Geometric methods in engineering applications

Xianfeng Gu, Yalin Wang, Hsiao Bing Cheng, Li Tien Cheng, Shing Tung Yau

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Abstract

In this work, we introduce two sets of algorithms inspired by the ideas from modern geometry. One is computational conformal geometry method, including harmonic maps, holomorphic 1-forms and Ricci flow. The other one is optimization method using affine normals. In the first part, we focus on conformal geometry. Conformal structure is a natural structure of metric surfaces. The concepts and methods from conformal geometry play important roles for real applications in scientific computing, computer graphics, computer vision and medical imaging fields. This work systematically introduces the concepts, methods for numerically computing conformal structures inspired by conformal geometry. The algorithms are theoretically rigorous and practically efficient. We demonstrate the algorithms by real applications, such as surface matching, global conformal parameterization, conformal brain mapping etc. In the second part, we consider minimization of a real-valued function f over Rn+1 and study the choice of the affine normal of the level set hypersurfaces of f as a direction for minimization. The affine normal vector arises in affine differential geometry when answering the question of what hypersurfaces are invariant under unimodular affine transformations. It can be computed at points of a hypersurface from local geometry or, in an alternative description, centers of gravity of slices. In the case where f is quadratic, the line passing through any chosen point parallel to its affine normal will pass through the critical point of f. We study numerical techniques for calculating affine normal directions of level set surfaces of convex f for minimization algorithms.

Original languageEnglish (US)
Title of host publicationMathematics and Computation, a Contemporary View: The Abel Symposium 2006 - Proceedings of the 3rd Abel Symposium
Pages1-19
Number of pages19
DOIs
StatePublished - 2008
Externally publishedYes
Event3rd Abel Symposium 2006: Mathematics and Computation, a Contemporary View - Alesund, Norway
Duration: May 25 2006May 27 2006

Other

Other3rd Abel Symposium 2006: Mathematics and Computation, a Contemporary View
CountryNorway
CityAlesund
Period5/25/065/27/06

Fingerprint

Conformal Geometry
Engineering Application
Geometry
Hypersurface
Conformal Structure
Level Set
Affine geometry
Ricci Flow
Centre of gravity
Normal vector
Scientific Computing
Brain mapping
Computational Geometry
Medical Imaging
Harmonic Maps
Differential Geometry
Computer graphics
Numerical Techniques
Slice
Computer Vision

ASJC Scopus subject areas

  • Computational Mathematics

Cite this

Gu, X., Wang, Y., Cheng, H. B., Cheng, L. T., & Yau, S. T. (2008). Geometric methods in engineering applications. In Mathematics and Computation, a Contemporary View: The Abel Symposium 2006 - Proceedings of the 3rd Abel Symposium (pp. 1-19) https://doi.org/10.1007/978-3-540-68850-1_1

Geometric methods in engineering applications. / Gu, Xianfeng; Wang, Yalin; Cheng, Hsiao Bing; Cheng, Li Tien; Yau, Shing Tung.

Mathematics and Computation, a Contemporary View: The Abel Symposium 2006 - Proceedings of the 3rd Abel Symposium. 2008. p. 1-19.

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Gu, X, Wang, Y, Cheng, HB, Cheng, LT & Yau, ST 2008, Geometric methods in engineering applications. in Mathematics and Computation, a Contemporary View: The Abel Symposium 2006 - Proceedings of the 3rd Abel Symposium. pp. 1-19, 3rd Abel Symposium 2006: Mathematics and Computation, a Contemporary View, Alesund, Norway, 5/25/06. https://doi.org/10.1007/978-3-540-68850-1_1
Gu X, Wang Y, Cheng HB, Cheng LT, Yau ST. Geometric methods in engineering applications. In Mathematics and Computation, a Contemporary View: The Abel Symposium 2006 - Proceedings of the 3rd Abel Symposium. 2008. p. 1-19 https://doi.org/10.1007/978-3-540-68850-1_1
Gu, Xianfeng ; Wang, Yalin ; Cheng, Hsiao Bing ; Cheng, Li Tien ; Yau, Shing Tung. / Geometric methods in engineering applications. Mathematics and Computation, a Contemporary View: The Abel Symposium 2006 - Proceedings of the 3rd Abel Symposium. 2008. pp. 1-19
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