By Alexander I. Bobenko (eds.)

This is among the first books on a newly rising box of discrete differential geometry and a very good approach to entry this interesting sector. It surveys the interesting connections among discrete types in differential geometry and intricate research, integrable structures and functions in desktop graphics.

The authors take a better examine discrete versions in differential

geometry and dynamical structures. Their curves are polygonal, surfaces

are made up of triangles and quadrilaterals, and time is discrete.

Nevertheless, the variation among the corresponding soft curves,

surfaces and classical dynamical structures 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 quantity via its relevance for special effects and mathematical physics. This ebook is written by way of experts operating jointly on a standard study undertaking. it truly is approximately differential geometry and dynamical platforms, soft and discrete theories, and on natural arithmetic and its sensible functions. The interplay of those points is proven by way of concrete examples, together with discrete conformal mappings, discrete advanced research, discrete curvatures and precise surfaces, discrete integrable platforms, conformal texture mappings in special effects, and free-form architecture.

This richly illustrated publication will persuade readers that this new department of arithmetic is either attractive and helpful. it's going to entice graduate scholars and researchers in differential geometry, complicated research, mathematical physics, numerical tools, discrete geometry, in addition to special effects and geometry processing.

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**Additional info for Advances in Discrete Differential Geometry**

**Example text**

35(3), 193–214 (1998) 37. : Teichmüller space and fundamental domains of Fuchsian groups. Enseign. Math. (2) 45(1-2), 169–187 (1999) 38. : Circle patterns with the combinatorics of the square grid. Duke Math. J. 86, 347–389 (1997) 39. : DGD Gallery, Lawson’s surface uniformization. de/models/lawsons_surface_uniformization (2015) 40. : Critical percolation in the plane: conformal invariance, Cardy’s formula, scaling limits. C. R. Acad. Sci. Paris Sér. I Math. 333(3), 239–244 (2001) 41. : Conformal invariance in random cluster models.

53 Fig. 34 Left A surface glued from six squares. Right Fuchsian uniformization and fundamental domain For each representation we choose corresponding fundamental polygons that allow the comparison of the uniformization: • an octagon with canonical edge pairing aba b cdc d , • an octagon with opposite sides identified, abcda b c d , • a 12-gon that is adapted to the six-squares surface. All data presented in this section is available on the DGD Gallery webpage [39]. Hyperelliptic curve. We uniformize the hyperelliptic curve μ2 = λ6 − 1 as described in Sect.

To achieve this we choose points with normally distributed coordinates and project them to S 2 [31]. 4 Numerical Experiments Given the branch points of an elliptic curve, the modulus τ can be calculated in terms of hypergeometric functions. In this section, we compare the theoretical value of τ with the value τˆ that we obtain by the discrete uniformization method explained in Sect. 2. I. Bobenko et al. so the branch points λ1 , λ2 , λ3 , ∞ satisfy λ1 + λ2 + λ3 = 0, and g2 = −4(λ1 λ2 + λ2 λ3 + λ3 λ1 ), g3 = 4λ1 λ2 λ3 .