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Epicycloid Pedal Curve

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The pedal curve of an epicycloid

x=(a+b)cost-b[((a+b)t)/b]
(1)
y=(a+b)sint-bsin[((a+b)t)/b]
(2)

with pedal point at the origin is

x_p=1/2(a+2b){cost-cos[((a+b)t)/b]}
(3)
y_p=1/2(a+2b){sint-sin[((a+b)t)/b]}.
(4)
EpicycloidPedal

For an n-cusped epicycloid with (a,b)=(n,1), the pedal curve with pedal point at the origin is

x_p=1/2(n+2){cost-cos[(n+1)t]}
(5)
y_p=1/2(n+2){sint-sin[(n+1)t]}.
(6)

Noting that

r=(n+2)sin[1/2(nt)]
(7)
theta=-tan^(-1){cot[1/2(n+2)t]},
(8)

so solving for t gives

t=-2/(n+2)(theta+1/2pi)
(9)

and plugging in gives a polar equation of

r=(n+2)sin[n/(n+2)(theta+1/2pi)],
(10)

which is the equation of a rose curve (Lawrence 1972, p. 204).

See also

Epicycloid, Epicycloid Evolute, Hypocycloid Pedal Curve, Pedal Curve, Rose Curve

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References

Lawrence, J. D. A Catalog of Special Plane Curves. New York: Dover, p. 204, 1972.

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Epicycloid Pedal Curve

Cite this as:

Weisstein, Eric W. "Epicycloid Pedal Curve." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/EpicycloidPedalCurve.html

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