Differential Geometry

Advances in Discrete Differential Geometry by Alexander I. Bobenko (eds.)

By Alexander I. Bobenko (eds.)

This is without doubt one of the first books on a newly rising box of discrete differential geometry and a very good technique to entry this fascinating zone. It surveys the interesting connections among discrete versions in differential geometry and intricate research, integrable platforms and purposes in machine graphics.

The authors take a better 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 variation among the corresponding gentle curves,

surfaces and classical dynamical platforms with non-stop time can not often be visible. this can be the paradigm of structure-preserving discretizations. present advances during this box are prompted to a wide volume through its relevance for special effects and mathematical physics. This ebook is written by means of experts operating jointly on a standard learn venture. it truly is approximately differential geometry and dynamical structures, delicate and discrete theories, and on natural arithmetic and its functional purposes. The interplay of those elements is confirmed by means of concrete examples, together with discrete conformal mappings, discrete complicated research, discrete curvatures and particular surfaces, discrete integrable platforms, conformal texture mappings in special effects, and free-form architecture.

This richly illustrated booklet will persuade readers that this new department of arithmetic is either appealing and valuable. it is going to attract 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 info for Advances in Discrete Differential Geometry

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The spherical surfaces are Möbius-normalized to achieve rotational symmetry Figure 1 (top) and Fig. 12 show examples of discrete conformal maps to polyhedral surfaces inscribed in a sphere that were calculated using this method. 5). If all vertices v ∈ V are contained in the unit sphere S 2 ⊂ R3 , then there is a Möbius transformation T of S 2 such that the center of mass of the transformed vertices is the origin [43], T (v) = 0. v∈V The Möbius transformation T is uniquely determined up to post-composition with a rotation around the origin.

10 show examples. The bottom row visualizes a discrete con- Fig. 10 Mapping surfaces with holes to slit surfaces. In all images, the left and right parts of the boundary are identified by a horizontal translation. Preimages of horizontal lines visualize the flow of an incompressible inviscid fluid around the hole in a channel with periodic boundary conditions. Top row A cylinder with a triangular hole is mapped to a cylinder with a slit. One vertex of the triangle and the midpoint of the opposite side are mapped to the endpoints of the slit.

16 Mapping the hexagonal torus C/(Z + τ Z), τ = 21 + i 23 (left) to a double cover of the sphere (right). Because the regular triangulation of the torus on the left is symmetric with respect to the elliptic involution, its image projects to a triangulation of the sphere seen on the right Discrete Conformal Maps: Boundary Value Problems, Circle Domains . . 35 Fig. 17 Mapping the square torus C/(Z + i Z) (left) to a double cover of the sphere (right). 3 Choosing Points on the Sphere The uniformization procedure for elliptic curves described in Sect.

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