Definition

The Deltoid curve, or tricuspoid, is a member of a family of curves called Hypocycloids.  A hypocycloid is a curve produced when a fixed point on a small circle is traced while the small circle rolls within a larger circle, and the deltoid curve is a 3-cusped hypocycloid .  The Deltoid curve is produced when the ratio of the radii of the circles is either 3/1 or 3/2. If this ratio is rational the curve joins itself at the point of its origin, and the radius of the small circle is equal to the number of cusps that the curve has.  If the ratio is irrational, the curve never closes and fills the interior of the circle. For simplicity I will only work with the case where the ratio of the radii is 3/1 and the small circles radius is one unit.
   

History
 
 Euler first studied the deltoid curve while working with an optical problem in 1745, and Steiner studied the deltoid curve in 1856. Although Euler studied this curve first it is sometimes called Steiner's Hypocycloid.
 
                                                                  
        Euler                                                                            Steiner
 

 Formula

When the radius of the larger circle is three times larger than the radius of the smaller circle their circumferences will also have the same ratio of three to one.  The small circle will rotate three times while rolling within the larger circle.  The angle from the center of the small circle to the fixed point, which I will call theta, will cover a range from 0 to negative 6*pi while the small circle rotates three times clockwise on its journey within the larger circle.
 
   

The angle from the center of the large circle to the center of the small circle, which I will call alpha, will range that of a normal circle [0 to 2*pi]. The angles alpha and theta are directly proportional to eachother as given in the following equation:

(large radius -small radius)*(alpha)=(small radius)*(theta)

Solving for theta gives:

(theta) =  (alpha)*(large radius -small radius) / (small radius)

In the case of the deltoid curve this becomes:

(theta) = -2*(alpha)

Theta is equal to negative-two times alpha because of the small circle is rotating clockwise.

 
As the fixed point on the smaller circle traces out the Deltoid curve the x-values of the fixed point are composed of the first two thirds of the x-value of the large circle plus the x-value taken from the small circle as if it were centered at the origin.

x = (2/3)*3cos(alpha)+cos(theta)

Which simplifies to:

x = 2cos(alpha)+cos(-2*alpha)
 

The y-values are found by adding the y-values found as the fixed point rotates around the small circle, as if it were centered at the origin, to two thirds of the y-values found for the larger circle. The small circle is rotating clockwise, so its angle is negative.

y=(2/3)*3sin(alpha)+sin(theta)
 
Which simplifies to:

y=2sin(alpha)+sin(-2*alpha)
 
 

All of this combines to give the parametric equation for the deltoid curve.

alpha = [0: 2*pi]

x = 2cos(alpha)+cos(-2*alpha)

y=2sin(alpha)+sin(-2*alpha)
 

References

Gray, A. Modern Differential Geometry of Curves and Surfaces. Boca Raton, FL: CRC Press, p. 53, 1993.

Lawrence, J. D. A Catalog of Special Plane Curves. New York: Dover, pp. 131-135, 1972.

Lee, X. "Deltoid.'' http://www.best.com/~xah/SpecialPlaneCurves_dir/Deltoid_dir/deltoid.html.

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

Yates, R. C. "Deltoid.'' A Handbook on Curves and Their Properties. Ann Arbor, MI: J. W. Edwards,
pp. 71-74, 1952.