  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.
