In geometry, the -ellipse is a generalization of the ellipse allowing more than two

foci Focus, or its plural form foci may refer to: Arts * Focus or Focus Festival, former name of the Adelaide Fringe arts festival in South Australia Film *''Focus'', a 1962 TV film starring James Whitmore * ''Focus'' (2001 film), a 2001 film based ...

. -ellipses go by numerous other names, including multifocal ellipse, polyellipse, egglipse, -ellipse, and Tschirnhaus'sche Eikurve (after

Ehrenfried Walther von Tschirnhaus Ehrenfried Walther von Tschirnhaus (or Tschirnhauß, ; 10 April 1651 – 11 October 1708) was a German mathematician, physicist, physician, and philosopher. He introduced the Tschirnhaus transformation and is considered by some to have been the ...

). They were first investigated by

James Clerk Maxwell James Clerk Maxwell (13 June 1831 – 5 November 1879) was a Scottish mathematician and scientist responsible for the classical theory of electromagnetic radiation, which was the first theory to describe electricity, magnetism and light ...

in 1846. Given focal points in a plane, an -ellipse is the

locus Locus (plural loci) is Latin for "place". It may refer to: Entertainment * Locus (comics), a Marvel Comics mutant villainess, a member of the Mutant Liberation Front * ''Locus'' (magazine), science fiction and fantasy magazine ** ''Locus Award' ...

of points of the plane whose sum of distances to the foci is a constant . In formulas, this is the set :

\left\.

The 1-ellipse is the

circle A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is con ...

, and the 2-ellipse is the classic ellipse. Both are

algebraic curve In mathematics, an affine algebraic plane curve is the zero set of a polynomial in two variables. A projective algebraic plane curve is the zero set in a projective plane of a homogeneous polynomial in three variables. An affine algebraic plane ...

s of degree 2. For any number of foci, the -ellipse is a

closed Closed may refer to: Mathematics * Closure (mathematics), a set, along with operations, for which applying those operations on members always results in a member of the set * Closed set, a set which contains all its limit points * Closed interval, ...

convex curve In geometry, a convex curve is a plane curve that has a supporting line through each of its points. There are many other equivalent definitions of these curves, going back to Archimedes. Examples of convex curves include the convex polygons, ...

. The curve is

smooth Smooth may refer to: Mathematics * Smooth function, a function that is infinitely differentiable; used in calculus and topology * Smooth manifold, a differentiable manifold for which all the transition maps are smooth functions * Smooth algebraic ...

unless it goes through a focus. The ''n''-ellipse is in general a subset of the points satisfying a particular

algebraic equation In mathematics, an algebraic equation or polynomial equation is an equation of the form :P = 0 where ''P'' is a polynomial with coefficients in some field, often the field of the rational numbers. For many authors, the term ''algebraic equation'' ...

. If ''n'' is odd, the algebraic degree of the curve is

2^n

, while if ''n'' is

even Even may refer to: General * Even (given name), a Norwegian male personal name * Even (surname) * Even (people), an ethnic group from Siberia and Russian Far East **Even language, a language spoken by the Evens * Odd and Even, a solitaire game wh ...

the degree is

2^n - \binom.

J. Nie, P.A. Parrilo, B. Sturmfels:
J. Nie, P. Parrilo, B.St.: "Semidefinite representation of the k-ellipse", in ''Algorithms in Algebraic Geometry'', I.M.A. Volumes in Mathematics and its Applications, 146, Springer, New York, 2008, pp. 117-132
/ref> ''n''-ellipses are special cases of spectrahedra.

References

{{reflist, 30em, refs= Z.A. Melzak and J.S. Forsyth (1977): "Polyconics 1. polyellipses and optimization", ''Q. of Appl. Math.'', pages 239–255, 1977. P.V. Sahadevan (1987): "The theory of egglipse—a new curve with three focal points", ''International Journal of Mathematical Education in Science and Technology'' 18 (1987), 29–39. {{MR, 872599; {{Zbl, 613.51030. J. Sekino (1999): "''n''-Ellipses and the Minimum Distance Sum Problem", ''American Mathematical Monthly'' 106 #3 (March 1999), 193–202. {{MR, 1682340; {{Zbl, 986.51040.

(1846):
Paper on the Description of Oval Curves
Feb 1846, from ''The Scientific Letters and Papers of James Clerk Maxwell: 1846-1862''

See also

References

Further reading