# Fibers of multi-graded rational maps and orthogonal projection onto rational surfaces

## TL;DR

This paper introduces an algebraic method to compute orthogonal projections onto rational surfaces in 3D space, using syzygy modules and elimination matrices for efficient numerical calculations.

## Contribution

A novel algebraic approach leveraging syzygy modules and elimination matrices to efficiently compute orthogonal projections onto rational surfaces.

## Key findings

- Method enables fast numerical projection calculations.
- Matrices depend linearly on spatial variables.
- Approach is robust under genericity assumptions.

## Abstract

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 pre-computed so that the orthogonal projections of points are approximately computed by means of fast and robust numerical linear algebra calculations.