Focal Surfaces of Discrete Geometry
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Date
2007
Journal Title
Journal ISSN
Volume Title
Publisher
The Eurographics Association
Abstract
The differential geometry of smooth three-dimensional surfaces can be interpreted from one of two perspectives: in terms of oriented frames located on the surface, or in terms of a pair of associated focal surfaces. These focal surfaces are swept by the loci of the principal curvatures radii. In this article, we develop a focal-surfacebased differential geometry interpretation for discrete mesh surfaces. Focal surfaces have many useful properties. For instance, the normal of each focal surface indicates a principal direction of the corresponding point on the original surface. We provide algorithms to robustly approximate the focal surfaces of a triangle mesh with known or estimated normals. Our approach locally parameterizes the surface normals about a point by their intersections with a pair of parallel planes.We show neighboring normal triplets are constrained to pass simultaneously through two slits, which are parallel to the specified parametrization planes and rule the focal surfaces. We develop both CPU and GPU-based algorithms to efficiently approximate these two slits and, hence, the focal meshes. Our focal mesh estimation also provides a novel discrete shape operator that simultaneously estimates the principal curvatures and principal directions.
Description
@inproceedings{10.2312:SGP/SGP07/023-032,
booktitle = {Geometry Processing},
editor = {Alexander Belyaev and Michael Garland},
title = {{Focal Surfaces of Discrete Geometry}},
author = {Yu, Jingyi and Yin, Xiaotian and Gu, Xianfeng and McMillan, Leonard and Gortler, Steven},
year = {2007},
publisher = {The Eurographics Association},
ISSN = {1727-8384},
ISBN = {978-3-905673-46-3},
DOI = {10.2312/SGP/SGP07/023-032}
}