A Perturbation Approach to Inverting Projective Degenerate Quadrics and Recovering Lost Geometric Information

This article presents a general perturbation approach to retrieving geometric information about the duals of degenerate quadrics obscured by the singular matrices that represent them. By embedding singular quadrics within a one-parameter family of neighbouring non-degenerate quadrics, and then applying standard techniques from analytic perturbation theory, we rigorously isolate the finite remainder from the singular pole residue to extract well-defined pseudo-distance metrics and polarity maps, and reveal how the degenerate limit of this family of non-degenerate quadrics retains an ``asymptotic memory'' of the directions of the perturbation matrices.