Classical and Discrete Differential Geometry: Theory, Applications and Algorithms

This book introduces differential geometry and cutting-edge findings from the discipline by incorporating both classical approaches and modern discrete differential geometry across all facets and applications, including graphics and imaging, physics and networks.

With curvature as the centerpiece, the authors present the development of differential geometry, from curves to surfaces, thence to higher dimensional manifolds; and from smooth structures to metric spaces, weighted manifolds and complexes, and to images, meshes and networks. The first part of the book is a differential geometric study of curves and surfaces in the Euclidean space, enhanced while the second part deals with higher dimensional manifolds centering on curvature by exploring the various ways of extending it to higher dimensional objects and more general structures and how to return to lower dimensional constructs. The third part focuses on computational algorithms in algebraic topology and conformal geometry, applicable for surface parameterization, shape registration and structured mesh generation.
The volume will be a useful reference for students of mathematics and computer science, as well as researchers and engineering professionals who are interested in graphics and imaging, complex networks, differential geometry and curvature.

本書介紹了微分幾何學的基礎知識和該學科的最新研究成果，結合了傳統方法和現代離散微分幾何學的應用，包括圖形和影像、物理和網絡等方面。

以曲率為核心，作者們介紹了微分幾何學的發展，從曲線到曲面，再到更高維度的流形；從光滑結構到度量空間、加權流形和複合體，以及圖像、網格和網絡。書的第一部分是對歐幾里得空間中曲線和曲面的微分幾何學研究，第二部分則涉及以曲率為中心的更高維度流形，探討了將曲率概念擴展到更高維度對象和更一般結構的各種方法，以及如何回到低維度結構。第三部分專注於代數拓撲和共形幾何的計算算法，適用於表面參數化、形狀配准和結構化網格生成。

本書將對數學和計算機科學專業的學生，以及對圖形和影像、複雜網絡、微分幾何和曲率感興趣的研究人員和工程專業人員提供有用的參考資料。

David Xianfeng Gu is a SUNY Empire Innovation Professor of Computer Science and Applied Mathematics at State University of New York at Stony Brook, USA. His research interests focus on generalizing modern geometry theories to discrete settings and apply them in engineering and medical fields and recently on geometric views of optimal transportation theory. He is one of the major founders of an interdisciplinary field, Computational Conformal Geometry.

Emil Saucan is Associate Professor of Applied Mathematics at Braude College of Engineering, Israel. His main research interest is geometry in general (including Geometric Topology), especially Discrete and Metric Differential Geometry and their applications to Imaging and Geometric Design, as well as Geometric Modeling. His recent research focuses on various notions of discrete Ricci curvature and their practical applications.

David Xianfeng Gu是美國紐約州立大學石溪分校（State University of New York at Stony Brook）的SUNY Empire Innovation計劃教授，專攻計算機科學和應用數學。他的研究興趣集中在將現代幾何理論推廣到離散環境中，並應用於工程和醫學領域，最近還涉及到最優運輸理論的幾何觀點。他是跨學科領域「計算共形幾何」的主要創始人之一。

Emil Saucan是以色列布勞德工程學院（Braude College of Engineering）應用數學的副教授。他的主要研究興趣是幾何學（包括幾何拓撲），特別是離散和度量微分幾何及其在影像和幾何設計以及幾何建模中的應用。他最近的研究集中在離散黎曲曲率的各種概念及其實際應用。