Advances in Discrete Differential Geometry by Alexander I. Bobenko (eds.) PDF
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 technique to entry this fascinating quarter. It surveys the interesting connections among discrete versions in differential geometry and complicated research, integrable structures and purposes in machine graphics.
The authors take a more in-depth examine discrete types 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 soft curves,
surfaces and classical dynamical structures with non-stop time can not often be noticeable. this is often the paradigm of structure-preserving discretizations. present advances during this box are inspired to a wide volume through its relevance for special effects and mathematical physics. This e-book is written through experts operating jointly on a standard examine venture. it truly is approximately differential geometry and dynamical structures, soft and discrete theories, and on natural arithmetic and its useful purposes. The interplay of those features is proven by way of concrete examples, together with discrete conformal mappings, discrete complicated research, discrete curvatures and unique surfaces, discrete integrable platforms, conformal texture mappings in special effects, and free-form architecture.
This richly illustrated ebook will persuade readers that this new department of arithmetic is either attractive and necessary. it's going to entice graduate scholars and researchers in differential geometry, advanced research, mathematical physics, numerical equipment, discrete geometry, in addition to special effects and geometry processing.
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Additional resources for Advances in Discrete Differential Geometry
2. (We will also consider tori in the form of Schottky uniformization in Sect. ) The belief that discrete conformal maps approximate conformal maps is not based on a proven theorem but on experimental evidence like the Wente torus example of Sect. 1 and the numerical experiments of Sect. 4. 1 Immersed Tori First we consider a simple example with quadrilateral faces. Figure 13 (left) shows a coarse discretization of a torus. The faces are isosceles trapezoids, so they are inscribed in circles. 1 with prescribed total angle Θ = 2π at all vertices.
I. Bobenko et al. Fig. 33 Uniformization of Lawson’s surface. Left Triangulated model , with the boundary of the fundamental domain shown in brown and the axes of the generators shown in blue. Right Fuchsian uniformizations and fundamental domains. Canonical domain (top), opposite sides domain (middle), and 12-gon (bottom) Discrete Conformal Maps: Boundary Value Problems, Circle Domains . . 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.
Math. 228(3), 1590–1630 (2011) 9. : Universality in the 2D Ising model and conformal invariance of fermionic observables. Invent. Math. 189(3), 515–580 (2012) 10. : Über partielle Differentialgleichungen der mathematischen Physik. Math. Ann. 100, 32–74 (1928) 11. : Discrete potential theory. Duke Math. J. 20, 233–251 (1953) 12. : Basic properties of discrete analytic functions. Duke Math. J. 23, 335–363 (1956) 13. : Distributed and lumped networks. J. Math. Mech. 8, 793–826 (1959) 14. : Potential theory on a rhombic lattice.
Advances in Discrete Differential Geometry by Alexander I. Bobenko (eds.)