Cynosure对《Discrete Differential Geometry》的笔记(1)

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一本讲几何的书，图却少的可怜
<<Modern Approaches to Discrete Curvature (2017)>>里的这段评论倒是不错：
The starting point of these methods is the Steiner formula, which links the(extrinsic) curvatures to the area of the offsets of the surface under study, a methodmuch used by Alexandrov. Interestingly enough, this approach turns out to bethe starting point also of quadrangle/offset formalism for surfaces, developed inChap. 8. This approach, derived from the integrable systems theory of geometricpartial differential equations, is based on the following idea: A specific choice ofparametrization (such as conformal, asymptotic, isothermic, etc.) transforms thestudy of a given (sub-)manifold into the study of a specific (simpler) PDE. In thediscrete setup though, there is no such thing as a change of parametrization. On thecontrary, it seems that one handles the surface itself instead of a parametrization,  as is the rule in differential geometry (although not in geometric measure theory,as the reader should notice). But even though changes of variables are hardlypossible, special parametrizations do exist, e.g., conformal, isothermic, etc., inspiredby Thurston’s work in complex analysis (circle patterns). This leads to a beautifultheory of quadrangle-based discrete surfaces with special offsets, which allows todefine Gaussian and mean curvature through offsets (à la Steiner) and minimal aswell as constant mean or Gaussian curvature surfaces. The analogy with the smoothcases is justified by the convergence and the existence of similar structural properties(transformations of the space of solutions). In this approach, the geometric properties of the moduli space characterizes the discrete differential geometry. Thesespecial surfaces are—as expected—solution to an integrable system given by a Laxpair, analog to the smooth one, or described by conserved quantities [6]. In spite oftheir abstraction, these definitions often correspond to intuitive three-dimensionalgeometry; for instance, for K-nets, Steiner-defined curvature coincides with theproduct of orthogonal osculating circles. And logically but remarkably enough, thistheoretical development has direct applications in architecture, in the constructionof free form surfaces with constrained faces of fixed width and parallelism (see[24]). Surfaces made of panels of fixed width or with beams of fixed breadth arespecific types of geometric quad-surfaces: Koebe and conical meshes. This leads toa new paradigm for discrete surfaces: instead of considering the vertices/edges/facesof one surface, one considers the surface together with its offset or equivalentlythe surface with its “Gauss map.” By doing so, one avoids the tricky problem ofdefining the normals (thus guaranteeing convergence), but more than that, we havea consistent theory, even at a low discretization level [5]. Such indeed is the goal ofdiscrete differential geometry. 
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