Related papers: Harmonic Shears and Numerical Conformal Mappings
In this paper, we study harmonic mappings by using the shear construction, introduced by Clunie and Sheil-Small in 1984. We consider two classes of conformal mappings, each of which maps the unit disk D univalently onto a domain which is…
In 1984, a simple and useful univalence criterion for harmonic functions was given by Clunie and Sheil-Small, which is usually called the shear construction. However, the application of this theorem is limited to the planar harmonic…
Harmonic mappings have long intrigued researchers due to their intrinsic connection with minimal surfaces. In this paper, we investigate shearing of two distinct classes of univalent conformal mappings which are convex in horizontal…
We propose a novel meshless method to compute harmonic maps and conformal maps for surfaces embedded in the Euclidean 3-space, using point cloud data only. Given a surface, or a point cloud approximation, we simply use the standard cubic…
We construct sense-preserving univalent harmonic mappings which map the unit disk onto a domain which is convex in the horizontal direction, but with varying dilatation. Also, we obtain minimal surfaces associated with such harmonic…
New algorithms are presented for numerical conformal mapping based on rational approximations and the solution of Dirichlet problems by least-squares fitting on the boundary. The methods are targeted at regions with corners, where the…
We use spherical cap harmonic (SCH) basis functions to analyse and reconstruct the morphology of scanned genus-0 rough surface patches with open edges. We first develop a novel one-to-one conformal mapping algorithm with minimal area…
The aim of this study is to analyze the properties of harmonic fields in the vicinity of rough boundaries where either a constant potential or a zero flux is imposed, while a constant field is prescribed at an infinite distance from this…
Computing spherical harmonic decompositions is a ubiquitous technique that arises in a wide variety of disciplines and a large number of scientific codes. Because spherical harmonics are defined by integrals over spheres, however, one must…
Finding surface mappings with least distortion arises from many applications in various fields. Extremal Teichm\"uller maps are surface mappings with least conformality distortion. The existence and uniqueness of the extremal…
An iterative numerical method to compute the conformal mapping in the context of propagating water waves over uneven topographies is investigated. The map flattens the fluid domain onto a canonical strip in which computations are performed.…
Many authors have studied the numerical computation of conformal mappings (numerical conformal mapping), and there are nowadays several efficient numerical schemes. Among them, Amano's method offers a straightforward numerical procedure for…
We introduce a conceptually simple and efficient algorithm for seamless parametrization, a key element in constructing quad layouts and texture charts on surfaces. More specifically, we consider the construction of parametrizations with…
In the early 1980's an elementary algorithm for computing conformal maps was discovered by R. K\"uhnau and the first author. The algorithm is fast and accurate, but convergence was not known. Given points z_0,...,z_n in the plane, the…
We propose a harmonic surface mapping algorithm (HSMA) for electrostatic pairwise sums of an infinite number of image charges. The images are induced by point sources within a box due to a specific boundary condition which can be…
Let $f=h+\overline{g}$ be a harmonic univalent map in the unit disk $\mathbb{D}$, where $h $ and $g$ are analytic. We obtain an improved estimate for the second coefficient of $h$. This indeed is the first qualitative improvement after the…
This paper solves the problem of computing conformal structures of general 2-manifolds represented as triangle meshes. We compute conformal structures in the following way: first compute homology bases from simplicial complex structures,…
A general criterion in terms of the Schwarzian derivative is given for global univalence of the Weierstrass--Enneper lift of a planar harmonic mapping. Results on distortion and boundary regularity are also deduced. Examples are given to…
An algorithm for the computation of global discrete conformal parametrizations with prescribed global holonomy signatures for triangle meshes was recently described in [Campen and Zorin 2017]. In this paper we provide a detailed analysis of…
Motivated by geometry processing for surfaces with non-trivial topology, we study discrete harmonic maps between closed surfaces of genus at least two. Harmonic maps provide a natural framework for comparing surfaces by minimizing…