Astroid

Astroid

The hypocycloid construction of the astroid.

Astroid as the common envelope of a family of traced ellipses, where a + b = const .

An astroid is a particular mathematical 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.^[1] 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 with 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.^[2]^[3] 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.

Equations

If the radius of the fixed circle is a then the equation is given by^[4]

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

This implies that an astroid is also a superellipse.

Parametric equations are

x=a\cos ^{3}t={a \over 4}(3\cos t+\cos 3t),

y=a\sin ^{3}t={a \over 4}(3\sin t-\sin 3t).

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^[5]

r={\frac {a}{(\cos ^{{2/3}}\theta +\sin ^{{2/3}}\theta )^{{3/2}}}}.

The astroid is a real locus of a plane algebraic curve of genus zero. It has the equation^[6]

(x^{2}+y^{2}-a^{2})^{3}+27a^{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:

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}(x^{2/3}+y^{2/3})+y^{2}=a^{2}\,

x^{2}+y^{2}-a^{2}=-3x^{2/3}y^{2/3}(x^{2/3}+y^{2/3})\,

Cube both sides again:

(x^{2}+y^{2}-a^{2})^{3}=-27x^{2}y^{2}(x^{2/3}+y^{2/3})^{3}\,

But since:

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

It follows that

(x^{2/3}+y^{2/3})^{3}=a^{2}.\,

Therefore:

(x^{2}+y^{2}-a^{2})^{3}=-27x^{2}y^{2}a^{2}\,

or

(x^{2}+y^{2}-a^{2})^{3}+27x^{2}y^{2}a^{2}=0.\,

Metric properties

Area enclosed^[7]: ${\frac {3}{8}}\pi a^{2}$

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

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 $\textstyle x^{2}y^{2}=x^{2}+y^{2}.$ The evolute of an astroid is an astroid twice as large.

References

↑ Yates
↑ J. J. v. Littrow (1838). "§99. Die Astrois". Kurze Anleitung zur gesammten Mathematik. Wien. p. 299.
↑ Loria, Gino (1902). Spezielle algebraische und transscendente ebene kurven. Theorie und Geschichte. Leipzig. p. 224.
↑ Yates, for section
↑ Mathworld
↑ A derivation of this equation is given on p. 3 of http://xahlee.info/SpecialPlaneCurves_dir/Astroid_dir/astroid.pdf
↑ Yates, for section

J. Dennis Lawrence (1972). A catalog of special plane curves. Dover Publications. pp. 4–5, 34–35, 173–174. ISBN 0-486-60288-5.
Wells D (1991). The Penguin Dictionary of Curious and Interesting Geometry. New York: Penguin Books. pp. 10&ndash, 11. ISBN 0-14-011813-6.
R.C. Yates (1952). "Astroid". A Handbook on Curves and Their Properties. Ann Arbor, MI: J. W. Edwards. pp. 1 ff.

External links

Hazewinkel, Michiel, ed. (2001) [1994], "Astroid", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4
Weisstein, Eric W. "Astroid". MathWorld.
"Astroid" at The MacTutor History of Mathematics archive
"Astroid" at The Encyclopedia of Remarkable Mathematical Forms
Article on 2dcurves.com
Visual Dictionary Of Special Plane Curves, Xah Lee
Bars of an Astroid by Sándor Kabai, The Wolfram Demonstrations Project.

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[1] Yates

[2] J. J. v. Littrow (1838). "§99. Die Astrois". Kurze Anleitung zur gesammten Mathematik. Wien. p. 299.

[3] Loria, Gino (1902). Spezielle algebraische und transscendente ebene kurven. Theorie und Geschichte. Leipzig. p. 224.

[4] Yates, for section

[5] Mathworld

[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

Astroid

Equations

Derivation of the polynomial equation

Metric properties

Properties

See also

References

External links