Cartesian equation:
x2/3 + y2/3 = a2/3
or parametrically:
x = a cos3(t), y = a sin3(t)

Click below to see one of the Associated curves.

Definitions of the Associated curves Evolute
Involute 1 Involute 2
Inverse curve wrt origin Inverse wrt another circle
Pedal curve wrt origin Pedal wrt another point
Negative pedal curve wrt origin Negative pedal wrt another point
Caustic wrt horizontal rays Caustic curve wrt another point

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The astroid was first discussed by Johann Bernoulli in 1691-92. It also appears in Leibniz's correspondence of 1715. It is sometimes called the tetracuspid for the obvious reason that it has four cusps.

The astroid only acquired its present name in 1836 in a book published in Vienna. It has been known by various names in the literature, even after 1836, including cubocycloid and paracycle.

The length of the astroid is 6a and its area is 3πa2/8.

The gradient of the tangent T from the point with parameter p is -tan(p). The equation of this tangent T is

x sin(p) + y cos(p) = a sin(2p)/2
Let T cut the x-axis and the y-axis at X and Y respectively. Then the length XY is a constant and is equal to a.

It can be formed by rolling a circle of radius a/4 on the inside of a circle of radius a.
It can also be formed as the envelope produced when a line segment is moved with each end on one of a pair of perpendicular axes. It is therefore a glissette.

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Xah Lee

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JOC/EFR/BS January 1997

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