Astroid
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In mathematics, an astroid is a particular type of roulette curve: a hypocycloid with four cusps. Specifically, it is the locus of a point on a circle as it rolls inside a fixed circle with four times the radius.<ref>Yates</ref> By double generation, it is also the locus of a point on a circle as it rolls inside a fixed circle with 4/3 times the radius. It can also be defined as the envelope of a line segment of fixed length that moves while keeping an end point on each of the axes. It is therefore the envelope of the moving bar in the Trammel of Archimedes.
Its modern name comes from the Greek word for "star". It was proposed, originally in the form of "Astrois", by Joseph Johann von Littrow in 1838.<ref>Template:Cite book</ref><ref>Template:Cite book</ref> 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.
EquationsEdit
If the radius of the fixed circle is a then the equation is given by<ref>Yates, for section</ref> <math display="block">x^{2/3} + y^{2/3} = a^{2/3}. </math> This implies that an astroid is also a superellipse.
Parametric equations are <math display="block"> \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} </math>
The pedal equation with respect to the origin is <math display="block">r^2 = a^2 - 3p^2,</math>
the Whewell equation is <math display="block">s = {3a \over 4} \cos 2\varphi,</math> and the Cesàro equation is <math display="block">R^2 + 4s^2 = \frac{9a^2}{4}.</math>
The polar equation is<ref>{{#invoke:Template wrapper|{{#if:|list|wrap}}|_template=cite web |_exclude=urlname, _debug, id |url = https://mathworld.wolfram.com/{{#if:Astroid%7CAstroid.html}} |title = Astroid |author = Weisstein, Eric W. |website = MathWorld |access-date = |ref = Template:SfnRef }}</ref> <math display="block">r = \frac{a}{\left(\cos^{2/3}\theta + \sin^{2/3}\theta\right)^{3/2}}.</math>
The astroid is a real locus of a plane algebraic curve of genus zero. It has the equation<ref>A derivation of this equation is given on p. 3 of http://xahlee.info/SpecialPlaneCurves_dir/Astroid_dir/astroid.pdf</ref> <math display="block">\left(x^2 + y^2 - a^2\right)^3 + 27 a^2 x^2 y^2 = 0. </math>
The astroid is, therefore, a real algebraic curve of degree six.
Derivation of the polynomial equationEdit
The polynomial equation may be derived from Leibniz's equation by elementary algebra: <math display="block">x^{2/3} + y^{2/3} = a^{2/3}. </math>
Cube both sides: <math display="block">\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}</math>
Cube both sides again: <math display="block">\left(x^2 + y^2 - a^2\right)^3 = -27 x^2 y^2 \left(x^{2/3} + y^{2/3}\right)^3</math>
But since: <math display="block">x^{2/3} + y^{2/3} = a^{2/3} \,</math>
It follows that <math display="block">\left(x^{2/3} + y^{2/3}\right)^3 = a^2.</math>
Therefore: <math display="block">\left(x^2 + y^2 - a^2\right)^3 = -27 x^2 y^2 a^2</math> or <math display="block">\left(x^2 + y^2 - a^2\right)^3 + 27 x^2 y^2 a^2 = 0. </math>
Metric propertiesEdit
- Area enclosed<ref>Yates, for section</ref>
- <math>\frac{3}{8} \pi a^2</math>
- Length of curve
- <math>6a</math>
- Volume of the surface of revolution of the enclose area about the x-axis.
- <math>\frac{32}{105}\pi a^3</math>
- Area of surface of revolution about the x-axis
- <math>\frac{12}{5}\pi a^2</math>
PropertiesEdit
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 <math display="inline"> x^2 y^2 = x^2 + y^2.</math> 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.<ref>Template:Cite journal</ref>
See alsoEdit
- 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