Discrete Koenigs nets and discrete isothermic surfaces

Details Discrete Koenigs nets and discrete isothermic surfaces Discrete Koenigs nets and discrete isothermic surfaces

2007-09-21

arxiv.org/abs/0709.3408v1

Intern. Math. Research Notices, 2009, No. 11, p. 1976-2012.

Mathematics

Differential Geometry

30 pages, 11 figures

Scientific paper

10.1093/imrn/rnp008

We discuss discretization of Koenigs nets (conjugate nets with equal Laplace invariants) and of isothermic surfaces. Our discretization is based on the notion of dual quadrilaterals: two planar quadrilaterals are called dual, if their corresponding sides are parallel, and their non-corresponding diagonals are parallel. Discrete Koenigs nets are defined as nets with planar quadrilaterals admitting dual nets. Several novel geometric properties of discrete Koenigs nets are found; in particular, two-dimensional discrete Koenigs nets can be characterized by co-planarity of the intersection points of diagonals of elementary quadrilaterals adjacent to any vertex; this characterization is invariant with respect to projective transformations. Discrete isothermic nets are defined as circular Koenigs nets. This is a new geometric characterization of discrete isothermic surfaces introduced previously as circular nets with factorized cross-ratios.

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