| 释义 |
DeltoidA 3-cusped Hypocycloid, also called a Tricuspoid, which has  or  , where  is theRadius of the large fixed Circle and  is the Radius of the small rolling Circle. The deltoidwas first considered by Euler  in 1745 in connection with an optical problem. It was also investigated by Steinerin 1856 and is sometimes called MacTutor Archive). The equation of the deltoid is obtained bysetting  in the equation of the Hypocycloid, yielding the parametric equations
The Arc Length, Curvature, and Tangential Angle are
As usual, care must be taken in the evaluation of  for  . Since the form given above comes from an integralinvolving the Absolute Value of a function, it must be monotonic increasing. Each branch can betreated correctly by defining
 | (6) |
 is the Floor Function, giving the formula
 | (7) |
The total Arc Length is computed from the general Hypocycloid equation
 | (8) |
 | (9) |
 | (10) |
 | (11) |
 ,  in which it cuts the curve again, isconstant and equal to  . If you draw Tangents at and , they are at Right Angles.
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. Lockwood, E. H. ``The Deltoid.'' Ch. 8 in A Book of Curves. Cambridge, England: Cambridge University Press, pp. 72-79, 1967. Macbeth, A. M. ``The Deltoid, I.'' Eureka 10, 20-23, 1948. Macbeth, A. M. ``The Deltoid, II.'' Eureka 11, 26-29, 1949. Macbeth, 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. Yates, R. C. ``Deltoid.'' A Handbook on Curves and Their Properties. Ann Arbor, MI: J. W. Edwards, pp. 71-74, 1952.
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