geometry Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...

, the -ellipse is a

generalization A generalization is a form of abstraction whereby common properties of specific instances are formulated as general concepts or claims. Generalizations posit the existence of a domain or set of elements, as well as one or more common characteri ...

of the

ellipse In mathematics, an ellipse is a plane curve surrounding two focus (geometry), focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. It generalizes a circle, which is the special ty ...

allowing more than two foci. -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 physicist and mathematician who was responsible for the classical theory of electromagnetic radiation, which was the first theory to describe electricity, magnetism an ...

in 1846. Given focal points in a plane, an -ellipse is the locus 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 point (geometry), points in a plane (mathematics), plane that are at a given distance from a given point, the Centre (geometry), centre. The distance between any point of the circle and the centre is cal ...

, 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 cu ...

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

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 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 example, x^5-3x+1=0 is an algebraic equati ...

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

2^n

, while if ''n'' is even 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