Index 403 mixed area, 171 M¨ obius differential geometry, 13 Clifford algebra model, 15 M¨ obius geometry, 97, 116, 341 M¨ obius group, 341 M¨ obius pair of tetrahedra, 67, 121 M¨ obius transformation, 13, 16, 239, 348 Moutard equation, 8, 11, 25, 60, 61, 65, 71, 162 Moutard net, 8, 21 convergence, 198 Moutard representative, 8, 25, 60, 155 Moutard transformation, 9, 12, 26, 63, 66, 72, 81, 295 convergence, 198 multidimensional consistency, xvii, xviii, 37, 63, 70, 87, 91, 129, 151, 162 principle, 87 multiplicative one-form, 50 multiratio equation, 285 net with equal invariants, 7 normal shift, 16, 105 orthogonal coordinate system, 13 orthogonal net, 13 Pappus theorem, 57, 68, 363 parallel net, 5, 47 parallel surface, 17, 27, 170, 174, 176 parallelogram embedding, 315 parametrized surfaces, xvii, xviii Pascal theorem, 365 pencil of conics, 270 Penrose tiling, 232 permutability of transformations B¨ acklund transformation, 22, 133 Darboux transformation, 24, 152 F-transformation, 5, 39 Moutard transformation, 10, 64 Ribaucour transformation, 18, 92 Weingarten transformation, 12, 73 Pl¨ ucker coordinates of a line, 353 Pl¨ ucker line geometry, 118, 353 Pl¨ ucker quadric, 119, 354 polarity, 335 polyhedral surface, 170 principal contact element net, 107, 177 principal curvature sphere, 17, 107 principal curvatures, 17, 174 principal directions, 17 projective differential geometry, 3 projective duality, 332 projective geometry, 331 projective space, 331 projective transformations, 32 pseudospherical surface, 20 Q∗-net, 33, 79 Q-congruence of spheres, 110, 165 Q-net, 32 convergence, 197 in a quadric, 101 quad-graph, 34, 89, 219 quadrangular set, 80, 362 quadric, 99, 334 quadrilateral surface, 235 quadrirational map, 267 R-congruence of spheres, 109, 113, 118 relative invariant, 246 rhombic embedding, 232, 298, 317 quasicrystallic, 232 ramified, 299 Ribaucour transformation, 17, 23, 92, 101, 108 convergence, 204 rotation coeﬃcients, 3, 14, 40, 94 S-isothermic net, 161 of touching spheres, 164 Schwarzian discrete KP equation, 285 sine-Gordon equation, 21, 210 discrete, 145 singular solution, 242 solitons, xv spectral parameter, 139, 160 sphere congruence principally parametrized, 19 spinor frame, 15, 98 spinor representation, 349 star-triangle map, 63, 273 Steiner formula, 17, 174 stereographic projection, 342 strip, 228 surface with constant Gaussian curvature, 20, 26, 175, 176, 179 surface with constant mean curvature, 26, 175, 176, 179 Sym formula, 180, 211 system on quad-graph, 223 T-net, 61, 71 in a quadric, 127, 161, 238 in a sphere, 131 in light cone, 155 tetrahedron property, 243, 245 three-leg form, 257, 281 transformation group principle, 87 W-congruence, 11, 119 Weierstrass invariant, 246 Weierstrass representation, 178, 183 Weingarten surface, 27, 176

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