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By Alexander I. Bobenko (eds.)
This is likely one of the first books on a newly rising box of discrete differential geometry and a very good method to entry this interesting sector. It surveys the attention-grabbing connections among discrete versions in differential geometry and complicated research, integrable platforms and functions in machine graphics.
The authors take a better examine discrete versions in differential
geometry and dynamical platforms. Their curves are polygonal, surfaces
are made up of triangles and quadrilaterals, and time is discrete.
Nevertheless, the adaptation among the corresponding delicate curves,
surfaces and classical dynamical platforms with non-stop time can not often be noticeable. this can be the paradigm of structure-preserving discretizations. present advances during this box are prompted to a wide volume by means of its relevance for special effects and mathematical physics. This booklet is written by way of experts operating jointly on a typical learn undertaking. it really is approximately differential geometry and dynamical platforms, delicate and discrete theories, and on natural arithmetic and its useful purposes. The interplay of those elements is tested via concrete examples, together with discrete conformal mappings, discrete advanced research, discrete curvatures and specific surfaces, discrete integrable structures, conformal texture mappings in special effects, and free-form architecture.
This richly illustrated e-book will persuade readers that this new department of arithmetic is either appealing and priceless. it's going to attract graduate scholars and researchers in differential geometry, complicated research, mathematical physics, numerical equipment, discrete geometry, in addition to special effects and geometry processing.
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Extra resources for Advances in Discrete Differential Geometry
This allows us to uniformize the model with corresponding fundamental domains. Six-squares surface. Figure 34 (left) shows a surface glued from six squares, which is conformally equivalent to Lawson’s surface and the hyperelliptic curve. Edges with the same marking are glued together. We calculate a uniformization using the triangulation with vertices added in the centers of the squares as shown. An adapted fundamental domain for this square-tiled translational surface arranges all squares around a single vertex, see Fig.
G = A, B, C, D, . . ∈ PSL(2, R) | . . DC D C B A B A = 1 43 (64) be the corresponding presentation of the uniformization group, see Fig. 25 (left). Then the axes of the generators A and B intersect in a point p0 . Choosing p0 as the base point of a new fundamental polygon as shown in Fig. 25 (right) renders it convex and uniquely determined for the given group and presentation. Fundamental polygons with opposite sides identified. When we consider the geometric characterization of hyperelliptic surfaces in Sect.
B Straighten the edges between vertices that are identified with more than one partner (shown in red). c Axes of the edge-pairing translations are shown in blue. d, e Two cut-and-paste operations lead to a fundamental polygon with one vertex class and opposite edges identified. The axes intersect in one point (see Sect. 4). We move this point to the origin. f Tiling the hyperbolic plane with fundamental polygons Discrete Conformal Maps: Boundary Value Problems, Circle Domains . . G = A, B, C, D, .
Advances in Discrete Differential Geometry by Alexander I. Bobenko (eds.)