Contents vii 3.2. Q-nets in quadrics 99 3.3. Discrete line congruences in quadrics 101 3.4. Conical nets 103 3.5. Principal contact element nets 106 3.6. Q-congruences of spheres 110 3.7. Ribaucour congruences of spheres 113 3.8. Discrete curvature line parametrization in Lie, obius and Laguerre geometries 115 3.9. Discrete asymptotic nets in Pl¨ ucker line geometry 118 3.10. Exercises 120 3.11. Bibliographical notes 123 Chapter 4. Special Classes of Discrete Surfaces 127 4.1. Discrete Moutard nets in quadrics 127 4.2. Discrete K-nets 130 4.2.1. Notion of a discrete K-net 130 4.2.2. acklund transformation 133 4.2.3. Hirota equation 133 4.2.4. Discrete zero curvature representation 139 4.2.5. Discrete K-surfaces 139 4.2.6. Discrete sine-Gordon equation 142 4.3. Discrete isothermic nets 145 4.3.1. Notion of a discrete isothermic net 145 4.3.2. Cross-ratio characterization of discrete isothermic nets 147 4.3.3. Darboux transformation of discrete isothermic nets 151 4.3.4. Metric of a discrete isothermic net 152 4.3.5. Moutard representatives of discrete isothermic nets 155 4.3.6. Christoffel duality for discrete isothermic nets 156 4.3.7. 3D consistency and zero curvature representation 158 4.3.8. Continuous limit 160 4.4. S-isothermic nets 161 4.5. Discrete surfaces with constant curvature 170 4.5.1. Parallel discrete surfaces and line congruences 170 4.5.2. Polygons with parallel edges and mixed area 170 4.5.3. Curvatures of a polyhedral surface with a parallel Gauss map 173 4.5.4. Q-nets with constant curvature 175 4.5.5. Curvature of principal contact element nets 177
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