Trochoid
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.gif) |
(5)
|
where 
is an incomplete elliptic
integral of the second kind and 
is the floor function.
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trochoid
