A trochoid is the locus of a point at a distance 
from the center of a circle of
radius 
rolling on a fixed line. A trochoid
has parametric equations
 |  |  |
(1)
|
 |  |  |
(2)
|
If 
, the trochoid is known as a curtate cycloid; if 
, it is a cycloid; and if 
, the curve
is a prolate cycloid.
The arc length function, curvature, and tangential angle are given by
 |  |  |
(3)
|
 |  |  |
(4)
|
 |  | ![-t/2+(pi|a-b|)/(2(a-b))-tan^(-1)[(a-b)/(a+b)cot(t/2)]+pi|_t/(2pi)_|,](/images/equations/Trochoid/Inline20.svg) |
(5)
|
where 
is an incomplete elliptic
integral of the second kind and 
is the floor function.
The term trochoid is also used for other classes of roulettes. The Encyclopedia of Mathematics includes epitrochoids and hypotrochoids under this name (Sokolov). In gearing terminology, it can also describe the path of a point rigidly attached to a circle rolling along a curved base, or at a fixed offset from a line rolling along the convex side of a base curve (Oberg et al. 2020, p. 2207). When the tracing point lies on the rolling line itself, its path is an involute.
