Astroid

Equations

If the radius of the fixed circle is a then the equation is given by x^{2/3} + y^{2/3} = a^{2/3}. This means that an astroid is also a superellipse.

Parametric equations are \begin{align} x = a\cos^3 t &= \frac{a}{4} \left( 3\cos \left(t\right) + \cos \left(3t\right)\right), \[2ex] y = a\sin^3 t &= \frac{a}{4} \left( 3\sin \left(t\right) - \sin \left(3t\right) \right). \end{align}

The pedal equation with respect to the origin is r^2 = a^2 - 3p^2,

the Whewell equation is s = {3a \over 4} \cos 2\varphi, and the Cesàro equation is R^2 + 4s^2 = \frac{9a^2}{4}.

The polar equation is r = \frac{a}{\left(\cos^{2/3}\theta + \sin^{2/3}\theta\right)^{3/2}}.

The astroid is a real locus of a plane algebraic curve of genus zero. It has the equation \left(x^2 + y^2 - a^2\right)^3 + 27 a^2 x^2 y^2 = 0.

The astroid is, therefore, a real algebraic curve of degree six.

Derivation of the polynomial equation

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

Cube both sides: \begin{align} x^{6/3} + 3x^{4/3}y^{2/3} + 3x^{2/3}y^{4/3} + y^{6/3} &= a^{6/3} \[1.5ex] x^2 + 3x^{2/3}y^{2/3} \left(x^{2/3} + y^{2/3}\right) + y^2 &= a^2 \[1ex] x^2 + y^2 - a^2 &= -3x^{2/3}y^{2/3} \left(x^{2/3} + y^{2/3}\right) \end{align}

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

But since: x^{2/3} + y^{2/3} = a^{2/3} ,

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

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

Metric properties

;Area enclosed :\frac{3}{8} \pi a^2 ;Length of curve :6a ;Volume of the surface of revolution of the enclose area about the x-axis. :\frac{32}{105}\pi a^3 ;Area of surface of revolution about the x-axis :\frac{12}{5}\pi a^2 ;Radius of the outscribed circle: a ;Radius of the inscribed circle: 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.{{cite journal

References

References

1. Yates
2. J. J. v. Littrow. (1838). "Kurze Anleitung zur gesammten Mathematik".
3. Loria, Gino. (1902). "Spezielle algebraische und transscendente ebene kurven. Theorie und Geschichte".
4. Yates, for section
5. "Astroid".
6. A derivation of this equation is given on p. 3 of http://xahlee.info/SpecialPlaneCurves_dir/Astroid_dir/astroid.pdf
7. Yates, for section