Focal surfaces of discrete geometry

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 foc...

• Type: Conference
• Year: 2007
• Where: SGP
• Authors: Jingyi Yu, Xiaotian Yin, Xianfeng Gu, Leonard McMillan, Steven J. Gortler
• Keywords: Associated Focal Surfaces, Computer Graphics, Focal Surfaces, SGP 2007, Smooth Three-dimensional Surfaces