Abstract
The theoretical basis of Floater's parameterization technique for triangulated surfaces is simultaneously a generalization (to non-barycentric weights) and a specialization (to a plane near-triangulation, which is an embedding of a planar graph with the property that all bounded faces are – possibly curved – triangles) of Tutte's Spring Embedding Theorem. Extensions of this technique cover surfaces with holes and periodic surfaces. The proofs presented previously need advanced concepts, such as rather involved results from graph theory or the theory of discrete 1-forms and consistent perturbations, or are not directly applicable to the above-mentioned extensions. We present a particularly simple geometric derivation of Tutte's theorem for plane near-triangulations and various extensions thereof, using solely the Euler formula for planar graphs. In particular, we include the case of meshes possessing a cylindrical topology – which has not yet been addressed explicitly but possesses important applications to periodic spline surface fitting – and we correct a minor inaccuracy in a previous result concerning Floater-type parameterizations for genus-1 meshes.
Highlights
In a seminal paper, Floater (1997) introduced a powerful parameterization method for triangulated surfaces patches based on graph theory
We present a simple geometric derivation of Tutte’s theorem for plane near-triangulations and various extensions thereof, using solely the Euler formula for planar graphs
We prove that an edge collapsing operations (ECOs) of a degenerate edge preserves both the convex combination mapping property and the convex boundary property
Summary
In a seminal paper, Floater (1997) introduced a powerful parameterization method for triangulated surfaces patches based on graph theory. Floater (2003) proves a specialization – which can be seen as a discrete version of the Radó-Kneser-Choquet theorem – of Tutte’s result to triangulations, i.e., to embeddings of planar graphs with the property that all bounded faces are triangles (with straight edges) He derives improvements of it, by relaxing the assumptions regarding the strict convexity of the boundary and the property of 3-connectedness. It is shown that the angle excess argument can deal with degenerate triangles too, once the degenerate edges have been ruled out via ECOs. In particular, we include the single periodic case – i.e., meshes possessing a cylindrical topology – and we correct a minor inaccuracy in a previous result (Theorem 3.3 of Gortler et al, 2006) concerning Floater-type parameterizations for meshes of genus 1, i.e., doubly periodic surfaces. The subsequent sections discuss the different variants of Tutte’s Theorem and an application to periodic spline surface fitting
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