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Astroid Pedal Curve

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AstroidPedal

An astroid pedal curve with pedal point at the center of an astroid

x=acos^3t
(1)
y=asin^3t
(2)

is the quadrifolium

x_p=acostsin^2t
(3)
y_p=acos^2tsint.
(4)

For a>0 and 0<=t<=2pi, this parametrization traverses the four lobes clockwise, so its signed area and geometric area are

A_s=-(pia^2)/8,
(5)

and

A=(pia^2)/8,
(6)

respectively. Since

x_p^'(t)^2+y_p^'(t)^2=a^2[1-3/4sin^2(2t)],
(7)

the total arc length is

L=4aE((sqrt(3))/2),
(8)

where E(k) is the complete elliptic integral of the second kind with modulus k.

See also

Arc Length, Astroid, Pedal Curve, Quadrifolium, Signed Area

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Cite this as:

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

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