Mathematics of Surfaces XII: 12th IMA International Conference, Sheffield, UK, September 4-6, 2007, Proceedings (Lecture Notes in Computer Science, 4647)

This book constitutes the refereed proceedings of the 12th IMA International Conference on the Mathematics of Surfaces, held in Sheffield, UK in September 2007. The papers cover a range of ideas from underlying theoretical tools to industrial uses of surfaces. Research is reported on theoretical aspects of surfaces as well as more practical topics. Title Page Preface Organization Table of Contents Regularity Criteria for the Topology of Algebraic Curves and Surfaces Introduction Polynomial Equations Bernstein Basis Representation Univariate Subdivision Solver Multivariate Bernstein Subdivision Solver Example Planar Curves Regularity Criterion for Planar Curves Tests of Regularity Algorithm of Subdivision Space Curves Regularity Criterion for Curves in 3D Tests of Regularity Algorithm of Connection Algorithm of Subdivision Surfaces Regularity Criterion for Surfaces The Polar Variety The Singular Locus of the Polar Curve Conclusion References Quadrangle Surface Tiling Through Contouring Introduction Previous Work Local Quadrangulation as Contouring Towards Global Contouring Enforcing Singular Continuity of Forms Properties of Singular Continuity Results Conclusion References Surfaces with Piecewise Linear Support Functions over Spherical Triangulations Introduction Shell Structures for Architectural Design Shell Structures Faceted Glass Shell Support Function and Dual Surface The Polarity with Respect to the Unit Sphere The Dual Surface Support Functions of Surfaces Piecewise Linear Support Functions Piecewise Linear Functions on Spherical Triangulations The Dual Mesh Quasi–Convex Polygons The Primal Mesh Examples TangentMeshes Asymptotic Behavior of the Vertices of the Primal Mesh Asymptotic Behavior of the Facets of the Dual Mesh Conclusion References A Developable Surface of Uniformly Negative Internal Angle Deficit Introduction Definitions Cycles in the Cut Graph, Angle Deficit and Developability A Developable Surface of Uniformly Negative Internal Angle Deficit A Black-Box Solution for Determining the Unfoldability of a Coolinoid A Functional Expression of the Unfolding of a Coolinoid Developability of the Coolinoid Future Work Conclusions References Rational Maximal Parametrisations of Dupin Cyclides Introduction Paths and Curves Rational Parametrisations of the Circle Optimized Maximal B ́ezier Parametrisations of S1 Optimal Quartic B ́ezier Circles Optimal Quintic B ́ezier Circles An Optimized Degree Six Rational Circle The Parametrisation of Ring Dupin Cyclides by Trigonometric Functions Rational Maximal Parametrisations of D Induced Parametrisations Induced Maximal Rational Parametrisations of Dupin Cyclides Induced Maximal, Bi-quartic, Rational Parametrisation of D on (0, 1) × (0, 1) Induced Bi-quintic Maximal Rational Parametrisations of D on (0, 1) × (0, 1) Induced Bi-sextic Maximal Parametrisation of D on (0, 1) × (0, 1) with Iso-parametric Curves ‘Almost’ Identical to Those of the Trigonometric Parametrisation Sub-maximal Parametrisations of D References Discrete Harmonic Functions from Local Coordinates Introduction Discrete Laplacian on Triangulations Discrete Harmonic Functions Local Coordinates Generalized Barycentric Coordinates in Polygons Natural Neighbor Coordinates Comparison Experimental Approximation Quality Varying Positions Conclusion References Computing the Topology of an Arrangement of Quartics Introduction Preliminaries: Subresultants and Special Points of a Real Algebraic Plane Curve On the Topology of a Quartic Curve General Setup x–Extremal Points Locatable Singularities Unlocatable Singularities On the Topology of an Arrangement with Two Quartics General Setup Checking Conditions on Intersection Points Intersections Singular–Singular At Least One of the Curves Is Regular at the Event Point One Curve Events AnExample Conclusions References Non-uniform B-Spline Subdivision Using Refine and Smooth Background Contents Notation Factorising Non-uniform Knot Insertions Oslo Knot Insertion Understanding Uniform Refine and Smooth Our Generalised Refine and Smooth Two Stencils to Consider Factorisable Knot