“Cartesian Ovals”的概念、定义、习题解题方法及翻译-数学辞典

单词

Cartesian Ovals

释义

Cartesian Ovals

A curve consisting of two ovals which was first studied by Descartes in 1637. It is the locus of a point whose distances from two Foci and in two-center Bipolar Coordinates satisfy

(1)

where

are Positive Integers,

is a Positive real, and

and

are the distancesfrom

and

. If

, the oval becomes an Ellipse. In Cartesian Coordinates, the Cartesian ovals can be written

(2)

(3)


	(4)

Now define

			(5)
			(6)

and set

. Then

(7)

is the distance between

and

, and the equation

(8)

is used instead, an alternate form is

(9)

The curves possess three Foci. If

, one Cartesian oval is a central Conic,while if

, then the curve is a Limaçon and the inside oval touches the outside one. Cartesianovals are Anallagmatic Curves.

References

Cundy, H. and Rollett, A. Mathematical Models, 3rd ed. Stradbroke, England: Tarquin Pub., p. 35, 1989.

Lawrence, J. D. A Catalog of Special Plane Curves. New York: Dover, pp. 155-157, 1972.

Lockwood, E. H. A Book of Curves. Cambridge, England: Cambridge University Press, p. 188, 1967.

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

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