Fibers of Multi-Graded Rational Maps and Orthogonal Projection onto Rational Surfaces

We contribute a new algebraic method for computing the orthogonal projections of a point onto a rational algebraic surface embedded in the three-dimensional projective space. This problem is first turned into the computation of the finite fibers of a generically finite dominant rational map: a congruence of normal lines to the rational surface. Then, an in-depth study of certain syzygy modules associated to such a congruence is presented and applied to build elimination matrices that provide universal representations of its finite fibers, under some genericity assumptions. These matrices depend linearly in the variables of the-three dimensional space. They can be precomputed so that the orthogonal projections of points are approximately computed by means of fast and robust numerical linear algebra calculations.