Insertions B ́ezier End Conditions Systematic Development The Need for a Different Approach Schaefer’s Knot Insertion Algorithm A Factorisation for B ́ezier End Conditions Conclusions References Scattered Data Fitting on Surfaces Using Projected Powell-Sabin Splines Introduction Preliminaries Projection Atlas Projected Gradients Consistent Triangulations Interpolation and Data Fitting An Interpolation Method A Two-Stage Data Fitting Method Numerical Examples Scattered Data Fitting on the Unit Sphere Ring Type Surfaces Remarks References Implicit Boundary Control of Vector Field Based Shape Deformations Introduction Related Work Deformation Blending Implicit Boundaries Smooth Blending Piecewise Deformation Integration in Space Time Applications Rotation Translation Implementation and Performance Conclusion References Tuning Subdivision Algorithms Using Constrained Energy Optimization Introduction Energy Minimization Concept Constrained Energy Minimization Constrained Energy Minimization for the Algorithm of Catmull-Clark Conclusion References Description of Surfaces in Parallel Coordinates by Linked Planar Regions Introduction Overview of Parallel Coordinates Duality in the Plane Lines in $\mathbb{R}^N$ Planes, Hyperplanes and Recursion Curves Surface Representation – Formulation Boundary Contours Developable Surfaces Reconstruction Resolution of Ambiguity Higher Dimensions Ruled Surfaces More General Surfaces References Discrete Surface Ricci Flow: Theory and Applications Introduction Conformal Geometry Ricci Flow Outline Previous Work Mappings Holomorphic Forms Metrics Ricci Flow Theoretical Background Riemannian Metric Gaussian Curvature Conformal Metric Deformation Universal Covering Space Riemann Surface Computational Methodology Harmonic Maps Holomorphic 1-Forms Ricci Flow Discrete Ricci Flow Theories on Discrete Ricci Flow Applications Global Conformal Parameterization Surface Matching Computing General Geometric Structures Manifold Splines Conclusion and Future Works References Guided $C^2$ Spline Surfaces with V-Shaped Tessellation Introduction A V-Shaped Tessellation Map Transition $\rtrans$ from $\rlin$ to $\ct$ Surface Cap in $\R^3$ Discussion References MOS Surfaces: Medial Surface Transforms with Rational Domain Boundaries Introduction Preliminaries Minkowski Space $\Rkosinka^{3,1}$ and Homogeneous Coordinates Medial Surface Transform Tangent Planes of Sheets of Medial Surface Transforms and Envelope Formula Tangent Planes of 2-Surfaces in $\Rkosinka^{3,1}$ Envelope Formula Sheets of Medial Surface Transforms Defining Rational Envelopes MOS Surfaces Characterizing the PILT Vectors of MOS Surfaces MOS Surfaces in Hyperplanes Classification Construction Example General MOS Surfaces Existence and Examples Towards the Construction of General MOS Surfaces Conclusion References Mean Value B ́ezier Surfaces Introduction MeanValueB ́ezier Surfaces Results and Conclusion References Curvature Estimation over Smooth Polygonal Meshes Using the Half Tube Formula Introduction Background and Related Work The Algorithm TestsandExamples Conclusion and Future Work References Segmenting Periodic Reliefs on Triangle Meshes Introduction Related Work Algorithm Overview Algorithm Details Whole Relief Segmentation Coarse Relief Unit Extraction Refinement Cut Localization Results Discussions Conclusions and Future Work References Estimation of End Curvatures from Planar Point Data Introduction Differential Geometry of Plane Curves Error Analysis of Curvature Vector Estimate Improved Estimate of Normal Vector Improved Estimate of Curvature Vector Numerical Validation Conclusions References Inversion, Degree and Reparametrization for Rational Surfaces Introduction Inversion of a Rational Surface Parametrization Algorithm and Examples Computation of the Degree of Rational Surface Parametrizations Preliminaries and Terminology Computation of the Degree of a Rational Map Algorithm and Examples Proper Reparametrization for Surfaces Algorithm of Proper Reparametrization for Surfaces References Discrete Surfaces in Isotropic Geometry Introduction Previous Work Contributions and Overview Fundamentals Isotropic Geometry Laguerre Geometry Principal Meshes in Isotropic Geometry Meshes with Planar Faces