# Anticycloid Curves I: A Rolling Circle

Anticycloid Curves I: A Rolling Circle

The curve traced out by a point on the rim of a circle rolling along a straight line is called a cycloid. Now change the situation: the point moves on a straight line when the circle rolls on a suitable trace. What is the shape of this trace? We call this curve an anticycloid. If is the distance from the point to the center of the circle, a parametrization of an anticycloid is given by

c

dτ,-

t

∫

0

1-ccos(τ)

1+-2ccos(τ)

2

c

1+-2ccos(t)

2

c

with

t

∫

0

1-ccos(τ)

1+-2ccos(τ)

2

c

t

2

4c

2

(c-1)

t

2

4c

2

(c-1)

where is an elliptic integral of the first kind and is an elliptic integral of the second kind.

F(ϕm)

E(ϕm)