These lectures develop the theory of quasiconformal mappings from scratch, give a selfcontained treatment of the beltrami equation, and cover the basic properties. One paper is longer and will be discussed by two of the participants. The theory of quasiconformal mappings is nearly 80 years old and has been. They testify to the continued vitality of the interplay between classical and modern. Thompson mississippi state university mississippi state, mississippi i. Thus, in order to study distortion of shape, one must extend the discussion to mappings which are not conformal, but at the same time keep the amount of distortion bounded. Genus zero surface conformal mapping and its application to brain surface mapping pdf xianfeng gu, yalin wang, tony f. He also coined the term quasiconformalinhis1935workonuberlagerungs.
Exploring mesh generation and quality enhancement with open source codes. Taken together, the articles provide the reader with a panorama of activity in these areas, drawn by a number of leading figures in the field. Analytical foundations of the theory of quasiconformal mappings in fin b. The first, written by earle and kra, describes further developments in the theory of teichmuller spaces and provides many references to the vast literature on teichmuller spaces and quasiconformal mappings. The angular distortion of a quasiconformal map can be represented by beltrami differentials. Quasiconformal mappings and complex dynamics bgsmath. Quasiconformal mappings on certain classes of domains in metric spaces we remember the following lemma. Next, we shall give some results about domains and balls in geodesic metric spaces. Proceedings of the 9th international conference, numgrid 2018 voronoi 150, celebrating the 150th anniversary. Let e be a connected subset of a topological space x. Current meth ods for triangular mesh mappings cannot, in general, control the. The surface meshes are visualized using the software. Thompson, and shingtung yau ieee transactions on medical imaging, vol. We will justify in section 3 below why this condition cannot.
Quasiconformal mapping encyclopedia of mathematics. Contrary to plane quasiconformal maps, little is known on factoring. Integergrid maps for reliable quad meshing inria sophia antipolis. Surface mapping plays an important role in geometric processing, which induces both area and angular distortions. Also i would like to know about motivation and history behind this concept i am a beginner of this subject.
Geometry modeling grid generation stanford university. The angular distortion of a qc mapping can be represented by the beltrami differentials. Conformal and quasiconformal maps home page topics for the participants lectures. Bibliography includes bibliographical references pages 419425 and index. Quasiconformal mappings with sobolev boundary values kariastala,mariobonk,andjuhaheinonen abstract. Lars ahlforss lectures on quasiconformal mappings, based on a course he gave at harvard university in the spring term of 1964, was first published in 1966 and was soon recognized as the classic it was shortly destined to become. Grotzsch in 1929 as mappings that are as close to conformal as possible, but that can, for example, take a square in the plane and map it to a nonsquare rectangle in the plane, mapping. After a first introduction to the two subjects we shall discuss how quasiconformal techniques play a crucial role in the understanding of holomorphic dynamical systems. Quasiconformal mappings for volumetric deformations in geometric modeling. I am looking for advice on the best mapping software for dungeon mapping. We can regard r as the interior of a riemann surface with boundary r whose boundary is as large as possible see 3. Introduction qc mappings of the whole plane qc mappings on domains the end. Mostows proof of his celebrat ed rigidity results in general rankone symmetric spaces required a quasiconformal mapping theory in subriemannian manifolds. The quasiconformal constant is sometimes called the quasiconformal norm and sometimes the quasiconformal dilatation.
If the angular distortion is bounded, the mapping is called a quasiconformal map. Numerical geometry, grid generation and scientific. An introduction to the theory of higherdimensional quasiconformal mappings. In mathematical complex analysis, a quasiconformal mapping, introduced by grotzsch 1928 and named by ahlfors 1935, is a homeomorphism between plane domains which to first order takes small circles to small ellipses of bounded eccentricity. The theory of quasiconformal mappings of domains in for and for, provided that one is not considering general and, as a rule.
For the basic theory of quasiconformal mappings in euclidean spaces we refer to v1. This thesis discusses three different projects concerning quasiconformal mappings on planar surfaces. An open framework for unstructured grid generation william t. In this section, we introduce the concept of quasiconformal maps, a gen eralization. An attempt is made to formulate the general requirements on variational principles used for the construction of mappings in grid generation and geometric modeling. Iwaniec this paper gives an exposition of basic analytical properties of quasiconformal and quasiregular mappings of ndimensional domains. A quality tetrahedral mesh generator and threedimensional. This process is experimental and the keywords may be updated as the learning algorithm improves.
