Focal Surfaces of Discrete Geometry
Eurographic Symposium on Geometry Processing
Jingyi Yu, Xiaotian Yin, Xianfeng Gu, Leonard McMillan, and Steven Gortler
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 estimate focal surfaces of a triangle mesh robustly, 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.