Discrete Koenigs nets and discrete isothermic surfaces
Abstract
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 noncorresponding 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, twodimensional discrete Koenigs nets can be characterized by coplanarity 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 crossratios.
 Publication:

arXiv eprints
 Pub Date:
 September 2007
 arXiv:
 arXiv:0709.3408
 Bibcode:
 2007arXiv0709.3408B
 Keywords:

 Mathematics  Differential Geometry;
 53A20;
 53A05 (Primary);
 51B10;
 52C35 (Secondary)
 EPrint:
 30 pages, 11 figures