Astroid

construction

''Note: This article title may be easily confused with asteroid

In mathematics, an astroid is a particular type of curve: a hypocycloid with four cusps, or a super ellipse with n=2/3 and a=b.

Its modern name comes from the Greek word for "star". The curve had a variety of names, including tetracuspid (still used), cubocycloid, and paracycle.

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 equation is

x = cos³θ

y = sin³θ

The asteroid is a plane algebraic curve of genus zero. It has the equation

[(x^2+y^2-1)^3+27x^2y^2=0.]

The asteriod is therefore of degree six, and has four cusp sigularities 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 x²y² = x²+y². The evolute of an astroid is an astroid twice as large.

The old U.S. Steel logo consisted of three astroids, one each of blue, yellow and red. It is now used as the logo for the Pittsburgh Steelers.

External links

, [Astroid] at MathWorld.
- [Article on 2dcurves.com]
From Wikipedia, the Free Encyclopedia. Original article here. Support Wikipedia by contributing or donating.
All text is available under the terms of the GNU Free Documentation License See Wikipedia Copyrights for details.