## Astroid

A 4-cusped 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)

In Cartesian Coordinates,
 (3)

In Pedal Coordinates with the Pedal Point at the center, the equation is
 (4)

The Arc Length, Curvature, and Tangential Angle are

 (5) (6) (7)

As usual, care must be taken in the evaluation of for . Since (5) comes from an integral involving the Absolute Value of a function, it must be monotonic increasing. Each Quadrant can be treated correctly by defining
 (8)

where is the Floor Function, giving the formula
 (9)

The overall Arc Length of the astroid can be computed from the general Hypocycloid formula
 (10)

with ,
 (11)

The Area is given by
 (12)

with ,
 (13)

The Evolute of an Ellipse is a stretched Hypocycloid. The gradient of the Tangent from the point with parameter is . The equation of this Tangent is
 (14)

(MacTutor Archive). Let cut the x-Axis and the y-Axis at and , respectively. Then the length is a constant and is equal to .

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)

which can be written
 (16)

The equation of the Envelope is given by the simultaneous solution of
 (17)

which is
 (18) (19)

Noting that
 (20) (21)

allows this to be written implicitly as
 (22)

the equation of the astroid, as promised.

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)

Using
 (25)

then gives
 (26) (27)

Solving (26) for , plugging into (27) and squaring then gives
 (28)

Rearranging produces the equation
 (29)

the equation of a (Quadrant of an) Ellipse with Semimajor and Semiminor Axes of lengths and .

The astroid is also the Envelope of the family of Ellipses

 (30)

illustrated above.

References

Lawrence, J. D. A Catalog of Special Plane Curves. New York: Dover, pp. 172-175, 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. 52-61, 1967.

MacTutor History of Mathematics Archive. Astroid.'' http://www-groups.dcs.st-and.ac.uk/~history/Curves/Astroid.html.

Yates, R. C. Astroid.'' A Handbook on Curves and Their Properties. Ann Arbor, MI: J. W. Edwards, pp. 1-3, 1952.