Computer Aided Geometric Design 28 (2011) 215–232 Contents lists available at ScienceDirect Computer Aided Geometric Design www.elsevier.com/locate/cagd Ridge extraction of a smooth 2-manifold surface based on vector field ✩ WuJun Che a,∗ , XiaoPeng Zhang a , Yi-Kuan Zhang a , Jean-Claud Paul b,c , Bo Xu a a LIAMA-NLPR-Digital Content Technology Research Center, Institute of Automation, CAS, Beijing 100190, PR China b School of Software, Tsinghua University, Beijing 100084, PR China c INRIA, France article info abstract Article history: Received 20 August 2009 Received in revised form 28 February 2011 Accepted 19 March 2011 Available online 2 April 2011 Keywords: Implicit surface Invariant feature Line of curvature Parametric surface Principal curvature Principal direction Ridge Umbilical point Vector field This paper presents a general scheme to compute ridges on a smooth 2-manifold surface from the standpoint of a vector field. A ridge field is introduced. Starting with an initial ridge, which may or may not be umbilical, a ridge line is then traced by calculating an associated integral curve of this field in conjunction with a new projection procedure to prevent it from diverging. This projection is the first that can optimize a ridge guess to lie on a ridge line uniquely and accurately. In order to follow this scheme, we not only develop practical ridge formulae but also address their corresponding computational procedures for an analytical surface patch, especially for an implicit surface. In contrast to other existing methods, our new approach is mathematically sound and characterized by considering the full geometric structures and topological patterns of ridges on a generic smooth surface. The resulting ridges are accurate in the numerical sense and meet the requirement of high accuracy with complete topology. Although the objective of this paper is to develop a mathematically sound framework for ridges on a smooth surface, we give a comprehensive review of relevant works on both meshes and smooth surfaces for readers.  2011 Elsevier B.V. All rights reserved. 1. Introduction Due to quickly developing computer technology and the extensive availability of 3D models, time-consuming computer- aided analysis of high-order differential geometry becomes more and more doable, increasing interest in techniques of 3D shape analysis in recent years. A set of geometric features consists of points describing a non-trivial region of a local surface with prescribed geometric properties, providing a solid foundation for shape analysis, design and recognition. Significant features usually involve high-order surface derivatives such that it is a difficult task to extract them precisely and robustly. Therefore, feature analysis and detection is currently a subject of intensive research in which invariant features play an important role. 1.1. Ridge Koenderink (1990) recognized that ridges are significant features of a surface shape, conveying most of the shape’s virtually essential characteristics. Several definitions for ridges in image and shape analysis can also be found in the lit- erature (Eberly et al., 1994; Porteous, 2001). A generally acceptable definition is via principal curvature extrema along ✩ This paper has been recommended for acceptance by G.E. Farin. * Corresponding author. E-mail addresses: chewj@liama.ia.ac.cn (W.J. Che), xpzhang@nlpr.ia.ac.cn (X.P. Zhang), y-k.zhang@163.com (Y.-K. Zhang), paul@tsinghua.edu.cn (J.-C. Paul), xubo@hitic.ia.ac.cn (B. Xu). 0167-8396/$ – see front matter  2011 Elsevier B.V. All rights reserved. doi:10.1016/j.cagd.2011.03.005