Crossing knot lines in composition of freeform B-spline geometry

Since the first publications on composition of splines (DeRose et al., 1993; Elber, 1992), it was clear that the composition, T(S), of a tensor product B-spline function S with another B-spline function T is no longer a tensor product B-spline if S crosses a knot line in the domain of T. The reason can be easily explained for the fact that the new function T(S) has a finite continuity that is governed by the knot line of T that is not, in general, an iso-parametric direction of the resulting composition T(S). In this work, we propose two approaches to circumvent this long standing difficulty and reconstruct precise representations for T(S) that are either tensor products or trimmed geometry. We focus our discussion on surface-trivariate compositions, T(S), while we present examples and results, for both reconstruction approaches, for microstructure surfaces and trivariates embedded in surface and trivariate deformation functions.