vi Contents 1.4.5. Ribaucour transformations of orthogonal nets 17 1.5. Principally parametrized sphere congruences 19 1.6. Surfaces with constant negative Gaussian curvature 20 1.7. Isothermic surfaces 22 1.8. Surfaces with constant mean curvature 26 1.9. Bibliographical notes 28 Chapter 2. Discretization Principles. Multidimensional Nets 31 2.1. Discrete conjugate nets (Q-nets) 32 2.1.1. Notion and consistency of Q-nets 32 2.1.2. Transformations of Q-nets 38 2.1.3. Alternative analytic description of Q-nets 40 2.1.4. Continuous limit 42 2.2. Discrete line congruences 43 2.3. Discrete Koenigs and Moutard nets 47 2.3.1. Notion of dual quadrilaterals 47 2.3.2. Notion of discrete Koenigs nets 49 2.3.3. Geometric characterization of two-dimensional discrete Koenigs nets 54 2.3.4. Geometric characterization of three-dimensional discrete Koenigs nets 56 2.3.5. Function ν and Christoffel duality 58 2.3.6. Moutard representative of a discrete Koenigs net 60 2.3.7. Continuous limit 60 2.3.8. Notion and consistency of T-nets 61 2.3.9. Transformations of T-nets 63 2.3.10. Discrete M-nets 65 2.4. Discrete asymptotic nets 66 2.4.1. Notion and consistency of discrete asymptotic nets 66 2.4.2. Discrete Lelieuvre representation 70 2.4.3. Transformations of discrete A-nets 72 2.5. Exercises 73 2.6. Bibliographical notes 82 Chapter 3. Discretization Principles. Nets in Quadrics 87 3.1. Circular nets 88 3.1.1. Notion and consistency of circular nets 88 3.1.2. Transformations of circular nets 92 3.1.3. Analytic description of circular nets 93 3.1.4. obius-geometric description of circular nets 96
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