Lattice Geometry of the Discrete Darboux, KP, BKP and CKP Equations. Menelaus’ and Carnot’s Theorems

Abstract

Möbius invariant versions of the discrete Darboux, KP, BKP and CKP equations are derived by imposing elementary geometric constraints on an (irregular) lattice in a three-dimensional Euclidean space. Each case is represented by a fundamental theorem of plane geometry. In particular, classical theorems due to Menelaus and Carnot are employed. An interpretation of the discrete CKP equation as a permutability theorem is also provided.