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“N-Symmetry Direction Fields on Surfaces of Arbitrary Genus”

Nicolas Ray, Bruno Vallet, Wan-Chiu Li and Bruno Lévy

Tech report, 2006

Abstract: A wide class of 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. These two direction fields can be smoothed in fundamentally different ways, according to the definition of continuity along an elementary displacement on the surface: inverting the directions may be allowed or not, and 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 smoothing algorithms are limited to use non-optimum local relaxation procedures. In this paper, we introduce 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 derive a N-symmetry generalization of the Poincaré-Hopf theorem. Based on this theorem, we explain how to represent discrete N-symmetry direction fields on meshes. We use this representation to derive a highly efficient algorithm to smooth the field with constrained singularities and directions.

## BibTex reference

@TECHREPORT{ray:NSDF:2006,
AUTHOR = "Nicolas Ray and Bruno Vallet and Wan-Chiu Li and Bruno Lévy",
TITLE = "N-Symmetry Direction Fields on Surfaces of Arbitrary Genus",
INSTITUTION = "INRIA - ALICE",
YEAR = "2006",
}

## Supplemental material, links, hindsight ...

This tech-report was published later as an
article in ACM Tranasctions on Graphics and presented at ACM SIGGRAPH.