402 Index discrete exponential function, 301 discrete Green’s function, 303 discrete Gronwall lemma, 194 discrete KdV equation, 237 in associative algebra, 238 discrete KP equation, 285 discrete Laplace equation, 217, 291 discrete Laplace type equation, 217, 257 integrable, 260 discrete logarithmic function, 304, 323 discrete nets, xvii, xviii discrete one-form, 50 discrete pseudosphere, 180 double cross-ratio equation, 278 double of a graph, 219, 256, 293 dual graph, 217, 292, 293 dual kagome lattice, 232 dual quadrilaterals, 47, 171 Dupin cyclide, 113, 340 Eisenhart hexahedron, 2 elementary flip, 234 elementary hexahedron, 34, 57, 62, 89, 100, 112, 115 extended frame, 180, 211 F-transformation, 4, 38, 45, 74 convergence, 197 focal A-net, 120 focal net, 45 functional tetrahedron equation, 275 fundamental transformation, 4, 38 Gauss map, 21, 131 generalized, 170 Gaussian curvature, 17, 20, 174 general position, 34, 43, 66 Goursat problem, 3, 188 graph, 215, 293 harmonic function, 291, 293 hierarchy of commuting flows, xvi, 209 Hirota equation, 135, 141, 213, 221, 315 in associative algebra, 236 Hirota-Miwa equation, 81, 277 holomorphic function, 291, 292, 294 hyperbolic system, 187 approximation, 191 consistency, 189 covergence of solutions, 192 essential dimension, 189 incidence theorem, xviii, 357 initial value problem, 228 integrability, xv, xviii integrable system, xv, xviii, 210, 213 on graphs, 216 inverse spectral method, 214 isomonodromic solution, 306, 321, 323 isothermic net, 145 cross-ratio characterization, 150 geometric characterization, 146, 147 metric coeﬃcient, 152 isothermic surface, 22, 27, 178 conformal metric, 23 cross-ratio characterization, 148 isotropic line, 102, 119, 338, 354 K-net, 130 K-surface, 20, 139, 211, 213 convergence, 205 Koebe polyhedron, 177 Koenigs conic, 79 Koenigs net, 7, 49, 145, 165, 175, 176 algebraic characterization, 51, 59 geometric characterization, 54, 56 planar, 82 point equation, 7, 59 Koenigs transformation, 10 labelling of edges, 129, 150, 154, 162, 232 Laguerre geometry, 105, 116, 350 Blaschke cylinder model, 350 cyclographic model, 350 Laguerre transformation, 350 Lam´ e system, 16 Laplace invariant, 77, 79 Laplace transformation, 76 Laplacian, 293 Lelieuvre normal field, 11, 21, 70, 131 Lelieuvre representation, 11, 70 Lie geometry, 16, 107, 116, 335 Lie quadric, 107, 337 Lie sphere transformation, 339 light cone, 13, 344 line congruence, 43, 170 in a quadric, 102 list A, 253 list H, 253 list Q, 241, 253 loop algebra, 210 loop group, 211 Lorentz-harmonic map, 21 Lorentz-harmonic net, 140 M-net, 65 in a sphere, 140 in light cone, 161 mean curvature, 17, 174 Menelaus theorem, 357 generalized, 54, 360 minimal surface, 26, 27, 175, 178 Minkowski space, 13, 161, 344 Miquel theorem, 90, 367

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