2-Dimension from the Topological Viewpoint by Barmak J.A., Minian E.G.

By Barmak J.A., Minian E.G.

Show description

Read or Download 2-Dimension from the Topological Viewpoint PDF

Best geometry and topology books

Induccion en la Geometria

L. a. INDUCCION EN GEOMETRIA de I. L. GOLOVINA

Geometry and Spectra of Compact Riemann Surfaces

This vintage monograph is a self-contained advent to the geometry of Riemann surfaces of continuous curvature –1 and their size and eigenvalue spectra. It makes a speciality of topics: the geometric thought of compact Riemann surfaces of genus more than one, and the connection of the Laplace operator with the geometry of such surfaces.

Extra resources for 2-Dimension from the Topological Viewpoint

Example text

Discrete S-isothermic surfaces are therefore composed of tangent spheres and tangent circles, with the spheres and circles intersecting orthogonally. The class of discrete S-isothermic surfaces is obviously invariant under M¨obius transformations. D/ ! R3 . The discrete isothermic surface obtained is called the central extension of the discrete S-isothermic surface. All its faces are orthogonal kites. An important fact of the theory of isothermic surfaces (smooth and discrete) is the existence of a dual isothermic surface [6].

The construction of discrete S-isothermic “round spheres” is based on their relation to circle packings in S 2 . The following theorem is central in this theory. 1. For every polytopal1 cellular decomposition of the sphere, there exists a pattern of circles in the sphere with the following properties. There is a circle corresponding to each face and to each vertex. The vertex circles form a packing with two circles touching if and only if the corresponding vertices are adjacent. Likewise, the face circles form a packing with circles touching if and only if the corresponding faces are 1 We call a cellular decomposition of a surface polytopal if the closed cells are closed discs, and two closed cells intersect in one closed cell if at all.

1 (Christoffel). , a part of the unit sphere. Taking this as a definition for smooth minimal surfaces and discretizing all notions in a convenient way, we are led to the following definition suggested in [2]. 2. A discrete minimal surface is defined to be a discrete S-isothermic surface (made of touching spheres and orthogonal intersecting circles; see [1, 2] for more details) such that all spheres of the dual discrete S-isothermic surface intersect one fixed additional sphere orthogonally. This fixed sphere is taken to be the unit sphere S 2 .

Download PDF sample

Rated 4.86 of 5 – based on 22 votes