Geometric elements and classification of quadrics in rational Bézier form

• Closed formulas for geometric elements of quadrics in rational Bézier form in terms of their weights and control points, using algebraic projective geometry.
• The results are derived for Bézier triangles, but are applicable to tensor product patches.
• Closed, coordinate-free formulas for implicit equations for quadrics in terms of their weights and control points.
• Affine classification of quadrics in Bézier form.

In this paper we classify and derive closed formulas for geometric elements of quadrics in rational Bézier 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 characterized within this framework.