Computational quasiconformal geometry has been developed, which extends the. An introduction of computational conformal geometry for pattern recognition and geometric modeling. Note also that orthogonality of line intersections in the wplane are preserved in the zplane. A conformal map is a function which preserves the angles. Conformal map preserves both angles and shape of in nitesimal small gures but not necessarily their size. Quasiconformal qc theory is an important topic in complex analysis, which studies geometric patterns of deformations between shapes. Numerical conformal mapping methods based on faber series. Computational vacuum forming with conformal mapping. If is antiholomorphic conjugate to a holomorphic function, it preserves angles but reverses their orientation in the literature, there is another definition of conformal. Eigenvectors, eigenvalues, and optimization 126 chapter 8. Numerical conformal mappings and capacity computation. Computational and applied mathematics elsevier journal of computational and applied mathematics 83 1997 205236 numerical conformal mapping methods based on faber series thomas k. A conformal map is distinguishable from other mappings between complex planes by.
In the pictures we will also see what happens at the critical points. Researchers in mathematics, physics, medicine, computer science and many. Conformal map, in mathematics, a transformation of one graph into another in such a way that the angle of intersection of any two lines or curves remains unchanged. It covers some basic concepts from algebraic topology, differential geometry and rieman surface. Map the network to a circle domain, all boundaries are circles. As a generalization of conformal maps, quasiconformal maps are of great interest in mathematical study and real applications. To the novice, it may seem that this subject should merely be a simple reworking of standard. Numerical conformal mapping and analytic continuation 127 the notation above is for the scalar product, a b re ab, and the dot above x indicates dds. It is the intersection of complex analysis, riemann surface theory, algebraic geometry, differential geometry and algebraic topology. First we display the input image, rendered semitransparently, over the input axes of the conformal map, along with a black ellipse and a red line along the real axis. Computational generation and conformal fabrication of. Discrete fourier analysis, cauchy integrals, construction of conformal maps, univalent functions peter henrici isbn. Numerical conformal mappings and capacity computation masters thesis submitted in partial ful. Computational conformal geometry is an interdisciplinary field between mathematics and computer science.
Citeseerx computational conformal geometry and its. This new volume presents thorough introductions to the theoretical. It introduces readers to conformal geometry theory and discusses implementation issues from an engineering perspective. Computational conformal geometry quickly developed after thurston et al.
Conformal maps preserve both angles and the shapes of infinitesimally small. Then the conformal map has many unexpected properties, for instance almost all the boundary is mapped onto almost nothing and vice versa. The most common example is the mercator map, a twodimensional representation of the surface of the earth that preserves compass directions. Computational conformal geometry, xianfeng david gu and shingtung yau, international press, 2008 catalog description an introduction of computational conformal geometry for pattern recognition and geometric modeling.
Thus, the problem of mapping one sector to another reduces to that simpler case, by precomposing and postcomposing with rotations. This book offers an essential overview of computational conformal geometry applied to fundamental problems in specific engineering fields. Conventional computational complex analysis methods focus on conformal mappings on planar domains. Computational conformal mapping edition by prem kythe. Computational topology application canonical homotopy class representative under. Other conformal maps, sometimes called orthomorphic projections, preserve angles but not. Computational conformal geometry, xianfeng david gu and shingtung yau, international press, 2008. Sectors with edges elsewhere than the positive real axis can be rotated, by map z. Computational conformal mapping 1998th edition by prem kythe author isbn.
Computational conformal geometry is an emerging interdisciplinary field, with applications to algebraic topology, differential geometry and riemann surface theories applied to geometric modeling, computer graphics, computer vision, medical imaging, visualization, scientific computation, and many other engineering fields. It will be assumed for convenience that xs is twice differentiate, for then an iteration of 2. Computational conformal geometry focuses on developing the computational metho. Computational conformal geometry is an interdisciplinary. Exhibiting the map as a holomorphic map shows that it preserves angles. General principle we now introduce the notion of conformal mapping. This is a unique monograph on numerical conformal mapping that gives a comprehensive account of the theoretical, computational and application aspects of the problems of determining conformal modules of quadrilaterals and of mapping conformally onto. Examples include feature recognition, segmentation, or shape analysis. Pdf conformal mapping download full pdf book download. The angle between any intersecting curves is preserved under conformal mapping.
