By Alexander I. Bobenko and Yuri B. Suris
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 significance in natural arithmetic but in addition from its functions in special effects, theoretical physics, structure, and numerics. really by surprise, the very easy constructions of discrete differential geometry turn into concerning the idea of integrable structures. one of many major targets of this e-book is to bare this integrable constitution of discrete differential geometry. For a given tender geometry you could recommend many alternative discretizations. Which one is the simplest? This publication solutions this query through supplying primary discretization rules and using them to varied concrete difficulties. It seems that clever theoretical discretizations are extraordinary additionally via their sturdy functionality in purposes. The meant viewers of this publication is threefold. it's a textbook on discrete differential geometry and integrable structures appropriate for a one semester graduate path. nevertheless, it really is addressed to experts in geometry and mathematical physics. It displays the hot development in discrete differential geometry and includes many unique effects. The 3rd team of readers at which this booklet is concentrated is shaped through experts in geometry processing, special effects, architectural layout, numerical simulations, and animation. they could locate right here solutions to the query ``How will we discretize differential geometry?'' bobbing up of their particular box. necessities for interpreting this e-book comprise typical undergraduate history (calculus and linear algebra). No wisdom of differential geometry is anticipated, even though a few familiarity with curves and surfaces will be priceless.
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Extra info for Discrete Differential Geometry: Integrable Structure
It can be demonstrated that in the case (i) the surface f+ is with necessity a Mobius transformation of f; we will not consider this trivial case further. In the case (ii) one gets proper Darboux transformations. , does not depend on u E ]R2. It is called a parameter of the Darboux transformation. 76). As usual, we regard iterating a Darboux transformation as adding a third (discrete) dimension to a two-dimensional isothermic net. The main classical result on Darboux transformations is the following theorem, which assures that one can add several discrete dimensions in a consistent way.
54). 1. 9. (Koenigs transformation) Two Koenigs nets f, f+ : ]R2 ~ ]RN are said to be related by a Koenigs transformation if the focal points F(l), F(2) of the line congruence (f f+) separate the points f, f+ harmonically. 31) by a suitable choice of Moutard lifts y, y+. 10. (Permutability of Moutard transformations) 1) Let y be a Moutard net, and let y(1) and y(2) be two of its Moutard transforms. Then there exists a one-parameter family of Moutard nets y(12) that are Moutard transforms of both y(1) and y(2).
Weingarten transformation = Moutard transformation for Lelieuvre normals) The Lelieuvre normal fields n, n+ of a Weingarten pair f, f+ of A-surfaces are Moutard transforms of one another. A Weingarten transform f+ of a given A-surface f is reconstructed from a Moutard transform n+ of the Lelieuvre normal field n. , two smooth functions Pi l13i of one variable. 10. 15. (Permutability of Weingarten transformations) 1) Let f be an A-surface, and let f(1) and f(2) be two of its Weingarten transforms.