1.2. Koenigs and Moutard nets 9 (M2) a smooth function q12 : R2 → R that has the meaning of the coeﬃcient of the Moutard equation. 1.2.2. Transformations of Koenigs and Moutard nets. Moutard in- vented a remarkable analytic device for transforming Moutard nets. Definition 1.8. (Moutard transformation) Two Moutard nets y, y+ : R2 → RN are called Moutard transforms of one another if they satisfy (lin- ear) differential equations ∂1y+ + ∂1y = p1(y+ − y), (1.30) ∂2y+ − ∂2y = p2(y+ + y), (1.31) with some functions p1, p2 : R2 → R (or similar equations with all plus and minus signs interchanged, which is also equivalent to renaming the coordi- nate axes 1 ↔ 2). The functions p1, p2, specifying the Moutard transform, must satisfy (nonlinear) differential equations that express compatibility of (1.30), (1.31) with (1.29): ∂1p2 = ∂2p1 = −q12 + p1p2, (1.32) q+ 12 = −q12 + 2p1p2. (1.33) The following data determine a Moutard transform y+ of a given Moutard net y: (MT1) a point y+(0) ∈ RN (MT2) the values of the functions pi on the coordinate axes Bi for i = 1, 2, i.e., two smooth functions pi B i of one variable. If the Moutard nets y, y+ in RN+1 are considered as lifts of Koenigs nets f = [y], f + = [y+] in RN , then a geometric content of the Moutard transformation can be easily revealed. Introduce two surfaces F (1) , F (2) : R2 → RN with the homogeneous coordinates F (1) = [y+ + y], F (2) = [y+ − y]. Then for every u ∈ R2 the points F (1) , F (2) lie on the line (ff + ), and equations (1.30), (1.31) show that this line is tangent to both surfaces F (1) , F (2) . One says that these surfaces are focal surfaces of the line congruence (ff + ). Now an easy computation shows that on each such line the four points f, F (1) , f + , F (2) build a harmonic set, that is, (1.34) q(f, F (1) , f + , F (2) ) = −1, where q is the cross-ratio of four collinear points see (9.54).

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