This paper proposes a unified and consistent set of flexible tools to approximate important geometric attributes, including normal vectors and curvatures on arbitrary triangle meshes. We present a consistent derivation of these first and second order differential properties using averaging Voronoi cells and the mixed Finite-Element/Finite-Volume method, and compare them to existing formulations. Building upon previous work in discrete geometry, these operators are closely related to the continuous case, guaranteeing an appropriate extension from the continuous to the discrete setting: they respect most intrinsic properties of the continuous differential operators. We show that these estimates are optimal in accuracy under mild smoothness conditions, and demonstrate their numerical quality. We also present applications of these operators, such as mesh smoothing, enhancement, and quality checking, and show results of denoising in higher dimensions, such as for tensor images.

/pdf/discrete-differential-geometry-operators-for-triangulated-2-w8vp3jjpy1.pdf

Discrete Differential-Geometry Operators for Triangulated 2-Manifolds

Curves in the plane and in space.- How much does a curve curve?.- Global properties of curves.- Surfaces in three dimensions.- Examples of surfaces.- The first fundamental form.- Curvature of surfaces.- Gaussian, mean and principal curvatures.- Geodesics.- Gauss' Theorema Egregium.- Hyperbolic geometry.- Minimal surfaces.- The Gauss-Bonnet theorem.

/pdf/elementary-differential-geometry-2dwja6huuz.pdf

Elementary Differential Geometry

This book provides a self-contained introduction to ordinary differential equations and dynamical systems suitable for beginning graduate students. The first part begins with some simple examples of explicitly solvable equations and a first glance at qualitative methods. Then the fundamental results concerning the initial value problem are proved: existence, uniqueness, extensibility, dependence on initial conditions. Furthermore, linear equations are considered, including the Floquet theorem, and some perturbation results. As somewhat independent topics, the Frobenius method for linear equations in the complex domain is established and Sturm-Liouville boundary value problems, including oscillation theory, are investigated. The second part introduces the concept of a dynamical system. The Poincare-Bendixson theorem is proved, and several examples of planar systems from classical mechanics, ecology, and electrical engineering are investigated. Moreover, attractors, Hamiltonian systems, the KAM theorem, and periodic solutions are discussed. Finally, stability is studied, including the stable manifold and the Hartman-Grobman theorem for both continuous and discrete systems. The third part introduces chaos, beginning with the basics for iterated interval maps and ending with the Smale-Birkhoff theorem and the Melnikov method for homoclinic orbits. The text contains almost three hundred exercises. Additionally, the use of mathematical software systems is incorporated throughout, showing how they can help in the study of differential equations.

/pdf/ordinary-differential-equations-and-dynamical-systems-4cj3x01ano.pdf

Ordinary Differential Equations and Dynamical Systems

Parameterization of discrete surfaces is a fundamental and widely-used operation in graphics, required, for instance, for texture mapping or remeshing. As 3D data becomes more and more detailed, there is an increased need for fast and robust techniques to automatically compute least-distorted parameterizations of large meshes. In this paper, we present new theoretical and practical results on the parameterization of triangulated surface patches. Given a few desirable properties such as rotation and translation invariance, we show that the only admissible parameterizations form a two-dimensional set and each parameterization in this set can be computed using a simple, sparse, linear system. Since these parameterizations minimize the distortion of different intrinsic measures of the original mesh, we call them Intrinsic Parameterizations. In addition to this partial theoretical analysis, we propose robust, efficient and tunable tools to obtain least-distorted parameterizations automatically. In particular, we give details on a novel, fast technique to provide an optimal mapping without fixing the boundary positions, thus providing a unique Natural Intrinsic Parameterization. Other techniques based on this parameterization family, designed to ease the rapid design of parameterizations, are also proposed.