Computational conformal mapping prem kythe download. Numerical computation of surface conformal mappings. Tom delillo wichita state u math dept numerical conformal mapping tutorial 2014 17 105. Applied and computational complex analysis, volume 3. Worked examples conformal mappings and bilinear transfor. Methods are presented for approximating the conformal map from the interior of various regions to the interior of simplyconnected target regions with a smooth boundary. Worked examples conformal mappings and bilinear transformations example 1 suppose we wish to. We report recent progress in the computation of conformal mappings from surfaces with arbitrary topologies to canonical domains.
Fourier series methods for numerical conformal mapping of. This barcode number lets you verify that youre getting exactly the right version or edition of a book. Applications of conformal geometry in brain mapping. Computational methods and function theory volume 000000, no. More formally, a map w fz 1 is called conformal or anglepreserving at z 0 if it preserves oriented angles between curves through z 0, as well as their orientation, i. Computational conformal geometry plays an important role in digital geometry processing. An advantage of conformal mapping via circle packing is the flexibility to map a region to a desired shape boundary angles, rather than boundary radii are preserved for a rectangle. In mathematics, a conformal map is a function that locally preserves angles, but not necessarily lengths more formally, let and be open subsets of. Pdf numerical computation of surface conformal mappings. The theory of conformal mapping is studied because of its connections to physical applications and for its significance in mathematics. This paper presents a framework for computational generation and conformal fabrication of woven thinshell structures with arbitrary topology based on the foliation theory which decomposes a surface into a group of parallel leaves. Recently, quasiconformal geometry has attracted attention from different fields, such as applied mathematics, computer vision and medical imaging. Every homeomorphic surface map can be considered as a quasiconformal map, with its local nonconformal distortion given by its beltrami.
Communications in information and systems c 2009 international press vol. Computational conformal geometry softcover reissue of. Curvatures on the surface are important geometric invariants and are widely used in di. We assume the reader is familiar with the basics of complex numbers and complex arith.
Citeseerx document details isaac councill, lee giles, pradeep teregowda. Recently, computational qc geometry has been developed and has made significant contributions to medical imaging, computer graphics and computer vision. A thorough introduction to the conventional methods can be found in the books 1 and 2. Surfaces are represented as triangular meshes on computers. To know that the map is conformal, we also need to know that the curves in the mesh are moving at the same speed at any given point of intersection. Computational conformal geometry applications david gu1 1department of computer science. Conversely, if is an automorphism of 0, then fis also a conformal map from onto 0. I know a conformal mapping is one whose derivative. How can i find a conformal map from one domain onto another. This work introduces the fundamentals of computational conformal geometry, including theoretic foundation, computational algorithms, and engineering applications. The computational advantage can be gain by conformally mapping a simply connected domain onto the unit disk, the upper half plane, or a rectangle. This means that the nonuniqueness of fis given my the group of conformal automorphisms of 0. It may have nowhere a tangent as is the case for julia sets. If is an open subset of the complex plane, then a function.
The methods for the disk due to fornberg 1980 and the ellipse due to delillo and elcrat 1993 are reformulated so that they may be extended to other new computational regions. Conformal geometry has deep roots in pure mathematics. Two major computational methodologies are emphasized. Complex analysis and conformal mapping the term complex analysis refers to the calculus of complexvalued functions fz depending on a single complex variable z. Recent advances in computational conformal geometry 165 fig. To further explore the conformal mapping, we can place the input and transformed images on the pair of axes used in the preceding examples and superpose a set of curves as well.