1.5. Principally parametrized sphere congruences 19 2) Let f be an m-dimensional orthogonal net. Let f (1) , f (2) and f (3) be three of its Ribaucour transforms, and let three further orthogonal nets f (12) , f (23) and f (13) be given such that f (ij) is a simultaneous Ribaucour transform of f (i) and f (j) . Then generically there exists a unique orthogonal net f (123) that is a Ribaucour transform of f (12) , f (23) and f (13) . The net f (123) is uniquely defined by the condition that for every permutation (ijk) of (123) the corresponding points of f (i) , f (ij) , f (ik) and f (123) are concircular. The theory of discrete orthogonal nets will unify the theories of smooth orthogonal nets and of their transformations. 1.5. Principally parametrized sphere congruences The obius-geometric formalism is very convenient in description of hyper- sphere congruences. The classical case is, of course, that of two-parametric families of spheres in R3. Nonoriented spheres in R3 can be represented as elements of P(R 4,1 out ), where (1.60) R4,1 out = ˆ R4,1 : ˆ ˆ 0 is the space-like part of R4,1. Definition 1.21. (Principally parametrized sphere congruence) A map (1.61) S : R2 nonoriented spheres in R3 is called a principally parametrized sphere congruence if the corresponding map ˆ : R2 P(R4,1 out ) is a conjugate net, i.e., if for any lift of it to R4,1 out , (1.62) ∂1∂2ˆ span(ˆ s, ∂1ˆ ∂2ˆ). A principal parametrization exists and is unique for a generic congru- ence. The classical description of this is as follows. In an arbitrary paramet- rization of a congruence, consider two neighbors S(u + du1), S(u + du2) of a sphere S(u), obtained by infinitesimal shifts along both coordinate lines they intersect the original sphere along two circles C1(u) and C2(u). Thus, in the projective model of obius geometry of R3, based on the Minkowski space R4,1 of pentaspherical coordinates, these circles are de- scribed as L4,1 ( span(ˆ s, ∂1ˆ) )⊥ , resp. L4,1 ( span(ˆ s, ∂2ˆ) )⊥ i.e. their points are represented by elements ˆ L4,1 satisfying (1.63) C1 : ˆ ˆ = 0, ∂1ˆ ˆ = 0, resp. (1.64) C2 : ˆ ˆ = 0, ∂2ˆ ˆ = 0.
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