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Vector field processing on triangle meshes

Authors:

Fernando de Goes,

Mathieu Desbrun,

Yiying TongAuthors Info & Claims

SIGGRAPH '16: ACM SIGGRAPH 2016 Courses

Article No.: 27, Pages 1 - 49

https://doi.org/10.1145/2897826.2927303

Published: 24 July 2016 Publication History

Abstract

While scalar fields on surfaces have been staples of geometry processing, the use of tangent vector fields has steadily grown in geometry processing over the last two decades: they are crucial to encode both directions and sizing on surfaces as commonly required in tasks such as texture synthesis, non-photorealistic rendering, digital grooming, and meshing. There are, however, a variety of discrete representations of tangent vector fields on triangle meshes, and each approach offers different trade-offs among simplicity, efficiency, and accuracy depending on the targeted application.

This course reviews the three main families of discretizations used to design computational tools for vector field processing on triangle meshes: face-based, edge-based, and vertex-based representations. In the process of reviewing the computational tools offered by these representations, we go over a large body of recent developments in vector field processing in the area of discrete differential geometry. We also discuss the theoretical and practical limitations of each type of discretization, and cover increasingly-common extensions such as n-direction and n-vector fields.

While the course will focus on explaining the key approaches to practical encoding (including data structures) and manipulation (including discrete operators) of finite-dimensional vector fields, important differential geometric notions will also be covered: as often in Discrete Differential Geometry, the discrete picture will be used to illustrate deep continuous concepts such as covariant derivatives, metric connections, or Bochner Laplacians.

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cover image ACM Conferences

SIGGRAPH '16: ACM SIGGRAPH 2016 Courses

July 2016

1735 pages

ISBN:9781450342896

DOI:10.1145/2897826

Copyright © 2016 Owner/Author.

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Published: 24 July 2016

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Su ZTong YWei G(2024)Hodge decomposition of vector fields in Cartesian gridsSIGGRAPH Asia 2024 Conference Papers10.1145/3680528.3687602(1-10)Online publication date: 3-Dec-2024
https://dl.acm.org/doi/10.1145/3680528.3687602
King NSu HAanjaneya MRuuth SBatty C(2024)A Closest Point Method for PDEs on Manifolds with Interior Boundary Conditions for Geometry ProcessingACM Transactions on Graphics10.1145/367365243:5(1-26)Online publication date: 9-Aug-2024
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