A Glimpse into Discrete Diff erential Geometry
A Glimpse into Discrete Diff erential Geometry
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DOI：
10.1090/noti1578
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发表时间：
2017-11
影响因子：
--
通讯作者：
Keenan Crane;M. Wardetzky
中科院分区：
文献类型：
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作者：
Keenan Crane;M. Wardetzky
离散微分几何(DDG)是一门新兴的学科，它研究光滑几何对象的离散模拟，在解析描述和计算之间提供了重要的联系。近年来，它在计算解剖学/生物学、计算力学、工业设计、计算建筑和数字几何处理等应用问题上发掘了丰富多样的新观点。离散微分几何的基本原理是，像多面体这样的离散对象不仅仅是光滑对象的近似，而是本身就是微分几何对象。与传统的数值分析侧重于消除精化极限中的逼近误差(例如，通过采用越来越小的有限差分)不同，DDG强调所谓的“模拟”观点，即准确地保留系统的关键属性，而不考虑网格单元的大小。正如模拟机械系统的算法可能寻求精确地保持物理不变量，例如总能量或动量一样，离散几何的结构保持模型寻求精确地保持全局几何不变量，例如总曲率。更广泛地说，DDG专注于对不属于传统数值分析保护伞的对象进行离散化。本文概述了DDG中的一些主题。
The emerging field of discrete differential geometry (DDG) studies discrete analogues of smooth geometric objects, providing an essential link between analytical descriptions and computation. In recent years it has unearthed a rich variety of new perspectives on applied problems in computational anatomy/biology, computational mechanics, industrial design, computational architecture, and digital geometry processing at large. The basic philosophy of discrete differential geometry is that a discrete object like a polyhedron is not merely an approximation of a smooth one, but rather a differential geometric object in its own right. In contrast to traditional numerical analysis which focuses on eliminating approximation error in the limit of refinement (e.g., by taking smaller and smaller finite differences), DDG places an emphasis on the so-called “mimetic” viewpoint, where key properties of a system are preserved exactly, independent of how large or small the elements of a mesh might be. Just as algorithms for simulating mechanical systems might seek to exactly preserve physical invariants such as total energy or momentum, structurepreserving models of discrete geometry seek to exactly preserve global geometric invariants such as total curvature. More broadly, DDG focuses on the discretization of objects that do not naturally fall under the umbrella of traditional numerical analysis. This article provides an overview of some of the themes in DDG.