دانشنامه ریاضی و کامپیوتر

Deltoid

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 in the equation of the hypocycloid, where is the radius of the large fixed circle and is the radius of the small rolling circle, yielding the parametric equation s

			(1)
			(2)
			(3)
			(4)

The arc length, curvature, and tangential angle are

			(5)
			(6)
			(7)

The total arc length is computed from the general hypocycloid equation

The total arc length is computed from the general hypocycloid equation

(8)

With , this gives

(9)

The area is given by

(10)

with

(11)

The length of the tangent to the tricuspoid, measured between the two points , in which it cuts the curve again, is constant and equal to . If you draw tangents at and , 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

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.