Astroid

Derivation of the polynomial equation

The polynomial equation may be derived from Leibniz's equation by elementary algebra: $${\displaystyle x^{2/3}+y^{2/3}=a^{2/3}.}$$

Cube both sides: $${\displaystyle \begin{array}{cc}{x}^{6/3}+3x^{4/3}{y}^{2/3}+3x^{2/3}{y}^{4/3}+y^{6/3}& =a^{6/3}\\ x^2+3x^{2/3}{y}^{2/3}\left(x^{2/3}+y^{2/3}\right)+y^2& =a^2\\ x^2+y^2-a^2& =-3x^{2/3}{y}^{2/3}\left(x^{2/3}+y^{2/3}\right)\end{array}}$$

Cube both sides again: $${\displaystyle {\left(x^2+y^2-a^2\right)}^3=-27x^2{y}^2{\left(x^{2/3}+y^{2/3}\right)}^3}$$

But since: $${\displaystyle x^{2/3}+y^{2/3}=a^{2/3}}$$

It follows that $${\displaystyle {\left(x^{2/3}+y^{2/3}\right)}^3=a^2.}$$

Therefore: $${\displaystyle {\left(x^2+y^2-a^2\right)}^3=-27x^2{y}^2{a}^2}$$ or $${\displaystyle {\left(x^2+y^2-a^2\right)}^3+27x^2{y}^2{a}^2=0.}$$

Metric properties

Area enclosed

${\displaystyle \frac{3}{8}\pi a^2}$

Length of curve

${\displaystyle 6a}$

Volume of the surface of revolution of the enclose area about the x-axis.

${\displaystyle \frac{32}{105}\pi a^3}$

Area of surface of revolution about the x-axis

${\displaystyle \frac{12}{5}\pi a^2}$

Properties

The astroid has four cusp singularities in the real plane, the points on the star. It has two more complex cusp singularities at infinity, and four complex double points, for a total of ten singularities.

The dual curve to the astroid is the cruciform curve with equation $x^2{y}^2=x^2+y^2.$ The evolute of an astroid is an astroid twice as large.

The astroid has only one tangent line in each oriented direction, making it an example of a hedgehog.