Nonhomeomorphisms with bounded distortion are usually called quasiregular mappings. For the most part, we assume only the background provided by elementary graduate courses in real and complex analysis. Surface mapping plays an important role in geometric processing. Description of harmonic quasiconformal mappings anila duka, ma university ismail qemali, vlora, albania abstract in this paper we will discuss harmonic functions, conformal mappings and quasiconformal mappings and their applications, ect. Let be a c2 strictly convex jordan curve and let fbe an arbitrary parametriza. Chapter 7 quasiconformal maps and beltrami equation. The term quasiconformal mapping, as a rule, presupposes the mapping to be a homeomorphism. Workedgrids is a windows application that displays a map showing the amateur radio grid squares contacted and logged in using a thirdparty logging program. I am looking an introductory book on quasiconformal mappings for selfstudy. Workedgrids uses colors to display information on a perband basis. Anisotropic smoothing and solution adaption for unstructured grids. The manipulation of surface homeomorphisms is an important aspect in 3d modeling and surface processing. An efficient and novel numerical method for quasiconformal. An introduction to the theory of higherdimensional.
The papers range over a wide variety of topics in complex analysis, quasiconformal mappings, and complex dynamics. This result supports the above claim that the class of. Quasiconformal mappings and their applications covers conformal invariance and conformally invariant metrics, hyperbolictype metrics and hyperbolic geodesics, isometries of relative metrics, uniform spaces and gromov hyperbolicity, quasiregular mappings and quasiconformal mappings in nspace, universal teichmuller space and related topics, quasiminimizers and potential. Dungeon fog quick and easy high quality battle map creation. Proceedings of the 9th international conference, numgrid 2018. Grid generator is a handy little plugin for generating all sorts of custom grids, quickly and efficiently, in adobe photoshop. It grew out as a byproduct and a proper part of our work on the monograph 10 on analytic methods in the. Quasiconformal geometry and dynamics banach center publications, volume 48 institute of mathematics polish academy of sciences warszawa 1999 nonlinear analysis and quasiconformal mappings from the perspective of pdes tadeusz iwaniec department of mathematics, syracuse university syracuse, new york 244, u. Quasiconformal extension of harmonic mappings in the plane 619 an extra condition on the second complex dilatation. We make the convention that all quasiconformal mappings are sensepreserving. An elliptic, quasiorthogonal grid generation system is formulated based on quasi conformal mapping for arbitrary anisotropic long and skinny regions. Preprint 1 computing quasiconformal maps on riemann. Mappings with convex potentials and the quasiconformal jacobian problem kovalev, leonid v.
Preprint 1 computing quasiconformal maps on riemann surfaces. Quasiconformal harmonic mappings and close to convex domains 2 following kaplan 9, an analytic mapping f. This book offers a modern, uptodate introduction to quasiconformal mappings from an explicitly geometric perspective, emphasizing both the extensive developments in mapping theory during the past few decades and the remarkable applications of geometric function theory to other fields, including dynamical systems, kleinian groups, geometric topology, differential geometry, and geometric group. Introduction qc mappings of the whole plane qc mappings on domains the end measuring smoothness and integrability in rd 1 p. Topology and analysis conformal mappings in euclidean space the moduli of curve families rings and condensers quasiconformal mappings mapping problems the tukiavaisala extension theorem the mostow rigidity theorem and discrete mobius groups basic. Nonlinear analysis and quasiconformal mappings from the perspective of pdes tadeusz iwaniec department of mathematics, syracuse university syracuse, new york 244, u. I want to start work on a megadungeon for my home campaign, and doing it by hand on paper or in illustrator is more than i want. Introduction many recent advances in the finitedifference solution of elliptic equations have been limited to regions whose boundary contours coincide with coordinate lines of the cartesian coordinate system. Introduction qc mappings of the whole plane qc mappings on domains the end introduction. Regularity of planar quasiconformal mappings mart prats september 8th, 2016. Let q be a ball in r m whose interior meets s but whose boundary does not. We discuss some topics that are not included in the standard introductory graduate course in complex analysis.
The importance of quasiconformal mappings in complex analysis was realized by ahlfors and teich muller in the 1930s. On the quasiisometries of harmonic quasiconformal mappings. Let be a c2 strictly convex jordan curve and let fbe an arbitrary parametriza tion. A curvilinear coordinate system generated by a conformal mapping is very rigid in the sense that little control can be exerted over the distribution of the grid points.
The multiplication consists of rotation by the angle arga and. To enable the efficient generation of integergrid maps, in section. Inkarnate comprehensive inbrowser map making software. As the author notes, quasiconformal mappings were introduced by the mathematician h. In the area of quasiconformal maps, however, two re cent approaches. On removable sets for quasiconformal mappings 143 when mn, the hyperplanes are r n, and the set a below should be disre garded. Bounded distortion mapping spaces for triangular meshes acm. Contents introduction topology and analysis conformal mappings in euclidean space the moduli of curve families rings and condensers quasiconformal mappings mapping problems the tukiavaisala extension theorem the mostow rigidity theorem and discrete mobius groups basic notation bibliography index. For a highly irregular mesh, the conformal parameterization is difficult to compute, causing the technique.
