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单词 Strophoid
释义

Strophoid

Let be a curve, let be a fixed point (the Pole), and let be a second fixed point. Let and bepoints on a line through meeting at such that . The Locus of and is called thestrophoid of with respect to the Pole and fixed point . Let be represented parametrically by, and let and . Then the equation of the strophoid is

(1)
(2)

where
(3)

The name strophoid means ``belt with a twist,'' and was proposed by Montucci in 1846 (MacTutor Archive). The polar form for a general strophoid is
(4)

If , the curve is a Right Strophoid. The following table gives the strophoids of some commoncurves.

CurvePoleFixed PointStrophoid
linenot on lineon lineoblique strophoid
linenot on linefoot of Perpendicular origin to lineRight Strophoid
Circlecenteron the circumferenceFreeth's Nephroid

See also Right Strophoid


References

Lawrence, J. D. A Catalog of Special Plane Curves. New York: Dover, pp. 51-53 and 205, 1972.

Lockwood, E. H. ``Strophoids.'' Ch. 16 in A Book of Curves. Cambridge, England: Cambridge University Press, pp. 134-137, 1967.

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

Yates, R. C. ``Strophoid.'' A Handbook on Curves and Their Properties. Ann Arbor, MI: J. W. Edwards, pp. 217-220, 1952.


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