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Cassini Ovals


CassiniOvals

The Cassini ovals are a family of quartic curves, also called Cassini ellipses, described by a point such that the product of its distances from two fixed points a distance 2a apart is a constant b^2. The shape of the curve depends on b/a. If a<b, the curve is a single loop with an oval (left figure above) or dog bone (second figure) shape. The case a=b produces a lemniscate (third figure). If a>b, then the curve consists of two loops (right figure). Cassini ovals are anallagmatic curves.

CassiniOvalCurves

A series of ovals for values of b/a=0.1 to 1.5 are illustrated above.

The curve was first investigated by Cassini 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.

The Cassini ovals are defined in two-center bipolar coordinates by the equation

 r_1r_2=b^2,
(1)

with the origin at a focus. Even more incredible curves are produced by the locus of a point the product of whose distances from 3 or more fixed points is a constant.

The Cassini ovals have the Cartesian equation

 [(x-a)^2+y^2][(x+a)^2+y^2]=b^4
(2)

or the equivalent form

 (x^2+y^2+a^2)^2-4a^2x^2=b^4
(3)

and the polar equation

 r^4+a^4-2a^2r^2[1+cos(2theta)]=b^4.
(4)

Solving for r^2 using the quadratic equation gives

r^2=(2a^2cos(2theta)+/-sqrt(4a^4cos^2(2theta)-4(a^4-b^4)))/2
(5)
=a^2cos(2theta)+/-sqrt(a^4cos^2(2theta)+b^4-a^4)
(6)
=a^2cos(2theta)+/-sqrt(a^4[cos^2(2theta)-1]+b^4)
(7)
=a^2cos(2theta)+/-sqrt(b^4-a^4sin^2(2theta))
(8)
=a^2[cos(2theta)+/-sqrt((b/a)^4-sin^2(2theta))].
(9)

Let a torus of tube radius a be cut by a plane perpendicular to the plane of the torus's centroid. Call the distance of this plane from the center of the torus hole r, let a=r, and consider the intersection of this plane with the torus as r is varied. The resulting curves are Cassini ovals, with a lemniscate occurring at r=1/2. Cassini ovals are therefore toric sections.

If a<b, the curve has area

A=1/2r^2dtheta
(10)
=2(1/2)int_(-pi/4)^(pi/4)r^2dtheta
(11)
=a^2+b^2E((a^2)/(b^2)),
(12)

where the integral has been done over half the curve and then multiplied by two and E(x) is the complete elliptic integral of the second kind. If a=b, the curve becomes

r^2=a^2[cos(2theta)+sqrt(1-sin^2theta)]
(13)
=2a^2cos(2theta),
(14)

which is a lemniscate having area

 A=2a^2
(15)

(two loops of a curve sqrt(2) the linear scale of the usual lemniscate r^2=a^2cos(2theta), which has area A=a^2/2 for each loop). If a>b, the curve becomes two disjoint ovals with equations

 r=+/-asqrt(cos(2theta)+/-sqrt((b/a)^4-sin^2(2theta))),
(16)

where theta in [-theta_0,theta_0] and

 theta_0=1/2sin^(-1)[(b/a)^2].
(17)

See also

Cassini Surface, Lemniscate, Mandelbrot Set, Oval, Torus

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References

Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 221, 1987.Gray, A. "Cassinian Ovals." §4.2 in Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, pp. 82-86, 1997.Kabai, S. Mathematical Graphics I: Lessons in Computer Graphics Using Mathematica. Püspökladány, Hungary: Uniconstant, p. 145, 2002.Lawrence, J. D. A Catalog of Special Plane Curves. New York: Dover, pp. 153-155, 1972.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.Piziak, R. and Turner, D. "Exploring Gerschgorin Circles and Cassini Ovals." Mathematica Educ. 3, 13-21, 1994.Smith, D. E. History of Mathematics, Vol. 2: Special Topics of Elementary Mathematics. New York: Dover, p. 329, 1958.Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, pp. 25-26, 1991.Yates, R. C. "Cassinian Curves." A Handbook on Curves and Their Properties. Ann Arbor, MI: J. W. Edwards, pp. 8-11, 1952.

Cite this as:

Weisstein, Eric W. "Cassini Ovals." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/CassiniOvals.html

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