By Alexander I. Bobenko and Yuri B. Suris

ISBN-10: 0821847007

ISBN-13: 9780821847008

An rising box of discrete differential geometry goals on the improvement of discrete equivalents of notions and strategies of classical differential geometry. The latter seems as a restrict of a refinement of the discretization. present curiosity in discrete differential geometry derives not just from its value in natural arithmetic but additionally from its purposes in special effects, theoretical physics, structure, and numerics. fairly abruptly, the very uncomplicated buildings of discrete differential geometry change into on the topic of the speculation of integrable structures. one of many major ambitions of this e-book is to bare this integrable constitution of discrete differential geometry. For a given delicate geometry you possibly can recommend many various discretizations. Which one is the easiest? This booklet solutions this question by means of supplying basic discretization ideas and utilising them to various concrete difficulties. It seems that clever theoretical discretizations are extraordinary additionally by way of their strong functionality in purposes. The meant viewers of this booklet is threefold. it's a textbook on discrete differential geometry and integrable platforms compatible for a one semester graduate direction. however, it's addressed to experts in geometry and mathematical physics. It displays the new development in discrete differential geometry and comprises many unique effects. The 3rd team of readers at which this publication is concentrated is shaped by means of experts in geometry processing, special effects, architectural layout, numerical simulations, and animation. they could locate right here solutions to the query ``How can we discretize differential geometry?'' bobbing up of their particular box. must haves for studying this publication comprise commonplace undergraduate heritage (calculus and linear algebra). No wisdom of differential geometry is anticipated, even though a few familiarity with curves and surfaces will be useful.

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Additional info for Discrete Differential Geometry: Integrable Structure

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Suppose its two coordinate lines, n"131 and n"132 , are given. I and 12, provided the latter three points are known (and are in general position). 4) So, on each such step one has two free real parameters C2l, C12, attached to the elementary square e12 (u) of the lattice Z2. Thus, one can define a Q-surface f by prescribing its two coordinate lines f 1"131' f 1"132' and two real-valued functions C12, C2l defined on all elementary squares of Z2. I2, h) of a Q-surface can be assigned to elementary quadrilaterals of Z2, that is, to the vertices of the dual lattice (Z2)*.

Algebro-geometric orthogonal coordinate systems were constructed by Krichever (1997). A survey of integrable systems in Lie geometry is given in Ferapontov (2000b). Spinor frames for orthogonal nets were introduced in Bobenko-Hertrich-Jeromin (2001). A modern textbook on the Mobius surface theory including the theory of orthogonal nets is Hertrich-Jeromin (2003). A Lie-geometric description of Ribaucour transformations is given in Burstall-Hertrich-Jeromin (2006). 5: Principally parametrized sphere congruences.

Then there exists a one-parameter family of A-surfaces f(12) that are Weingarten transforms of both f(1) and f(2). 2) Let f be an A-surface. Let f(l), f(2) and f(3) be three of its Weingarten transforms, and let three further A-surfaces f(12), f(23) and f(13) be given such that f(i j ) is a simultaneous Weingarten transform of f(i) and f{j). Then generically there exists a unique A-surface f(123) that is a Weingarten transform of f(12), f(23) and f(13). The net f(123) is uniquely defined by the condition that its every point lies in the tangent planes to f(12), f(23) and f(13) at the corresponding points.

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Discrete Differential Geometry: Integrable Structure by Alexander I. Bobenko and Yuri B. Suris

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