Semidefinite Representation of the $k$-Ellipse

Details Semidefinite Representation of the $k$-Ellipse Semidefinite Representation of the $k$-Ellipse

2007-01-31

arxiv.org/abs/math/0702005v1

IMA Volumes in Mathematics and its Applications, Vol. 146, pp. 117-132, 2008

Mathematics

Algebraic Geometry

16 pages, 5 figures

Scientific paper

10.1007/978-0-387-75155-9_7

The $k$-ellipse is the plane algebraic curve consisting of all points whose sum of distances from $k$ given points is a fixed number. The polynomial equation defining the $k$-ellipse has degree $2^k$ if $k$ is odd and degree $2^k{-}\binom{k}{k/2}$ if $k$ is even. We express this polynomial equation as the determinant of a symmetric matrix of linear polynomials. Our representation extends to weighted $k$-ellipses and $k$-ellipsoids in arbitrary dimensions, and it leads to new geometric applications of semidefinite programming.

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