Our first curve is the Astroid (sometimes spelled asteroid). An astroid (sometimes spelled asteroid) is a particular mathematical curve: a hypocycloid with four cusps. Astroids are also superellipses. Its modern name comes from the Greek word for "star". The curve had a variety of names, including tetracuspid (still used), cubocycloid, and paracycle. It is nearly identical in form to the evolute of an ellipse. The astroid was first discussed by Johann Bernoulli in 1691-1692. 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 area of this curve would be $\displaystyle \frac{3}{8}\pi a^2$ using the cartesian equation The arc length of the curve would be $\displaystyle s\left ( t \right )=\frac{3}{2}\sin^2\left ( t \right )$ using the polar equation. The parameter can be calculated from the general hypocycloid formula which is $\displaystyle s_n=\frac{8a\left ( n-1 \right )}{n}$ . So in this case, we let $\displaystyle n=4$ so the parameter would be $\displaystyle 6a$. Click Read More or here, to view the interactive graphic. | Equation: Cartesian Equation: $\displaystyle x^{\frac{2}{3}}+y^{\frac{2}{3}}=a^{\frac{2}{3}}$ Parametric Equations: $\displaystyle x=a\cos^3\left ( t \right )$ $\displaystyle y=a\sin^3\left ( t \right )$ Polar Equation: $\displaystyle r=\frac{\left | \sec\theta \right |}{\left ( 1+\tan^{\frac{2}{3}}\theta \right )^{\frac{3}{2}}}$ (from $\displaystyle 0$ to $\displaystyle 2\pi$) Graphs: |

**Interactive Graphic:**

**References:**

"Astroid."

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*MathWorld*. Wolfram, n.d. Web. 25 Oct. 2013.