Curves

Astroid

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 $6a$ and its area is $3\pi a^{2}/8$.

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

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