| 释义 |
Cassini OvalsThe curves, also called Cassini Ellipses, described by a point such that the product of its distances from two fixedpoints a distance  apart is a constant  . The shape of the curve depends on  . If  , the curve is a singleloop with an Oval (left figure above) or dog bone (second figure) shape. The case  produces a Lemniscate(third figure). If  , then the curve consists of two loops (right figure). The curve was first investigated byCassini  in 1680 when he was studying the relative motions of the Earth  and the Sun. Cassini believed that the Sun traveled around the Earth on one of these ovals, with the Earth at one Focus of the oval.
Cassini ovals are Anallagmatic Curves. The Cassini ovals are defined in two-centerBipolar Coordinates by the equation
 | (1) |
The Cassini ovals have the Cartesian equation
 | (2) |
 | (3) |
 | (4) |
 using the Quadratic Equation gives
If , the curve has Area
 | (6) |
 is the complete EllipticIntegral of the Second Kind. If , the curve becomes
 | (7) |
 | (8) |
 the linear scale of the usual lemniscate  , which has area for each loop). If , the curve becomes two disjoint ovals with equations
 | (9) |
 and
 | (10) |
References
Gray, A. ``Cassinian Ovals.'' §4.2 in Modern Differential Geometry of Curves and Surfaces. Boca Raton, FL: CRC Press, pp. 63-65, 1993.Lawrence, J. D. A Catalog of Special Plane Curves. New York: Dover, pp. 153-155, 1972. Lee, X. ``Cassinian Oval.''http://www.best.com/~xah/SpecialPlaneCurves_dir/CassinianOval_dir/cassinianOval.html Lockwood, E. H. A Book of Curves. Cambridge, England: Cambridge University Press, pp. 187-188, 1967. MacTutor History of Mathematics Archive. ``Cassinian Ovals.''http://www-groups.dcs.st-and.ac.uk/~history/Curves/Cassinian.html. Yates, R. C. ``Cassinian Curves.'' A Handbook on Curves and Their Properties. Ann Arbor, MI: J. W. Edwards, pp. 8-11, 1952.
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