Implicit equations of non-degenerate rational Bezier quadric triangles

TL;DR

This paper derives explicit implicit equations for non-degenerate rational Bezier quadric triangles using projective geometry, control net, and weights, providing a coordinate-free approach for Steiner surfaces of degree two.

Contribution

It introduces a novel coordinate-free method to obtain implicit equations of rational Bezier quadric triangles based on control net and weights.

Findings

Derived bilinear forms for quadrics in a coordinate-free manner

Utilized projective geometry and control net for construction

Provided explicit implicit equations for non-degenerate quadrics

Abstract

In this paper we review the derivation of implicit equations for non-degenerate quadric patches in rational Bezier triangular form. These are the case of Steiner surfaces of degree two. We derive the bilinear forms for such quadrics in a coordinate-free fashion in terms of their control net and their list of weights in a suitable form. Our construction relies on projective geometry and is grounded on the pencil of quadrics circumscribed to a tetrahedron formed by vertices of the control net and an additional point which is required for the Steiner surface to be a non-degenerate quadric.