By Desbrun M., et al.
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Additional resources for Discrete differential geometry: An applied introduction
These differential forms are extensively used in mathematics, physics and engineering, as evidenced by the fact that operations like gradient, divergence, and curl can all be expressed in terms of forms only, as well as fundamental theorems like Green’s or Stokes. 2 A Formal Definition For concreteness, consider the n-dimensional Euclidean space Rn , n ∈ N and let M be an open region M ⊂ Rn ; M is also called an n-manifold. The vector space Tx M consists of all the (tangent) vectors at a point x ∈ M and can be identified with Rn itself.
W HITE , J. H. 1973. A Global Invariant of Conformal Mappings in Space. Proceedings of the American Mathematical Society 38, 1, 162–164. W ILLMORE , T. J. 2000. Surfaces in Conformal Geometry. Annals of Global Analysis and Geometry 18, 3-4, 255–264. , AND BAJAJ , C. L. 2003. Discrete Surface Modeling using Geometric Flows. Tech. , University of Texas. , AND B ELYAEV, A. G. 2002. Fair Triangle Mesh Generation with Discrete Elastica. In Geometric Modeling and Processing, IEEE Computer Society, 119–123.
The deviation from the ideal angles is measured with a quadratic objective just as in the work of Sheffer and de Sturler. The critical observation is that the sum of two α-angles across an edge is a conformal invariant (because it measures the angle of intersection between the circumcircles), while the α-angles themselves are not. In other words, the best one can hope for in a discrete conformal mapping is conservation of θ -angles, not α-angles. Formulating the problem in terms of given θ -angles then leads to a simple convex, and thus unique, minimization problem: the final circle pattern in the plane is completely determined by the θ -angles.
Discrete differential geometry: An applied introduction by Desbrun M., et al.