Umbilics and lines of curvature for shape interrogation

authored by: T. Maekawa, F. E. Wolter, N. M. Patrikalakis
Abstract: This paper describes a method to extract the generic features of free-form parametric surfaces for shape interrogation. The umbilical points, which are the singular points of the orthogonal net of lines of curvature, have generic features and may act like fingerprints for shape recognition. We investigate the generic features of the umbilics and behavior of lines of curvature which pass through an umbilic on a parametric free-form surface. Our method is based on a coordinate transformation to set the parametric surface in Monge form and on a Taylor expansion to compute the angles of the tangent lines to the lines of curvatures at an umbilic. We also develop a novel and practical criterion which assures the existence of local extrema of principal curvature functions at umbilical points. Finally, numerical experiments illustrate how the generic features of the umbilics can be applied for surface recognition.
Organisation(s): Institute of Applied Mathematics
External Organisation(s): Massachusetts Institute of Technology
Type: Article
Journal: Computer aided geometric design
Volume: 13
Pages: 133-161
No. of pages: 29
ISSN: 0167-8396
Publication date: 03.1996
Publication status: Published
Peer reviewed: Yes
ASJC Scopus subject areas: Modelling and Simulation, Automotive Engineering, Aerospace Engineering, Computer Graphics and Computer-Aided Design
Electronic version(s): https://doi.org/10.1016/0167-8396(95)00018-6 (Access: Closed)

BibTeX

@article{510d822fa8f941de8efbff01db63c4de,
title = "Umbilics and lines of curvature for shape interrogation",
abstract = "This paper describes a method to extract the generic features of free-form parametric surfaces for shape interrogation. The umbilical points, which are the singular points of the orthogonal net of lines of curvature, have generic features and may act like fingerprints for shape recognition. We investigate the generic features of the umbilics and behavior of lines of curvature which pass through an umbilic on a parametric free-form surface. Our method is based on a coordinate transformation to set the parametric surface in Monge form and on a Taylor expansion to compute the angles of the tangent lines to the lines of curvatures at an umbilic. We also develop a novel and practical criterion which assures the existence of local extrema of principal curvature functions at umbilical points. Finally, numerical experiments illustrate how the generic features of the umbilics can be applied for surface recognition.",
keywords = "CAD, CAGD, Lines of curvature, Monge form, Perturbation, Shape recognition, Umbilics, Vision",
author = "T. Maekawa and Wolter, {F. E.} and Patrikalakis, {N. M.}",
year = "1996",
month = mar,
doi = "10.1016/0167-8396(95)00018-6",
language = "English",
volume = "13",
pages = "133--161",
journal = "Computer aided geometric design",
issn = "0167-8396",
publisher = "Elsevier",
number = "2",
}