16 research outputs found
Smooth 2D Coordinate Systems on Discrete Surfaces
International audienceWe introduce a new method to compute conformal param- eterizations using a recent definition of discrete conformity, and estab- lish a discrete version of the Riemann mapping theorem. Our algorithm can parameterize triangular, quadrangular and digital meshes. It can be adapted to preserve metric properties. To demonstrate the efficiency of our method, many examples are shown in the experiment section
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The boundary value problem for discrete analytic functions
This paper is on further development of discrete complex analysis introduced
by R. Isaacs, J. Ferrand, R. Duffin, and C. Mercat. We consider a graph lying
in the complex plane and having quadrilateral faces. A function on the vertices
is called discrete analytic, if for each face the difference quotients along
the two diagonals are equal.
We prove that the Dirichlet boundary value problem for the real part of a
discrete analytic function has a unique solution. In the case when each face
has orthogonal diagonals we prove that this solution uniformly converges to a
harmonic function in the scaling limit. This solves a problem of S. Smirnov
from 2010. This was proved earlier by R. Courant-K. Friedrichs-H. Lewy and L.
Lusternik for square lattices, by D. Chelkak-S. Smirnov and implicitly by P.G.
Ciarlet-P.-A. Raviart for rhombic lattices.
In particular, our result implies uniform convergence of the finite element
method on Delaunay triangulations. This solves a problem of A. Bobenko from
2011. The methodology is based on energy estimates inspired by
alternating-current network theory.Comment: 22 pages, 6 figures. Several changes: Theorem 1.2 generalized,
several assertions added, minor correction in the proofs of Lemma 2.5, 3.3,
Example 3.6, Corollary 5.
Discrete complex analysis on planar quad-graphs
We develop a linear theory of discrete complex analysis on general
quad-graphs, continuing and extending previous work of Duffin, Mercat, Kenyon,
Chelkak and Smirnov on discrete complex analysis on rhombic quad-graphs. Our
approach based on the medial graph yields more instructive proofs of discrete
analogs of several classical theorems and even new results. We provide discrete
counterparts of fundamental concepts in complex analysis such as holomorphic
functions, derivatives, the Laplacian, and exterior calculus. Also, we discuss
discrete versions of important basic theorems such as Green's identities and
Cauchy's integral formulae. For the first time, we discretize Green's first
identity and Cauchy's integral formula for the derivative of a holomorphic
function. In this paper, we focus on planar quad-graphs, but we would like to
mention that many notions and theorems can be adapted to discrete Riemann
surfaces in a straightforward way.
In the case of planar parallelogram-graphs with bounded interior angles and
bounded ratio of side lengths, we construct a discrete Green's function and
discrete Cauchy's kernels with asymptotics comparable to the smooth case.
Further restricting to the integer lattice of a two-dimensional skew coordinate
system yields appropriate discrete Cauchy's integral formulae for higher order
derivatives.Comment: 49 pages, 8 figure
The convergence of discrete period matrices
We study compact polyhedral surfaces as Riemann surfaces and their discrete
counterparts obtained through quadrilateral cellular decompositions and a
linear discretization of the Cauchy-Riemann equation. By ensuring uniformly
bounded interior and intersection angles of diagonals, we establish the
convergence of discrete Dirichlet energies of discrete harmonic differentials
with equal black and white periods to the Dirichlet energy of the corresponding
continuous harmonic differential with the same periods. This convergence also
extends to the discrete period matrix, with a description of the blocks of the
complete discrete period matrix in the limit. Moreover, when the quadrilaterals
have orthogonal diagonals, we observe convergence of discrete Abelian integrals
of the first kind. Adapting the quadrangulations around conical singularities
allows us to improve the convergence rate to a linear function of the maximum
edge length.Comment: 33 pages, 3 figure