Anticycloid Curves I: A Rolling Circle

roll the circle

2π

adjust the red point

0.498

show the trace of the center

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

c

is the distance from the point to the center of the circle, a parametrization of an anticycloid is given by



t

∫

0

1-ccos(τ)

1+

2

c

-2ccos(τ)

dτ,-

1+

2

c

-2ccos(t)



with

t

∫

0

1-ccos(τ)

1+

2

c

-2ccos(τ)

dτ=(c+1)F

t

2

-

4c

2

(c-1)

-(c-1)E

t

2

-

4c

2

(c-1)

,

where

F(ϕm)

is an elliptic integral of the first kind and

E(ϕm)

is an elliptic integral of the second kind.