Geometric elements and classification of quadrics in rational B\'ezier form

TL;DR

This paper classifies and derives formulas for geometric elements of quadrics in rational Bézier form using control vertices and weights, extending to tensor product patches with projective geometry techniques.

Contribution

It introduces a coordinate-free method to analyze quadrics in rational Bézier form, enabling classification and geometric element computation from control points and weights.

Findings

Formulas for geometric elements like center and axes derived from control data.

Method distinguishes oval and ruled quadrics using a key coefficient.

Characterizes spheres and quadrics of revolution within the framework.

Abstract

In this paper we classify and derive closed formulas for geometric elements of quadrics in rational B\'ezier triangular form (such as the center, the conic at infinity, the vertex and the axis of paraboloids and the principal planes), using just the control vertices and the weights for the quadric patch. The results are extended also to quadric tensor product patches. Our results rely on using techniques from projective algebraic geometry to find suitable bilinear forms for the quadric in a coordinate-free fashion, considering a pencil of quadrics that are tangent to the given quadric along a conic. Most of the information about the quadric is encoded in one coefficient, involving the weights of the patch, which allows us to tell apart oval from ruled quadrics. This coefficient is also relevant to determine the affine type of the quadric. Spheres and quadrics of revolution are characterised within this framework.