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We discuss discretization of Koenigs nets (conjugate nets with equal Laplaceinvariants) and of isothermic surfaces. Our discretization is based on thenotion of dual quadrilaterals: two planar quadrilaterals are called dual, iftheir corresponding sides are parallel, and their non-corresponding diagonalsare parallel. Discrete Koenigs nets are defined as nets with planarquadrilaterals admitting dual nets. Several novel geometric properties ofdiscrete Koenigs nets are found; in particular, two-dimensional discreteKoenigs nets can be characterized by co-planarity of the intersection points ofdiagonals of elementary quadrilaterals adjacent to any vertex; thischaracterization is invariant with respect to projective transformations.Discrete isothermic nets are defined as circular Koenigs nets. This is a newgeometric characterization of discrete isothermic surfaces introducedpreviously as circular nets with factorized cross-ratios.

Discrete Koenigs nets and discrete isothermic surfaces