Copolymer

Astroid

The construction of the astroid.

An astroid (sometimes spelled asteroid) 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

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

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

• Cycloid
• Deltoid a curve with three cusps.
• Stoner–Wohlfarth astroid a use of this curve in magnetics.

References

• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
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• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534

External links

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• "Astroid" at The MacTutor History of Mathematics archive
• "Astroïde" at Encyclopédie des Formes Mathématiques Remarquables (in French)
• Article on 2dcurves.com
• Visual Dictionary Of Special Plane Curves, Xah Lee
• Bars of an Astroid by Sándor Kabai, The Wolfram Demonstrations Project.