



A hypotrochoid is a roulette traced by a point attached to a circle of radius rolling around the inside of a fixed circle of radius , where is a distance from the center of the interior circle. The parametric equations for a hypotrochoid are
(1)
 
(2)

A polar equation can be derived by computing
(3)
 
(4)

Here, the parameter is not the polar angle but is related to it by
(5)

To get cusps in the hypotrochoid, , because then rotations of bring the point on the edge back to its starting position.
Special cases of the hypotrochoid are summarized in the table below.
curve  special values 
ellipse  
hypocycloid  
rose curve 
The arc length, curvature, and tangential angle are
(6)
 
(7)
 
(8)

where is an elliptic integral of the second kind.