It may be regarded as an epicycloid in which the rolling and fixed circles are equal in diameter, as the inverse of a parabola for its focus, or as the caustic produced by the reflection at a spherical surface of rays emanating from a point on the circumference.
In the particular case when the radii are in the ratio of I to 3 the epicycloid (curve a) will consist of three cusps external to the circle and placed at equal distances along its circumference.
The epicycloid shown is termed the "three-cusped epicycloid" or the "epicycloid of Cremona."
Leonhard Euler (Acta Petrop. 1784) showed that the same hypocycloid can be generated by circles having radii of; (a+b) rolling on a circle of radius a; and also that the hypocycloid formed when the radius of the rolling circle is greater than that of the fixed circle is the same as the epicycloid formed by the rolling of a circle whose radius is the difference of the original radii.
Therefore any epicycloid or hypocycloid may be represented by the equations p = A sin B+,' or p---A cos B,,G, s = A sin B11.