22 1. Classical Differential Geometry tangent planes at the corresponding points of a B¨ acklund pair is constant. Moreover, (1.40) yields that this constant angle is related to the constant distance between f and f + via |f + − f|2 = 1 − n, n+ 2 . The fact that n, n+ ∈ S2 allows one to express the coeﬃcients p1, p2 in eqs. (1.41), (1.42) in terms of the solutions themselves: p1 = ∂1n, n+− n, ∂1n+ 2 − 2 n, n+ = ∂1n, n+ 1 − n, n+ , (1.69) p2 = n, ∂2n+− ∂2n, n+ 2 + 2 n, n+ = −∂2n, n+ 1 + n, n+ . (1.70) With these expressions, (1.41), (1.42) become a compatible system of first order differential equations for n+ therefore the following data determine a B¨ acklund transform f + of the given K-surface f uniquely: (B) a point n+(0) ∈ S2. Theorem 1.27. (Permutability of B¨ acklund transformations) Let f be a K-surface, and let f (1) and f (2) be two of its B¨ acklund transforms. Then there exists a unique K-surface f (12) which is simultaneously a B¨acklund transform of f (1) and of f (2) . The points of the fourth surface f (12) lie in the intersection of the tangent planes to f (1) and to f (2) at the corresponding points, and are uniquely defined by the properties |f (12) − f (1) | = |f (2) − f| and |f (12) −f (2) | = |f (1) −f|, or, in terms of the Gauss maps, n(1), n(12) = n, n(2) and n(2), n(12) = n, n(1) . Equivalently, the Gauss map n(12) of f (12) can be characterized by the condition that n(12) − n is parallel to n(1) − n(2). We will see how the theory of discrete K-surfaces unifies the theories of smooth K-surfaces and of their B¨ acklund transformations. 1.7. Isothermic surfaces Classically, theory of isothermic surfaces and their transformations was con- sidered as one of the highest achievements of the local differential geometry. Definition 1.28. (Isothermic surface) A curvature line parametrized surface f : R2 → RN is called an isothermic surface if its first fundamental form is conformal, possibly upon a reparametrization of independent vari- ables ui → ϕi(ui) (i = 1, 2), i.e., if |∂1f|2/|∂2f|2 = α1(u1)/α2(u2) at every point u ∈ R2.

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