Multi-sided B-spline surfaces over curved, multi-connected domains

Trimmed Spline Surfaces with Accurate Boundary Control

“…[MR]) and also general fairness aspects are still waiting for a detailed assessment and analysis. Another line of research that has to be discussed in this context is documented in [VSVS20,VVSS20]. It is based on ideas similar those presented here, but there are also significant differences.…”

“…The basic principle and some technical details of ABC-surfaces were first presented in [Rei]. Roughly at the same time and independent of our research, another group developed similar ideas [VVSS20], which were recently brought to our attention. The formal similarities are obvious, but there are also differences, and one of them is significant.…”

“…First, the weighted base wb does not appear in that form 5 in [VVSS20]. Instead, a linear combination j b j p j of control points p j ∈ R 3 and Bsplines b j is used, where only B-splines with support completely contained in the domain of the surface are permitted so that there is no influence on the shape near the boundary.…”

“…We think of the base as a relatively accurate description of the entire surface, and of the ribbons as small, local correctives, used only to adapt b to the desired boundary behavior. By contrast, the ribbons in [VVSS20] are the major building blocks for the shape to be modeled. The interior terms b j p j , if included at all, are only used for variations inside if the shape defined by the ribbons alone does not yet meet expectations.…”

“…Second, and this difference is much more important, the surfaces constructed in [VSVS20,VVSS20] are transfinite in the sense that they do not possess a closed-form representation by means of elementary functions. In particular, these surfaces are not piecewise polynomial or rational and thus cannot be stored in standard file formats, as requested in many applications.…”