Curves

Astroid

Main
Cartesian equation:
x2/3+y2/3=a2/3x^{2/3} + y^{2/3} = a^{2/3}
or parametrically:
x=acos3(t),y=asin3(t)x = a \cos^{3}(t), y = a \sin^{3}(t)

Description

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 6a6a and its area is 3πa2/83πa^{2}/8.

The gradient of the tangent TT from the point with parameter pp is tan(p)-\tan(p). The equation of this tangent TT is
xsin(p)+ycos(p)=asin(2p)/2x \sin(p) + y \cos(p) = a \sin(2p)/2

Let TT cut the xx-axis and the yy-axis at XX and YY respectively. Then the length XYXY is a constant and is equal to aa.

It can be formed by rolling a circle of radius a/4a/4 on the inside of a circle of radius aa.
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.

Other Web site:

Xah Lee