Semidefinite Representation of the k-Ellipse

@article{Nie2008SemidefiniteRO,
title={Semidefinite Representation of the k-Ellipse},
author={Jiawang Nie and Pablo A. Parrilo and Bernd Sturmfels},
journal={arXiv: Algebraic Geometry},
year={2008},
pages={117-132}
}
• Published 31 January 2007
• Mathematics
• arXiv: Algebraic Geometry
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…
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