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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(ϕm)
is an elliptic integral of the first kind and
E(ϕm)
is an elliptic integral of the second kind.
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