Every homeomorphic surface map can be considered as a quasiconformal map, with its local nonconformal distortion given by its beltrami. This project was initiated because i could not find a logging program that provided the level of detail and mapping. Gardiner and jun hu winter, 2005, hri, allahabad, india. Extremal functions for plane quasiconformal mappings kurihara, shigenori and yamashita, shinji, journal of mathematics of kyoto university, 2003. Numgrid 2018, voronoi 150 conference proceedings on numerical geometry problems, computational grid generation methods, education in numerical. An efficient and novel numerical method for quasiconformal mappings of doubly connected domains. Computation of quasiconformal surface maps using discrete beltrami flow abstract. The proposed polyomino geomap conquers the limitations and combines their advantages. Which are more flexible than conformal mappings and this make them an easy tool. Today quasiconformal mappings are used eve rywhere in complex analysis of one variable. Computing quasiconformal maps using an auxiliary metric. U is called close to convex if there exists a univalent convex function.
Quasiconformal mapping singular integral operator riemann mapping beltrami equation distribution derivative these keywords were added by machine and not by the authors. Conformal mappings also do not exist in three dimensions except for trivial cases. In the same way that a quasiconformal mapping turns out to be a mobius transformation even without the a priori assumption that it is a homeomorphism, a quasiconformal mapping is locally homeomorphic as soon as its coefficient of quasiconformality is sufficiently close to 1. Current methods for triangular mesh mappings cannot, in general, control the worst case distortion of all triangles nor. The second, by shishikura, describes how quasiconformal mappings have revitalized the subject of complex dynamics. Grid square ham radio maps and great circle software calculator bearing category is a curation of 40 web resources on, tinylocator, ham atlas, workedgrids. Make maps for virtual tabletops, or run it on a tv for use as a dynamic grid map with fog of war. Grid generator provides full control for horizontal and vertical cell. This is a survey of some results in the constructive theory of functions of a complex variable, obtained by the author and his collaborators during the last 1520 years by an application of methods and results from modern geometric function theory and the theory of quasiconformal mappings. The generalisations to noninjective mappings was initiated with reshet. But early on, the theory found applications beyond the classical framework.
We found that the best possible estimates could be obtained either by using the sharp form of schottkys theorem or the sharp form of the distortion theorem for quasiconformal mappings. Application of conformal and quasiconformal mappings and. Many surface maps in our physical world are quasiconformal. Some of the papers will only be read partially, more detailed instructions are below. Many surface mappings in our physical world are quasiconformal. Let f be a riemann surface whose universal covering space is conformally equivalent to the unit disk. Quasiconformal mappings can be viewed as a generalization of conformal mappings. The measurable riemann mapping theorem and the stoilow factorization. Jul 14, 2006 lars ahlfors lectures on quasiconformal mappings, based on a course he gave at harvard university in the spring term of 1964, was first published in 1966 and was soon recognized as the classic it was shortly destined to become. Ndimensional quasiconformal qcf mappings petru caraman. In the first two projects we show that a priori weaker conditions still suffice to prove quasiconformality. This course will explore the interplay between quasiconformal geometry and holomorphic dynamics.
Lectures on quasiconformal mappings university lecture. Ahlfors used quasiconformal mappings in his geometric approach to nevanlinnas value distribution theory. The constant kmeasures how near a mapping is to being conformal, i. If the angular distortion is bounded, the mapping is called a quasiconformal mapping qcmapping. Relationship between quasisymmetric and quasiconformal maps.
Every participant will lecture on one of the following papers. Quasiconformal harmonic mappings onto a convex domain revisited 3. Conformal and quasiconformal mappings springerlink. Numerical geometry, grid generation and scientific computing. Quasiconformal mappings this gives the explanation we were after. Surface conformalquasiconformal parameterization with. We note that one of the most famous applications of the theory of higherdimensional quasiconformal mappings, mostows rigidity theorem 1967 74, came just. Jones nasa langley research center hampton, va 236812199 w. Computing quasiconformal maps using an auxiliary metric and. Quasiconformal geometry of monotone mappings page 3 of 18 theorem 20 implies that s is bilipschitz if and only if.
453 694 540 249 1055 621 1287 205 251 1194 788 316 457 185 1117 1492 867 1245 691 1428 1503 810 1213 1282 219 673 860 1165 524 90 575 243 1253 110