- Source: Cyclocycloid
A cyclocycloid is a roulette traced by a point attached to a circle of radius r rolling around, a fixed circle of radius R, where the point is at a distance d from the center of the exterior circle.
The parametric equations for a cyclocycloid are
x
(
θ
)
=
(
R
+
r
)
cos
θ
−
d
cos
(
R
+
r
r
θ
)
,
{\displaystyle x(\theta )=(R+r)\cos \theta -d\cos \left({R+r \over r}\theta \right),\,}
y
(
θ
)
=
(
R
+
r
)
sin
θ
−
d
sin
(
R
+
r
r
θ
)
.
{\displaystyle y(\theta )=(R+r)\sin \theta -d\sin \left({R+r \over r}\theta \right).\,}
where
θ
{\displaystyle \theta }
is a parameter (not the polar angle). And r can be positive (represented by a ball rolling outside of a circle) or negative (represented by a ball rolling inside of a circle) depending on whether it is of an epicycloid or hypocycloid variety.
The classic Spirograph toy traces out these curves.
See also
Centered trochoid
Cycloid
Epicycloid
Hypocycloid
Spirograph
External links
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