Semidefinite Representation of the k-Ellipse

By Jiawang Nie, Pablo A. Parrilo, Bernd Sturmfels. 2007

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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 \( 2^k - \left( {\begin{array}{*{20}c} k \\ {k/2} \\ \end{array} } \right) \) 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.

Subfields: Computational Mechanics, Computational Theory and Mathematics, Numerical Analysis

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