Abstract
We study local and global approximations of smooth nets of curvature lines and smooth conjugate nets by discrete nets (circular nets and planar quadrilateral nets, respectively) with edges of order ϵ. Both smooth and discrete geometries are described by integrable systems. It is shown that one can obtain an order ϵ2 approximation globally with points of the discrete nets on the smooth surface. A new simple geometric construction of principal directions of smooth surfaces is given.
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