plane curve created by a point P on a small circle rolling around inside the circumference of a larger circle.
The parametric equations are 
Cusps in the Hypocycloid Curve
Given that the larger circle’s diameter is a and the smaller circle’s diameter is b, we can define many unique hypocycloid curves. The ratio of a/b determines the number of cusps. For example, if the ratio is 3/1, the curve will have three cusps (i.e., it creates a deltoid curve).
If the ratio a/b is irrational, many more elaborate and interesting curves can be created, some of which resemble multi-petaled flowers or star shapes .
Check out this Desmos page, where you can change the values for a and b to create a variety of different shapes.
Many authors restrict a and b to integers. For example, for a rotating ring gear type epicycloid reducer (Shin) or designing hypocycloid gear assembly for internal combustion engines . In engineering, the profile of some gear teeth is a combination of hypocycloid and epicycloid .
History of the Hypocycloid Curve
Albrecht Durer (1471 to 1528) was the first to introduce the hypocycloid curve in his four-volume, 1,525 geometry tome The Art of Measurement with Compass and Straightedge . The curves were popularized in the latter half of the 20th Century with the invention of the Spirograph .
Petaled hypocycloid curve created with Desmos.
Spirograph image: Kungfuman,
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