Astroid

If the radius of the fixed circle is a then the equation is given byYates, for section

$$x^\{2/3\}\; +\; y^\{2/3\}\; =\; a^\{2/3\}.$$

This implies 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

$$s\; =\; \{3a\; \backslash over\; 4\}\; \backslash cos\; 2\backslash varphi,$$

and the Cesàro equation is

$$R^2\; +\; 4s^2\; =\; \backslash frac\{9a^2\}\{4\}.$$

The polar equation is{{MathWorld | urlname=Astroid | title=Astroid}}

$$r\; =\; \backslash frac\{a\}\{\backslash left(\backslash cos^\{2/3\}\backslash theta\; +\; \backslash sin^\{2/3\}\backslash theta\backslash right)^\{3/2\}\}.$$

The astroid is a real locus of a plane algebraic curve of genus zero. It has the equationA derivation of this equation is given on p. 3 of http://xahlee.info/SpecialPlaneCurves_dir/Astroid_dir/astroid.pdf

$$\backslash left(x^2\; +\; y^2\; -\; a^2\backslash right)^3\; +\; 27\; a^2\; x^2\; y^2\; =\; 0.$$

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

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:

$$\backslash begin\{align\}$$

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:

$$\backslash left(x^2\; +\; y^2\; -\; a^2\backslash right)^3\; =\; -27\; x^2\; y^2\; \backslash left(x^\{2/3\}\; +\; y^\{2/3\}\backslash right)^3$$

But since:

$$x^\{2/3\}\; +\; y^\{2/3\}\; =\; a^\{2/3\}\; \backslash ,$$

It follows that

$$\backslash left(x^\{2/3\}\; +\; y^\{2/3\}\backslash right)^3\; =\; a^2.$$

Therefore:

$$\backslash left(x^2\; +\; y^2\; -\; a^2\backslash right)^3\; =\; -27\; x^2\; y^2\; a^2$$

or

$$\backslash left(x^2\; +\; y^2\; -\; a^2\backslash right)^3\; +\; 27\; x^2\; y^2\; a^2\; =\; 0.$$

;Area enclosedYates, for section :$\backslash frac\{3\}\{8\}\; \backslash pi\; a^2$

;Length of curve :$6a$

;Volume of the surface of revolution of the enclose area about the x-axis. :$\backslash frac\{32\}\{105\}\backslash pi\; a^3$

;Area of surface of revolution about the x-axis :$\backslash frac\{12\}\{5\}\backslash pi\; a^2$

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

| last1 = Nishimura | first1 = Takashi

| last2 = Sakemi | first2 = Yu

| doi = 10.14492/hokmj/1319595861

| issue = 3

| journal = Hokkaido Mathematical Journal

| mr = 2883496

| pages = 361–373

| title = View from inside

| volume = 40

| year = 2011| doi-access = free

}}

• Cardioid – an epicycloid with one cusp
• Nephroid – an epicycloid with two cusps
• Deltoid – a hypocycloid with three cusps
• Stoner–Wohlfarth astroid – a use of this curve in magnetics
• Spirograph

{{Reflist}}

• {{Cite book | author=J. Dennis Lawrence | title=A catalog of special plane curves | url=https://archive.org/details/catalogspecialpl00lawr | url-access=limited | publisher=Dover Publications | year=1972 | isbn=0-486-60288-5 | pages=[https://archive.org/details/catalogspecialpl00lawr/page/n20 4]–5,34–35,173–174 }}
• {{Cite book | author = Wells D | year = 1991 | title = The Penguin Dictionary of Curious and Interesting Geometry | publisher = Penguin Books | location = New York | isbn = 0-14-011813-6 | pages = 10–11}}
• {{Cite book | author=R.C. Yates | title=A Handbook on Curves and Their Properties | location=Ann Arbor, MI | publisher=J. W. Edwards | pages=1 ff|chapter=Astroid| year=1952 }}

{{commons category|Astroid}}

• {{springer|title=Astroid|id=p/a013540}}
• [http://www-history.mcs.st-andrews.ac.uk/history/Curves/Astroid.html "Astroid" at The MacTutor History of Mathematics archive]
• [http://www.mathcurve.com/courbes2d.gb/astroid/astroid.shtml "Astroid" at The Encyclopedia of Remarkable Mathematical Forms]
• [https://web.archive.org/web/20170824133613/http://www.2dcurves.com/roulette/roulettea.html Article on 2dcurves.com]
• [http://xahlee.org/SpecialPlaneCurves_dir/Astroid_dir/astroid.html Visual Dictionary Of Special Plane Curves, Xah Lee]
• [http://demonstrations.wolfram.com/BarsOfAnAstroid/ Bars of an Astroid] by Sándor Kabai, The Wolfram Demonstrations Project.

Category:Sextic curves

Category:Roulettes (curve)