Curtate Cycloid

The path traced out by a fixed point at a Radius $b<a$, where $a$ is the Radius of a rolling Circle, sometimes also called a Contracted Cycloid.

$$x = a\phi-b\sin\phi$$ (1)

$$y = a-b\cos\phi.$$ (2)

The Arc Length from $\phi=0$ is

$$s=2(a+b)E(u),$$ (3)

where

$$\sin({\textstyle{1\over 2}}\phi)=\mathop{\rm sn}\nolimits u$$ (4)

$$k^2={4ab\over (a+c)^2},$$ (5)

and $E(u)$ is a complete Elliptic Integral of the Second Kind and $\mathop{\rm sn}\nolimits u$ is a Jacobi Elliptic Function.

See also Cycloid, Prolate Cycloid

References

Cundy, H. and Rollett, A. Mathematical Models, 3rd ed. Stradbroke, England: Tarquin Pub., 1989.

Wagon, S. Mathematica in Action. New York: W. H. Freeman, pp. 46-50, 1991.