Discrete complex analysis – the medial graph approach

@inproceedings{Bobenko2013DiscreteCA,
  title={Discrete complex analysis – the medial graph approach},
  author={Alexander I. Bobenko and Felix G{\"u}nther},
  year={2013}
}
We discuss a new formulation of the linear theory of discrete complex analysis on planar quad-graphs based on their medial graphs. It generalizes the theory on rhombic quad-graphs developed by Duffin, Mercat, Kenyon, Chelkak and Smirnov and follows the approach on general quad-graphs proposed by Mercat. We provide discrete counterparts of the most fundamental objects in complex analysis such as holomorphic functions, differential forms, derivatives, and the Laplacian. Also, we discuss discrete… 
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TLDR
The existence of a unique optimal solution for the minimization of discrete functionals that involve squared second order derivatives is proved and it is proved that second divided di erences (derivatives) tightly approximate intrinsic curvatures.

References

SHOWING 1-10 OF 22 REFERENCES
The boundary value problem for discrete analytic functions
Linear and nonlinear theories of discrete analytic functions. Integrable structure and isomonodromic Green’s function
Abstract Two discretizations, linear and nonlinear, of basic notions of the complex analysis are considered. The underlying lattice is an arbitrary quasicrystallic rhombic tiling of a plane. The
Potential theory on a rhombic lattice
Discrete Riemann Surfaces and the Ising Model
Abstract: We define a new theory of discrete Riemann surfaces and present its basic results. The key idea is to consider not only a cellular decomposition of a surface, but the union with its dual.
Discrete complex analysis on isoradial graphs
Approximation of conformal mappings by circle patterns
A circle pattern is a configuration of circles in the plane whose combinatorics is given by a planar graph G such that to each vertex of G there corresponds a circle. If two vertices are connected by
Discrete Complex Structure on Surfel Surfaces
This paper defines a theory of conformal parametrization of digital surfaces made of surfels equipped with a normal vector. The main idea is to locally project each surfel to the tangent plane,
Discrete Riemann Surfaces
We detail the theory of Discrete Riemann Surfaces. It takes place on a cellular decomposition of a surface, together with its Poincare dual, equipped with a discrete conformal structure. A lot of
Conformal invariance of domino tiling
Let U be a multiply connected region in R 2 with smooth boundary. Let P∈be a polyomino in ∈Z 2 approximating U as ∈ → 0. We show that, for certain boundary conditions on P∈, the height distribution
Universality in the 2D Ising model and conformal invariance of fermionic observables
It is widely believed that the celebrated 2D Ising model at criticality has a universal and conformally invariant scaling limit, which is used in deriving many of its properties. However, no
...
...