27 variants of Tutte's theorem for plane near-triangulations and an application to periodic spline surface fitting

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.