The distance function from a real algebraic variety

For any (real) algebraic variety X in a Euclidean space V endowed with a nondegenerate quadratic form q, we introduce a polynomial {EDpoly}_{X,u}(t^2) which, for any uin V, has among its roots the distance from u to X. The degree of EDpoly}_{X,u} is the Euclidean Distance degree of X. We prove a duality property when X is a projective variety, namely EDpoly}_{X,u}(t^2)=EDpoly}_{X^ee,u}(q(u)-t^2) where X^ee 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 Xcup(X^eecap Q)^ee.