BIRS Workshop Lecture Videos
Similarity maps and splines on surfaces Zorin, Denis
A variety of techniques were proposed to model smooth surfaces of arbitrary topology based on tensor product splines (e.g. subdivision surfaces, free-form splines, T-splines). Conversion of an input surface into such a representation is commonly achieved by constructing a global seamless parametrization, possibly aligned to a guiding cross-field and using this parametrization as a domain to construct the spline-based surface. (Informally, seamless parametrizations can be thought of as paramezations of surfaces cut to disks, with isoparametric line directions and spacing on the surface matching perfectly across the cuts). One major fundamental difficulty in designing robust algorithms for this task is the fact that for common types, e.g. subdivision surfaces (requiring a conforming domain mesh) or T-spline surfaces reliably obtaining a suitable parametrization that has the same topological structure (matching singularities and more generally rotations of parametric line directions along loops matching that of the cross-field) as the guiding field poses a major challenge. Even worse, not all fields do admit suitable parametrizations, and no concise conditions are known as to which fields do. I will discuss our recent work that addresses the problem by introducing two new concepts: (1) seamless similarity maps -- a relaxation of the seamless parametrization idea, allowing scale jumps across cuts (2) splines with half-edge knots, that relax the global knot interval consistency requirements on surfaces with nontrivial genus. It turns out, that for any given guiding field structure, a compatible parametrization of this kind exists and can be computed by a relatively simple algorithm; at the same time, for any such parametrization, a smooth piecewise rational surface with exactly the same structure as the input field can be constructed from it. This leads to fully automatic construction of high-order approximations of arbitrary surfaces, even with hiighly complex topology, potentially enabling, e.g., robust automatic conversion of surfaces to isogeometric form.
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