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Parameterization of Faceted Surfaces for Meshing using Angle-Based Flattening

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We propose a new method to compute planar triangulations of faceted surfaces for surface parameterization. In contrast to previous approaches that define the flattening problem as a mapping of the three-dimensional node locations to the plane, our method defines the flattening problem as a constrained optimization problem in terms of angles (only). After applying a scaling that derives from the ‘curvature’ at a node, we minimize the relative deformation of the angles in the plane with respect to their counterparts in the three-dimensional surface. This approach makes the method more stable and robust than previous approaches, which used node locations in their formulations. The new method can handle any manifold surface for which a connected, valid, two-dimensional parameterization exists, including surfaces with large curvature gradients. It does not require the boundary of the flat two-dimensional domain to be prede-fined or convex. We use only the necessary and sufficient constraints for a valid two-dimensional triangulation. As a result, the existence of a theoretical solution to the minimization procedure is guaranteed.

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Sheffer, A., de Sturler, E. Parameterization of Faceted Surfaces for Meshing using Angle-Based Flattening. EWC 17, 326–337 (2001).

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