
A 4cusped Hypocycloid which is sometimes also called a Tetracuspid, Cubocycloid, or Paracycle.
The parametric equations of the astroid can be obtained by plugging in or into the equations for a
general Hypocycloid, giving
(1)  
(2) 
(3) 
(4) 
The Arc Length, Curvature, and Tangential Angle are
(5)  
(6)  
(7) 
(8) 
(9) 
(10) 
(11) 
(12) 
(13) 
(14) 
The astroid can also be formed as the Envelope produced when a Line Segment is moved with each end on one of a
pair of Perpendicular axes (e.g., it is the curve enveloped by a ladder sliding against a wall or a garage door with the
top corner moving along a vertical track; left figure above). The astroid is therefore a Glissette. To see this, note
that for a ladder of length , the points of contact with the wall and floor are and
,
respectively. The equation of the Line made by the ladder with its foot at is therefore
(15) 
(16) 
(17) 
(18)  
(19) 
(20)  
(21) 
(22) 
The related problem obtained by having the ``garage door'' of length with an ``extension'' of length
move up and down a slotted track also gives a surprising answer. In this case, the position of the ``extended''
end for the foot of the door at horizontal position and Angle is given by
(23)  
(24) 
(25) 
(26)  
(27) 
(28) 
(29) 
The astroid is also the Envelope of the family of Ellipses
(30) 
See also Deltoid, Ellipse Envelope, Lamé Curve, Nephroid, Ranunculoid
References
Lawrence, J. D. A Catalog of Special Plane Curves. New York: Dover, pp. 172175, 1972.
Lee, X. ``Astroid.'' http://www.best.com/~xah/SpecialPlaneCurves_dir/Astroid_dir/astroid.html.
Lockwood, E. H. ``The Astroid.'' Ch. 6 in A Book of Curves. Cambridge, England: Cambridge University Press, pp. 5261, 1967.
MacTutor History of Mathematics Archive. ``Astroid.'' http://wwwgroups.dcs.stand.ac.uk/~history/Curves/Astroid.html.
Yates, R. C. ``Astroid.'' A Handbook on Curves and Their Properties. Ann Arbor, MI: J. W. Edwards, pp. 13, 1952.
© 19969 Eric W. Weisstein