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Astroid

Astroid

Object type: Plane curve

Definition

The astroid (not to be confused with the word 'asteroid') is the curve $$x^{2/3} + y^{2/3} = 1.$$

Parameterisation

The astroid is the image $\mathbf{r}(\left[0,2\pi\right[)$ where $$\mathbf{r}(t) = \basis\begin{pmatrix}\cos^3 t\\\sin^3 t\end{pmatrix}, \quad\quad\forall t\in\left[ 0,2\pi\right[.$$

Properties

The astroid is the set of points traced out by a fixed point on a circle as it rolls (without slipping) along the inside of another circle with radius four times as large.

The length of the curve is trivially found to be $$L = 6$$ and, using Green's theorem, the area enclosed by the astroid is found to be $$A = \frac{3}{2}\int_0^{2\pi} (\cos^2 t ~ \sin^4 t + \sin^2 t ~ \cos^4 t) ~ dt = \frac{3\pi}{8}.$$