Intrinsic Parameterizations of Surface Meshes

A method for concise, faithful approximation of complex 3D datasets is key to reducing the computational cost of graphics applications. Despite numerous applications ranging from geometry compression to reverse engineering, efficiently capturing the geometry of a surface remains a tedious task. In this paper, we present both theoretical and practical contributions that result in a novel and versatile framework for geometric approximation of surfaces. We depart from the usual strategy by casting shape approximation as a variational geometric partitioning problem. Using the concept of geometric proxies, we drive the distortion error down through repeated clustering of faces into best-fitting regions. Our approach is entirely discrete and error-driven, and does not require parameterization or local estimations of differential quantities. We also introduce a new metric based on normal deviation, and demonstrate its superior behavior at capturing anisotropy.

/pdf/variational-shape-approximation-30mutumbju.pdf

Variational shape approximation

From the Publisher:
The Second Edition combines a traditional approach with thesymbolic manipulation abilities of Mathematica to explain and develop the classical theory of curves and surfaces. You will learn to reproduce and study interesting curves and surfaces - many more than are included in typical texts - using computer methods. By plotting geometric objects and studying the printed result, teachers and students can understand concepts geometrically and see the effect of changes in parameters.
Modern Differential Geometry of Curves and Surfaces with Mathematica explains how to define and compute standard geometric functions, for example the curvature of curves, and presents a dialect of Mathematica for constructing new curves and surfaces from old. The book also explores how to apply techniques from analysis.
Although the book makes extensive use of Mathematica, readers without access to that program can perform the calculations in the text by hand. While single- and multi-variable calculus, some linear algebra, and a few concepts of point set topology are needed to understand the theory, no computer or Mathematica skills are required to understand the concepts presented in the text. In fact, it serves as an excellent introduction to Mathematica, and includes fully documented programs written for use with Mathematica.
Ideal for both classroom use and self-study, Modern Differential Geometry of Curves and Surfaces with Mathematica has been tested extensively in the classroom and used in professional short courses throughout the world.

Modern Differential Geometry of Curves and Surfaces with Mathematica

"Modern Differential Geometry of Curves and Surfaces" explains the mathematics of curves and surfaces and describes how to draw the pictures illustrating them using "Mathematica". The reader learns not only the classical concepts, ideas and methods of differential geometry, but also how to define, construct and compute standard functions. The reader also learns how to create new curves and surfaces from old ones. The book should be useful for both the classroom and self-study. Material is presented clearly with over 150 exercises, 175 "Mathematica" programmes and 225 geometric figures to thoroughly develop the topics presented. A brief tutorial explaining how to use "Mathematica" in differential geometry is included as well.

Modern Differential Geometry of Curves and Surfaces

From the Publisher: 
The Second Edition combines a traditional approach with thesymbolic manipulation abilities of Mathematica to explain and develop the classical theory of curves and surfaces. You will learn to reproduce and study interesting curves and surfaces - many more than are included in typical texts - using computer methods. By plotting geometric objects and studying the printed result, teachers and students can understand concepts geometrically and see the effect of changes in parameters. 
Modern Differential Geometry of Curves and Surfaces with Mathematica explains how to define and compute standard geometric functions, for example the curvature of curves, and presents a dialect of Mathematica for constructing new curves and surfaces from old. The book also explores how to apply techniques from analysis. 
Although the book makes extensive use of Mathematica, readers without access to that program can perform the calculations in the text by hand. While single- and multi-variable calculus, some linear algebra, and a few concepts of point set topology are needed to understand the theory, no computer or Mathematica skills are required to understand the concepts presented in the text. In fact, it serves as an excellent introduction to Mathematica, and includes fully documented programs written for use with Mathematica. 
Ideal for both classroom use and self-study, Modern Differential Geometry of Curves and Surfaces with Mathematica has been tested extensively in the classroom and used in professional short courses throughout the world.

Alfred Gray

Papers

Modern Differential Geometry of Curves and Surfaces with Mathematica

Modern Differential Geometry of Curves and Surfaces

Modern Differential Geometry of Curves and Surfaces with Mathematica, Third Edition

Modern Differential Geometry of Curves and Surfaces with Mathematica, Third Edition (Studies in Advanced Mathematics)

Introduction to Ordinary Differential Equations with Mathematica