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Discrete Differential Geometry: An Applied Introduction SIGGRAPH 2006 Discrete Differential Forms for Computational Modeling Mathieu Desbrun Eva Kanso — Yiying Tong Applied Geometry Lab Caltech Motivation The emergence of computers as an essential tool in scienti c research has shaken the very foundations of differential modeling. Indeed, the deeply-rooted abstraction of smoothness, or differentiability, seems to inherently clash with a computer ™s ability of storing only nite sets of numbers. While there has been a series of computational techniques that proposed discretizations of differential equations, the geometric structures they are simulating are often lost in the process. 1.1 The Role of Geometry in Science Geometry is the study of space and of the properties of shapes in space. Dating back to Euclid, models of our surroundings have been formulated using simple, geometric descriptions, formalizing apparent symmetries and experimental invariants. Consequently, geometry is at the foundation of many current physical theories: general relativity, electromagnetism (E&M), gauge theory as well as solid and ‚uid mechanics all have strong underlying geometrical structures. Einstein ™s theory for instance states that gravitational eld strength is directly proportional to the curvature of space-time. In other words, the physics of relativity is directly modelled by
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