TY - JOUR
T1 - The distance function from a real algebraic variety
AU - Ottaviani, Giorgio
AU - Sodomaco, Luca
PY - 2020/10
Y1 - 2020/10
N2 - 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)∨.
AB - 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)∨.
KW - Euclidean distance
KW - Euclidean Distance degree
KW - Euclidean Distance polynomial
KW - Isotropic quadric
KW - Polar degrees
KW - Real algebraic variety
UR - http://www.scopus.com/inward/record.url?scp=85089890037&partnerID=8YFLogxK
U2 - 10.1016/j.cagd.2020.101927
DO - 10.1016/j.cagd.2020.101927
M3 - Article
AN - SCOPUS:85089890037
SN - 0167-8396
VL - 82
JO - COMPUTER AIDED GEOMETRIC DESIGN
JF - COMPUTER AIDED GEOMETRIC DESIGN
M1 - 101927
ER -