Astroid: Definition from Answers.com

Not to be confused with asteroid.

Astroid

The construction of the astroid.

In mathematics, an astroid is a particular type of 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 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.

1 In popular culture
2 See also
3 References
4 External links

In popular culture

The Steelmark, originally created by U.S. Steel, consists of three astroids, one each of blue, yellow and red. It has been adapted as the logo for the Pittsburgh Steelers on helmets and merchandise since 1962. It can also be found on many steel products and canned goods as it has been "gifted" by U.S. Steel to the American Iron and Steel Institute for use on all steel products.

On the American television show The Price is Right astroids have been used as set decorations both behind the host, on contestant's bid booths and behind the prizes since the programs inception in 1968.

Former Formula One driver Michael Schumacher wore a series of race helmets from 1992 until his retirement in 2006, all of which featured a pattern of white astroids on the top.

References

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.

External links

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Astroid

Contents

In popular culture

See also

References

External links