The Upper Half of the Astroid $y = \left(1 - |x|^{2/3}\right)^{3/2}$

$y = \left(1 - |x|^{2/3}\right)^{3/2}$ is the upper half of the astroid¹. The name comes from the Greek word for star, and the curve is known for the four sharp points at its corners.

Since $x^{2/3}$ equals $|x|^{2/3}$ even for negative $x$, the formula behaves identically on both sides of the $y$-axis.

Domain and symmetry

• The domain is $-1 \leq x \leq 1$
• The range is $0 \leq y \leq 1$
• Even function
• It passes through the apex $(0, 1)$ and touches the $x$-axis at $(\pm 1, 0)$

Together with the lower half, the full curve is symmetric about the $x$-axis, the $y$-axis and the origin, and it maps onto itself under a $90^\circ$ rotation about the origin.

Parametrisation

Setting $x = \cos^3 t$ and $y = \sin^3 t$ gives $\cos^2 t + \sin^2 t = 1$, so this parametrisation traverses the whole curve exactly once. The upper half corresponds to $0 \leq t \leq \pi$.

Cusps and tangents

For $0 < x < 1$ the derivative is $y' = -\dfrac{\sqrt{1 - x^{2/3}}}{x^{1/3}}$.

The apex is therefore a cusp with a vertical tangent, where the two branches meet at a point rather than joining smoothly. All four corners are cusps of this kind.

Concavity

For $0 < x < 1$ we have $y'' > 0$, so the curve is concave up, and by symmetry the left half is too. There is no inflection point. The curve falls steeply from the apex and flattens out as it approaches the $x$-axis.

Perimeter and area

Both quantities are smaller here by exactly the amount the sides curve inward.

Envelope of a segment

Slide a segment of length $1$ so that its two endpoints stay on the $x$-axis and the $y$-axis. The family of such segments envelops precisely this astroid: if the intercepts are $(a, 0)$ and $(0, b)$ with $a^2 + b^2 = 1$, the line touches the curve at $(a^3, b^3)$. Equivalently, it is the outline swept by a ladder sliding down a wall.

Construction and history

The astroid is a hypocycloid: the path traced by a point on a circle of radius $\dfrac{1}{4}$ rolling without slipping inside a circle of radius $1$.

It appeared in the seventeenth century in Romer's study of gear teeth and was taken up by Johann Bernoulli and Leibniz; the name astroid only became standard in the nineteenth century.

1. Astroid, Wikipedia