The distance function from a real algebraic variety

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Journal Title
Journal ISSN
Volume Title
A1 Alkuperäisartikkeli tieteellisessä aikakauslehdessä
Date
2020-10
Major/Subject
Mcode
Degree programme
Language
en
Pages
Series
COMPUTER AIDED GEOMETRIC DESIGN, Volume 82
Abstract
For any (real) algebraic variety X in a Euclidean space V endowed with a nondegenerate quadratic form q, we introduce a polynomial EDpolyX,u(t2) which, for any u∈V, has among its roots the distance from u to X. The degree of EDpolyX,u is the Euclidean Distance degree of X. We prove a duality property when X is a projective variety, namely EDpolyX,u(t2)=EDpolyX∨,u(q(u)−t2) where X∨ is the dual variety of X. When X is transversal to the isotropic quadric Q, we prove that the ED polynomial of X is monic and the zero locus of its lower term is X∪(X∨∩Q)∨.
Description
Keywords
Euclidean distance, Euclidean Distance degree, Euclidean Distance polynomial, Isotropic quadric, Polar degrees, Real algebraic variety
Other note
Citation
Ottaviani , G & Sodomaco , L 2020 , ' The distance function from a real algebraic variety ' , Computer Aided Geometric Design , vol. 82 , 101927 . https://doi.org/10.1016/j.cagd.2020.101927