Discrete Curvature Theories and Applications

Handle URI:
http://hdl.handle.net/10754/618995
Title:
Discrete Curvature Theories and Applications
Authors:
Sun, Xiang ( 0000-0003-0242-0319 )
Abstract:
Discrete Di erential Geometry (DDG) concerns discrete counterparts of notions and methods in di erential geometry. This thesis deals with a core subject in DDG, discrete curvature theories on various types of polyhedral surfaces that are practically important for free-form architecture, sunlight-redirecting shading systems, and face recognition. Modeled as polyhedral surfaces, the shapes of free-form structures may have to satisfy di erent geometric or physical constraints. We study a combination of geometry and physics { the discrete surfaces that can stand on their own, as well as having proper shapes for the manufacture. These proper shapes, known as circular and conical meshes, are closely related to discrete principal curvatures. We study curvature theories that make such surfaces possible. Shading systems of freeform building skins are new types of energy-saving structures that can re-direct the sunlight. From these systems, discrete line congruences across polyhedral surfaces can be abstracted. We develop a new curvature theory for polyhedral surfaces equipped with normal congruences { a particular type of congruences de ned by linear interpolation of vertex normals. The main results are a discussion of various de nitions of normality, a detailed study of the geometry of such congruences, and a concept of curvatures and shape operators associated with the faces of a triangle mesh. These curvatures are compatible with both normal congruences and the Steiner formula. In addition to architecture, we consider the role of discrete curvatures in face recognition. We use geometric measure theory to introduce the notion of asymptotic cones associated with a singular subspace of a Riemannian manifold, which is an extension of the classical notion of asymptotic directions. We get a simple expression of these cones for polyhedral surfaces, as well as convergence and approximation theorems. We use the asymptotic cones as facial descriptors and demonstrate the practicability and accuracy of their applications in face recognition
Advisors:
Pottmann, Helmut ( 0000-0002-3195-9316 )
Committee Member:
Wonka, Peter ( 0000-0003-0627-9746 ) ; Wu, Ying ( 0000-0002-7919-1107 ) ; Bobenko, Alexander Ivanovich
KAUST Department:
Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division
Program:
Applied Mathematics and Computational Science
Issue Date:
25-Aug-2016
Type:
Dissertation
Appears in Collections:
Dissertations

Full metadata record

DC FieldValue Language
dc.contributor.advisorPottmann, Helmuten
dc.contributor.authorSun, Xiangen
dc.date.accessioned2016-08-29T05:41:05Z-
dc.date.available2016-08-29T05:41:05Z-
dc.date.issued2016-08-25-
dc.identifier.urihttp://hdl.handle.net/10754/618995-
dc.description.abstractDiscrete Di erential Geometry (DDG) concerns discrete counterparts of notions and methods in di erential geometry. This thesis deals with a core subject in DDG, discrete curvature theories on various types of polyhedral surfaces that are practically important for free-form architecture, sunlight-redirecting shading systems, and face recognition. Modeled as polyhedral surfaces, the shapes of free-form structures may have to satisfy di erent geometric or physical constraints. We study a combination of geometry and physics { the discrete surfaces that can stand on their own, as well as having proper shapes for the manufacture. These proper shapes, known as circular and conical meshes, are closely related to discrete principal curvatures. We study curvature theories that make such surfaces possible. Shading systems of freeform building skins are new types of energy-saving structures that can re-direct the sunlight. From these systems, discrete line congruences across polyhedral surfaces can be abstracted. We develop a new curvature theory for polyhedral surfaces equipped with normal congruences { a particular type of congruences de ned by linear interpolation of vertex normals. The main results are a discussion of various de nitions of normality, a detailed study of the geometry of such congruences, and a concept of curvatures and shape operators associated with the faces of a triangle mesh. These curvatures are compatible with both normal congruences and the Steiner formula. In addition to architecture, we consider the role of discrete curvatures in face recognition. We use geometric measure theory to introduce the notion of asymptotic cones associated with a singular subspace of a Riemannian manifold, which is an extension of the classical notion of asymptotic directions. We get a simple expression of these cones for polyhedral surfaces, as well as convergence and approximation theorems. We use the asymptotic cones as facial descriptors and demonstrate the practicability and accuracy of their applications in face recognitionen
dc.language.isoenen
dc.subjectDiscrete differential geometryen
dc.subjectcurvaturesen
dc.subjectArchitectural geometryen
dc.subjectline congruencesen
dc.subjectnormal cyclesen
dc.titleDiscrete Curvature Theories and Applicationsen
dc.typeDissertationen
dc.contributor.departmentComputer, Electrical and Mathematical Sciences and Engineering (CEMSE) Divisionen
thesis.degree.grantorKing Abdullah University of Science and Technologyen_GB
dc.contributor.committeememberWonka, Peteren
dc.contributor.committeememberWu, Yingen
dc.contributor.committeememberBobenko, Alexander Ivanovichen
thesis.degree.disciplineApplied Mathematics and Computational Scienceen
thesis.degree.nameDoctor of Philosophyen
dc.person.id113351en
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