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 [1]
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 [2].
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 [3]. In engineering, the profile of some gear teeth is a combination of hypocycloid and epicycloid [2].
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 [4]. The curves were popularized in the latter half of the 20th Century with the invention of the Spirograph [5].
References
Petaled hypocycloid curve created with Desmos.
Spirograph image: Kungfuman,
[1] Chen, K. et al. (1999). Mathematical Explorations with MATLAB. Cambridge University Press.
[2] Parhusip, H. Arts revealed in calculus and its extension. International Journal of Statistics and Mathematics. Vol. 1(3), pp. 016-023, August, 2014.
[3] Bhattacharyya, B. (2013). Engineering Graphics. I.K. International Publishing House
[4] Simoson AJ (2008) Albrecht Dürer’s trochoidal woodcuts. Probl Resour Issues Math Undergraduate Stud (PRIMUS) 18(6):489–499
[5] Tsiotras, P. & Castro, L. Chapter 6 The Artistic Geometry of Consensus Protocols.