Discrete improper affine spheres

Representation formula for discrete indefinite affine spheres

“…Let f : Z 2 → R 3 , (n, m) → f m n be a map. We call f a discrete indefinite affine sphere if it satisfies the following two properties ( [3], [2], [16]): We suppose that detF m n = 0, then there exist functions ω, A, B such that which can be regarded as a discrete analogue of (1.1). We write (2.5) g m n = 2 2 − δHω m n to have expressions ω m n g m n = detF m n and…”

“…It is a distinctive feature of integrable systems that we can discretize them while keeping their integrability. For example, a discrete Liouville equation was derived in [10] using the bilinear techniques, and the corresponding discrete improper affine sphere was introduced in [16]. As for the Tzitzeica equation, an integrable discrete model was proposed in [3], which can be written into the trilinear equation in terms of the τ function.…”

Area Distances of Convex Plane Curves and Improper Affine Spheres

“…In [5], these definitions are generalized to discrete improper affine spheres, indefinite and definite. We now describe these latter definitions, with a slight modification in the definite case.…”

THE GENERALIZATION OF SIERPINSKI CARPET AND MENGER SPONGE IN n-DIMENSIONAL SPACE

“…Recently, the expansion of computer graphics and applications in mathematical physics have given a great impulse to the issue of giving discrete equivalents of affine differential geometric objects [1]. In [2] a consistent definition of discrete affine spheres is proposed, both for definite and indefinite metrics, and in [7] a similar construction is done in the context of improper affine spheres.…”