N-Symmetry Direction Field Design

Nicolas Ray 1 Bruno Vallet 1 Wan-Chiu Li 1 Bruno Lévy 1
1 ALICE - Geometry and Lighting
INRIA Lorraine, LORIA - Laboratoire Lorrain de Recherche en Informatique et ses Applications
Abstract : Many algorithms in computer graphics and geometry processing use two orthogonal smooth direction fields (unit tangent vector fields) defined over a surface. For instance, these direction fields are used in texture synthesis, in geometry processing or in non-photorealistic rendering to distribute and orient elements on the surface. Such direction fields can be designed in fundamentally different ways, according to the symmetry requested: inverting a direction or swapping two directions may be allowed or not. Despite the advances realized in the last few years in the domain of geometry processing, a unified formalism is still lacking for the mathematical object that characterizes these generalized direction fields. As a consequence, existing direction field design algorithms are limited to use non-optimum local relaxation procedures. In this paper, we formalize N-symmetry direction fields, a generalization of classical direction fields. We give a new definition of their singularities to explain how they relate with the topology of the surface. Namely, we provide an accessible demonstration of the Poincare-Hopf theorem in the case of N-symmetry direction fields on 2-manifolds. Based on this theorem, we explain how to control the topology of N-symmetry direction fields on meshes. We demonstrate the validity and robustness of this formalism by deriving a highly efficient algorithm to design a smooth field interpolating user defined singularities and directions.
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Submitted on : Monday, October 20, 2008 - 9:26:05 AM
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Nicolas Ray, Bruno Vallet, Wan-Chiu Li, Bruno Lévy. N-Symmetry Direction Field Design. ACM Transactions on Graphics, Association for Computing Machinery, 2008, 27 (2), pp.Article 10. ⟨10.1145/1356682.1356683⟩. ⟨inria-00331900⟩

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