Discrete Riemann surfaces

Abstract

We detail the theory of Discrete Riemann Surfaces. It takes place on a cellular decomposition of a surface, together with its Poincaré dual, equipped with a discrete conformal structure. A lot of theorems of the continuous theory follow through to the discrete case, we will define the discrete analogs of period matrices, Riemann's bilinear relations, exponential of constant argument and series. We present the notion of criticality and its relationship with integrability.

Discrete Complex Structure on Surfel Surfaces

“…In forthcoming papers, this theory, already put to use in the context of polyhedral surfaces, will be implemented in the context of surfel surfaces: The first applications that come to mind are the recognition of digital surfaces, simple ones and of higher topology through the computation of their discrete period matrices [15], the analysis of vector fields on digital surfaces allowed by the Hodge theorem that decomposes vector fields into rotational and divergence free parts, the creation of vector fields with given circulation properties, in general the correct discrete treatment of partial differential equations on a digital surface which are solved by analytic functions in the continuous case, like incompressible fluid dynamics for example.…”

Discrete one-forms on meshes and applications to 3D mesh parameterization

“…Whitney showed how co-chains could be interpreted as continuous forms [48]. We will use the term one-form to emphasize this connection, as has been done before by others [5,32,21,18]. …”

“…Harmonic discrete one-forms over a mesh are studied by Mercat [32]. His analysis is based on simultaneously looking at a mesh and a perpendicular (also called semi-critical) dualization.…”

“…Edge weights can be used to define a Hodge-star operation (e.g. [32]), which among other things, results in a definition of co-closedness, which we use in this paper. One-forms that are both closed and co-closed are called harmonic.…”

“…Inspired by recent work on the theory of discrete one-forms [3,18,32] and their use in mesh parameterization [18] as well as related results in vector field visualization [36,45], we show how some properties of these one-forms on meshes can be used to prove the injectivity of a number of mesh parameterization algorithms.…”