TOPICS
Search

Hypocycloid Pedal Curve

DOWNLOAD Mathematica NotebookDownload Wolfram Notebook
HypocycloidPedal

The pedal curve for an n-cusped hypocycloid

x=a((n-1)cost+cos[(n-1)t])/n
(1)
y=a((n-1)sint-sin[(n-1)t])/n
(2)

with pedal point at the origin is the curve

x_p=a((n-2){cost-cos[(1-n)t]})/(2n)
(3)
y_p=a((n-2)cos[t(1-1/2n)]sin(1/2nt))/n.
(4)

Noting that

r=(n-2)sin[1/2(nt)]
(5)
theta=-tan^(-1){cot[1/2(2-n)t]},
(6)

so solving for t gives

t=-2/(n-2)(theta+1/2pi)
(7)

and plugging in gives a polar equation of

r=(n-2)sin[n/(n-2)(theta+1/2pi)],
(8)

which is the equation of a rose curve. In particular, the special cases n=3 and n=4 give a trifolium and quadrifolium, respectively.

See also

Epicycloid Pedal Curve, Hypocycloid, Pedal Curve, Quadrifolium, Trifolium

Explore with Wolfram|Alpha

Cite this as:

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

Subject classifications