Astroid

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Not to be confused with asteroid.
Astroid
The construction of the astroid.

An astroid is a particular mathematical curve: a hypocycloid with four cusps. Astroids are also superellipses: all astroids are scaled versions of the curve specified by the equation

x^{2/3} + y^{2/3} = 1. \,

Its modern name comes from the Greek word for "star". 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.

A circle of radius 1/4 rolls around inside a circle of radius 1 and a point on its circumference traces an astroid. A line segment of length 1 slides with one end on the x-axis and the other on the y-axis, so that it is tangent to the astroid (which is therefore an envelope). The polar equation is

r=\frac{|\sec(\theta)|}{(1+\tan^{2/3}(\theta))^{3/2}}

and the parametric equations are

x=\cos^3\theta,\qquad y=\sin^3\theta.

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

(x^2+y^2-1)^3+27x^2y^2=0. \,

The astroid is therefore of degree six, and 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.

An astroid created by a circle rolling inside a circle of radius a will have an area of \frac{3}{8} \pi a^2 and a perimeter of 6a.

See also[edit]

References[edit]

  • 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. 10–11. 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. 

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