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)