Multi-sided Bézier 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.…”

“…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.…”

“…For instance, it is in general impossible to find splines w that vanish on all but one of these boundary curves 6 . In [VSVS20,VVSS20], it is suggested to construct reparametrizations and weights as solutions of variational problems with the curves γ providing the boundary conditions. This requests a certain computational effort, but visually, the results appear to be excellent.…”

Constrained modeling of multi-sided patches

“…Although in some cases [3,11] the domain does not take an explicit geometric form, it is usually a 2D shape. In the beginning it was assumed to be a regular n-sided polygon [1], but later this gave place to arbitrary convex polygons [8], concave polygons [5], and even general curved domains [10].…”

“…is equal to π, i.e., when the boundaries have parallel tangents, most existing methods cannot be used (but cf. splits [10] and periodic ribbons [6]).…”