The pattern of lines of curvature is determined by the Jacobian.
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It can be viewed as a Cauchy problem for minimal surfaces, allowing one to find a surface if a geodesic, asymptote or lines of curvature is known.
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The transformation by reciprocal directions transforms oriented spheres into oriented spheres and oriented planes into oriented planes, leaving invariant the " tangential distance " of two cycles ( the distance between the points of each one of their common tangents ), and also conserves the lines of curvature.
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