Geometry prediction for high degree polygons
Martin Isenburg, Ioannis P. Ivrissimtzis, Stefan Gumhold, Hans‐Peter Seidel · 2005
The parallelogram rule is a simple, yet effective scheme to predict the position of a vertex from a neighboring triangle. It was introduced by Touma and Gotsman [1998] to compress the vertex positions of triangular meshes. Later, Isenburg and Alliez [2002] showed that this rule is especially efficient for quad-dominant polygon meshes when applied "within" rather than across polygons. However, for hexagon-dominant meshes the parallelogram rule systematically performs miss-predictions.In this paper we present a generalization of the parallelogram rule to higher degree polygons. We compute a Fourier decomposition for polygons of different degrees and assume the highest frequencies to be zero for predicting missing points around the polygon. In retrospect, this theory also validates the parallelogram rule for quadrilateral surface mesh elements, as well as the Lorenzo predictor [Ibarria et al. 2003] for hexahedral volume mesh elements.