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Deltoid


Deltoid
DeltoidFrames

A 3-cusped hypocycloid, also called a tricuspoid. The deltoid was first considered by Euler in 1745 in connection with an optical problem. It was also investigated by Steiner in 1856 and is sometimes called Steiner's hypocycloid (Lockwood 1967; Coxeter and Greitzer 1967, p. 44; MacTutor). The equation of the deltoid is obtained by setting n=a/b=3 in the equation of the hypocycloid, where a is the radius of the large fixed circle and b is the radius of the small rolling circle, yielding the parametric equations

x=[2/3cosphi-1/3cos(2phi)]a
(1)
=2bcosphi+bcos(2phi)
(2)
y=[2/3sinphi+1/3sin(2phi)]a
(3)
=2bsinphi-bsin(2phi).
(4)

The arc length, curvature, and tangential angle are

s(t)=(16)/9sin^2(3/4t)
(5)
kappa(t)=-3/8csc(3/2t)
(6)
phi(t)=-1/2t.
(7)

The total arc length is computed from the general hypocycloid equation

 s_n=(8a(n-1))/n.
(8)

With n=3, this gives

 s_3=(16)/3a.
(9)

The area is given by

 A_n=((n-1)(n-2))/(n^2)pia^2
(10)

with n=3,

 A_3=2/9pia^2.
(11)

The length of the tangent to the tricuspoid, measured between the two points P, Q in which it cuts the curve again, is constant and equal to 4a. If you draw tangents at P and Q, they are at right angles.

Rather surprisingly, the deltoid can act as a rotor inside an astroid and, in fact, the deltoid catacaustic is an astroid.


See also

Astroid, Deltoid Catacaustic, Deltoid Evolute, Deltoid Involute, Deltoid Pedal Curve, Deltoid Radial Curve, Hypocycloid, Simson Line, Steiner Deltoid

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References

Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 219, 1987.Coxeter, H. S. M. and Greitzer, S. L. Geometry Revisited. Washington, DC: Math. Assoc. Amer., p. 44, 1967.Gray, A. Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, p. 70, 1997.Lawrence, J. D. A Catalog of Special Plane Curves. New York: Dover, pp. 131-135, 1972.Lockwood, E. H. "The Deltoid." Ch. 8 in A Book of Curves. Cambridge, England: Cambridge University Press, pp. 72-79, 1967.MacBeath, A. M. "The Deltoid." Eureka 10, 20-23, 1948.MacBeath, A. M. "The Deltoid, II." Eureka 11, 26-29, 1949.MacBeath, A. M. "The Deltoid, III." Eureka 12, 5-6, 1950.MacTutor History of Mathematics Archive. "Tricuspoid." http://www-groups.dcs.st-and.ac.uk/~history/Curves/Tricuspoid.html.Patterson, B. C. "The Triangle: Its Deltoids and Foliates." Amer. Math. Monthly 47, 11-18, 1940.Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, p. 52, 1991.Yates, R. C. "Deltoid." A Handbook on Curves and Their Properties. Ann Arbor, MI: J. W. Edwards, pp. 71-74, 1952.

Cite this as:

Weisstein, Eric W. "Deltoid." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Deltoid.html

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