Conical and Circular Meshes in Isotropic Geometry Optimization Algorithms Isothermic Meshes and Their Dual Counterparts in $I^3$ Isotropic Curvatures in Meshes with Planar Faces References An Appropriate Geometric Invariant for the $C^2$-Analysis of Subdivision Surfaces Introduction The Extended Weingarten Map Asymptotic Expansions Conditions for Curvature Continuity References Curvature-Based Surface Regeneration Introduction Methodology Example Conclusions References Bounded Curvature Subdivision Without Eigenanalysis Background New Results Terminology and Notation Kinds of New Vertices Masks and Stencils Differences, Derivatives and Curvature At Regular Vertices Loop Catmull-Clark 4-8 Why? Why Zero Mean Second Differences for f- and e-Vertices? Why Set the New Mean Second Difference to the Old for v-Vertices? At Extraordinary Vertices Loop Catmull-Clark 4-8 Conclusions and Loose Ends References Facial Shape-from-Shading Using Principal Geodesic Analysis and Robust Statistics Introduction Shape-from-Shading A Statistical Surface Normal Model Preliminaries PGA of Needle-Maps Incorporating Principal Geodesics into SFS Robust Statistics Combining and Classifying Experimental Results Ground Truth Data Real World Data Conclusions References Statistical Methods for Surface Integration Introduction Surface Integration Related Work Exploiting a Statistical Model of Surface Height Coupling Height and Surface Normal Variation A Surface Height Model Coupling the Surface Normal and Height Models Fitting the Coupled Model to Surface Normal Data Imposing Surface Height Constraints on Surface Normals A Global Statistical Integrability Constraint Normals from Height Parameters Model-Based Integration Experiments Comparing Surface Normal and Height Models Surface Integration Conclusions References Skeleton Surface Generation from Volumetric Models of Thin Plate Structures for Industrial Applications Introduction Related Work Mesh Based Approach A Voxel-Based Approach Contouring Skeleton Surface Extracting Skeleton Cells Point Labeling $\chi$ Contouring Skeleton Meshes Junctions of Plates Characteristics of Skeleton Cells Implementation and Experimental Results Implementation Techniques Experimentation Conclusions References Parallel Tangency in $\mb{R}^3$ Introduction Some Basic Notions Setup for Disjoint Surface Pieces Setup for the Local Case Structure of the Set of Parallel Tangent Pairs Disjoint Surface Pieces The Local Case The Mid-Point Tangent Surface Disjoint Surface Pieces The Local Case Conclusions and Further Work References Condition Numbers and Least Squares Regression Introduction Condition Numbers for the Least Squares Problem A Simple Normwise Condition Number The Effective Condition Number Componentwise Condition Numbers Mixed Condition Numbers Inequalities Between the Condition Numbers Summary References Propagation of Geometric Tolerance Zones in 3D Introduction Tolerances of Fundamental Geometric Elements Tolerance Zones and Geometric Computations in 3D Skew Line Detection and Distance Computation Reflection and Line Projection in 2D Reflection and Plane Projection in 3D Rotation Conclusion and Future Work References Author Index This book constitutes the refereed proceedings of the 12th IMA International Conference on the Mathematics of Surfaces, held in Sheffield, UK in September 2007. The 22 revised full papers presented together with 8 invited papers were carefully reviewed and selected from numerous submissions. Among the topics addressed is the applicability of various aspects of mathematics to engineering and computer science, especially in domains such as computer aided design, computer vision, and computer graphics. The papers cover a range of ideas from underlying theoretical tools to industrial uses of surfaces. Research is reported on theoretical aspects of surfaces including topology, parameterization, differential geometry, and conformal geometry, and also more practical topics such as geometric tolerances, computing shape from shading, and medial axes for industrial applications. Other specific areas of interest include subdivision schemes, solutions of differential equations on surfaces, knot insertion, surface segmentation, surface deformation, and surface fitting