Geometric elements and classification of quadrics in rational Bezier form

In this paper we classify and derive closed formulas for geometric elements of quadrics in rational Bezier triangular form (such as the center, the tonic 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 characterized within this framework. (C) 2016 Elsevier B.V. All rights reserved.