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Abstract

Barycentric coordinates are commonly used to represent a point inside a polygon as an affine combination of the polygon's vertices and to interpolate data given at these vertices. While unique for triangles, various generalizations to arbitrary simple polygons exist, each satisfying a different set of properties. Some of these generalized barycentric coordinates do not have a closed form and can only be approximated by piecewise linear functions. In this paper we show that subdivision can be used to refine these piecewise linear functions without losing the key barycentric properties. For a wide range of subdivision schemes, this generates a sequence of piecewise linear coordinates which converges to non-negative and C1 continuous coordinates in the limit. The power of the described approach comes from the possibility of evaluating the C1 limit coordinates and their derivatives directly. We support our theoretical results with several examples, where we use Loop or Catmull–Clark subdivision to generate C1 coordinates, which inherit the favourable shape properties of harmonic coordinates or the small support of local barycentric coordinates.

5. Conclusions

Mesh subdivision is widely known in computer graphics as a technique for creating smooth surfaces with arbitrary topology by repeatedly refining an initial base mesh with simple local rules. In this paper we show that subdivision can also be used to construct barycentric coordinates with favourable properties. While the theory developed in Section 2 is general and works for a large class of subdivision schemes, we believe that Loop subdivision is the method of choice, for two reasons. On the one hand, it is simple and comes with well-understood boundary rules and exact evaluation routines. On the other hand, our examples confirm that the main shape of the limit coordinate functions b∞ i is dictated by the initial functions b0 i , and we do not expect other subdivision schemes to yield qualitatively better results. However, it still remains future work to develop a strategy for constructing initial triangulations T0, for which it can be formally proven that the refined triangulations Tk are regular in the interior, even in the limit. Note that this problem is not restricted to the construction of well-defined Loop coordinates, as it addresses the general question under which conditions the two-dimensional Loop mapping v: → is bijective. Another direction for future work is the extension of our approach to 3D by using volumetric subdivision schemes (Chang et al., 2002; Schaefer et al., 2004).