Nurbs Approximation of A-Splines and A-Patches

Given A spline curves and A patch surfaces that are implicitly de ned on triangles and tetrahedra we determine their NURBS representations We provide a trimmed NURBS form for A spline curves and a parametric tensor product NURBS form for A patch surfaces We concentrate on cubic A patches providing a C continuous surface that interpolates a given triangulation together with surface normals at the vertices In many cases we can generate cubic trimming curves that are rationally parametrizable on the triangular faces of the tetrahedra for the remaining faces we resort to using quadratic curves which are always rationally parametrizable to approximate the cubic trimming curves Introduction Low degree polynomial or algebraic surfaces can often have dual parametric and implicit representations Each form has its distinct advantages The parametric polynomial spline in B spline Basis B and Bernstein B ezier BB bases are cur rently overwhelmingly popular in commercial and industrial CAGD systems In this paper we show how to generate trimmed parametric B spline and BB spline representations for a collection of implicitly de ned algebraic surface patches intro duced in Refs Each implicit algebraic surface patch A patch is a smooth Research supported in part by NSF grants CCR KDI DMS and ACI yProject supported by NSFC bounded zero contour of a trivariate polynomial de ned within a tetrahedron for the barycentric B BB basis and within a box for the tensor product B BB basis see Ref Chap for details We also show how to convert the trimming curves of the input A patch collection into rational parametric form in the same basis as the surface conversion yielding standard trimmed NURBS representations As NURBS representations are e cient to compute and are a very common standard form for splines being able to represent A patches as NURBS is highly desirable Many low degree implicit curves or surfaces are rational i e convertible into rational parametric form All degree two curves conics are rational but only the subset of singular degree three cubics are rational i e elliptic cubics are non singular and not rationally parametrizable In general a necessary and su cient condition for the global rationality of an algebraic curve of arbitrary degree is given by the Cayley Reimann criterion a curve is rational if and only if g where g the genus of the curve is a measure of the de ciency of the curves singularities from its maximum allowable limit For surfaces all implicit quadratic and cubic surfaces can be rationally parametrizable except the elliptic cubic cylinders or cones A method for rationally parametrizing general quadratic curves and surfaces is given in Refs and These are all we need to rationally parametrize C quadratic A patches Similarly techniques for parametrizing rational cubic curves and surfaces have previously been given in Refs A proper subset of higher degree surface can be rationally parametrized with a necessary and su cient criterion given due to Castelnuovo Since it is not always possible to perform exact conversions to rational parametric form we appeal to approximate conversions when necessary However we preserve the continuity of the spline surface to be converted that is we construct trimmed NURBS representations of C cubic A splines and C cubic A patches The rest of the paper is organized as follows Section discusses the conversion of A splines curves which are also the boundary trimming curves of A patches given in implicit form to NURBS representation In Section we rst classify the cases of exact convertibility of C cubic A patch splines into trimmed NURBS form When exact convertibility is not possible we show how to generate fair approximate trimmed NURBS Section concludes the paper Details of the derivations and examples are presented in the Appendices NURBS Representation of A splines An A spline of degree n over the triangle p p p is de ned by