Focal surfaces of discrete geometry

DSpace/Manakin Repository

Focal surfaces of discrete geometry

Citable link to this page


Title: Focal surfaces of discrete geometry
Author: McMillan, Leonard; Yu, Jingyi; Yin, Xiaotian; Gu, Xianfeng; Gortler, Steven

Note: Order does not necessarily reflect citation order of authors.

Citation: Yu, Jingyi, Xiaotian Yin, Xianfeng Gu, Leonard McMillan, and Steven J. Gortler. 2007. Focal surfaces of discrete geometry. In Proceedings of the fifth eurographics symposium on geometry processing, July 4-6, 2007, Barcelona, Spain, ed. ACM Symposium on Geometry Processing, Alexander Belyaev, and Michael Garland, ed. ACM Symposium on Geometry Processing, Alexander Belyaev, and Michael Garland, 23-32. ACM conference proceedings series, vol. 257. Aire-la-ville, Switzerland: Eurographics Association.
Full Text & Related Files:
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-surface-based 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.
Published Version:
Other Sources:
Terms of Use: This article is made available under the terms and conditions applicable to Other Posted Material, as set forth at
Citable link to this page:
Downloads of this work:

Show full Dublin Core record

This item appears in the following Collection(s)


Search DASH

Advanced Search