Epicycloid and Hypocycloid
An epicycloid is a plane curve created by tracing a chosen point on the edge of a circle of radius r rolling on the outside of a circle of radius R. A hypocycloid is obtained similarly except that the circle of radius r rolls on the inside of the circle of radius R.
The parametric equations for the epicycloid and hypocycloid are:
xθ= R+r ⋅ cosθ +s⋅r ⋅ cosR+rr⋅θ
yθ= R+r ⋅ sinθ − r ⋅ sinR+rr⋅θ
where s=1 for the epicycloid and s=−1 for the hypocycloid.
Epitrochoid and hypotrochoid
Two related curves result when we include another parameter, L, which represents the ratio of pen length to the radius of the circle:
xθ= R+r ⋅ cosθ +s⋅L⋅r ⋅ cosR+rr⋅θ
yθ= R+r ⋅ sinθ − L⋅r ⋅ sinR+rr⋅θ
When s=1 and L ≠ 1 the curve is called an epitrochoid; when s=−1 and L ≠ 1, the curve is called a hypotrochoid.
Number of cusps
Let k = Rr.
If k is an integer, the curve has k cusps.
If k is a rational number, k = ab and k is expressed in simplest terms, then the curve has a cusps.
If k is an irrational number, then the curve never closes.
Fixed circle radius (R) =
Rolling circle radius (r) =
Ratio of Pen length/radius
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