The authors focus on using analytic methods in the study of some of the fundamental results and problems of polyhedral geometry: for instance, the Cauchy rigidity theorem, Thurston's circle packing theorem, rigidity of circle packing theorems, and Colin de Verdiere's variational principle. This present book is the first complete treatment of the vast, and expansively developed, field of polyhedral geometry.
The goal of this book is to present a unified mathematical treatment of diverse problems in mathematics, physics, computer science, and engineer ing using geometric algebra. Geometric algebra was invented by William Kingdon Clifford in as a unification and generalization of the works of Grassmann and Hamilton, which came more than a quarter of a century before.
Whereas the algebras of Clifford and Grassmann are well known in advanced mathematics and physics, they have never made an impact in elementary textbooks where the vector algebra of Gibbs-Heaviside still predominates.
The approach to Clifford algebra adopted in most of the ar ticles here was pioneered in the s by David Hestenes. Later, together with Garret Sobczyk, he developed it into a unified language for math ematics and physics. Sobczyk first learned about the power of geometric algebra in classes in electrodynamics and relativity taught by Hestenes at Arizona State University from to He still vividly remembers a feeling of disbelief that the fundamental geometric product of vectors could have been left out of his undergraduate mathematics education.
Geometric algebra provides a rich, general mathematical framework for the develop ment of multilinear algebra, projective and affine geometry, calculus on a manifold, the representation of Lie groups and Lie algebras, the use of the horosphere and many other areas. This book is addressed to a broad audience of applied mathematicians, physicists, computer scientists, and engineers. Author : Tobin A. Driscoll,Lloyd N. This book provides a comprehensive look at the Schwarz-Christoffel transformation, including its history and foundations, practical computation, common and less common variations, and many applications in fields such as electromagnetism, fluid flow, design and inverse problems, and the solution of linear systems of equations.
It is an accessible resource for engineers, scientists, and applied mathematicians who seek more experience with theoretical or computational conformal mapping techniques.
The most important theoretical results are stated and proved, but the emphasis throughout remains on concrete understanding and implementation, as evidenced by the 76 figures based on quantitatively correct illustrative examples.
There are over classical and modern reference works cited for readers needing more details. Author : Prem K. The subject of conformal mappings is a major part of geometric function theory that gained prominence after the publication of the Riemann mapping theorem — for every simply connected domain of the extended complex plane there is a univalent and meromorphic function that maps such a domain conformally onto the unit disk.
The Handbook of Conformal Mappings and Applications is a compendium of at least all known conformal maps to date, with diagrams and description, and all possible applications in different scientific disciplines, such as: fluid flows, heat transfer, acoustics, electromagnetic fields as static fields in electricity and magnetism, various mathematical models and methods, including solutions of certain integral equations. Conformal Maps and Geometry presents key topics in geometric function theory and the theory of univalent functions, and also prepares the reader to progress to study the SLE.
It succeeds admirably on both counts. Though Riemann mapping theorem is frequently explored, there are few texts that discuss general theory of univalent maps, conformal invariants, and Loewner evolution. This textbook provides an accessible foundation of the theory of conformal maps and their connections with geometry. It offers a unique view of the field, as it is one of the first to discuss general theory of univalent maps at a graduate level, while introducing more complex theories of conformal invariants and extremal lengths.
Conformal Maps and Geometry is an ideal resource for graduate courses in Complex Analysis or as an analytic prerequisite to study the theory of Schramm-Loewner evolution.
Understanding Geometric Algebra: Hamilton, Grassmann, and Clifford for Computer Vision and Graphics introduces geometric algebra with an emphasis on the background mathematics of Hamilton, Grassmann, and Clifford.
It shows how to describe and compute geometry for 3D modeling applications in computer graphics and computer vision. Unlike similar texts. Geometry processing, or mesh processing, is a fast-growing area of research that uses concepts from applied mathematics, computer science, and engineering to design efficient algorithms for the acquisition, reconstruction, analysis, manipulation, simulation, and transmission of complex 3D models.
Applications of geometry processing algorithms already cover a wide range of areas from multimedia, entertainment, and classical computer-aided design, to biomedical computing, reverse engineering, and scientific computing.
Over the last several years, triangle meshes have become increasingly popular, as irregular triangle meshes have developed into a valuable alternative to traditional spline surfaces. This work will focus on the methodology for computing the conformal structures of metric surfaces with complicated topologies. Conformal Transformation and Conformal Structure. Conformal geometry investigates quantities invariant under the angle preserving transformation group.
Primary 53A30; Secondary 52C Key words and phrases. Conformal mappings preserve angles. The conformal equivalence class of a Riemannian metric on a surface is called a conformal structure. A Riemann surface is a smooth surface together with a conformal structure. Thus, in a Riemann surface, one can measure angles, but not lengths. Each surface with a Riemannian metric is automatically a Riemann surface. Conformal mapping is the natural equiv- alence relation for Riemann surfaces.
The goal of conformal geometry is to classify Riemann surfaces up to conformal mappings or biholomorphism in complex geo- metric terminology. Theoretically, this is called the moduli space problem. Fundamental Tasks. The following computational problems are some of the most fundamental tasks for computational conformal geometry.
These prob- lems are intrinsically inter-dependent: 1 Conformal Structure Given a surface with a Riemannian metric, compute various representations of its intrinsic conformal structure.
These are called the conformal modulus of the Riemann surface. A funda- mental theorem for Riemann surfaces, called the uniformization theorem, says that each Riemannian metric is conformal to a completer Riemann- ian metric of constant Gaussian curvature. This metric is unique unless the surface is the 2-sphere or the torus.
Computing such metrics is of fundamental importance for computational conformal geometry. This can be reduced to compute the confor- mal mapping of each surface to a canonical shape, such as circular domain on the sphere, plane or hyperbolic space.
This is closely related to the quasi-conformal mapping problem. In this work, we will explain the methods for solving these fundamental prob- lems in detail. Merits of Conformal Geometry for Engineering Applications. The following are some of the major intrinsic reason: 1 Canonical Domain All metric surfaces can be conformally mapped to canonical domains of either sphere, plane or hyperbolic disk.
This helps to convert 3D geometric processing problems to 2D ones. This metric is valu- able for many geometric problems. For example, each homotopy class of a none trivial loop has a unique closed geodesic representative in a hy- perbolic metric. Furthermore, one can design a Riemannian metric with prescribed curvatures, which is useful for geometric modeling purposes.
These structures are crucial for geometric modeling applications. Surface registration and comparison are the most fundamental tasks in computer vision and medical imaging.
Un- der this type of coordinates, the Riemannian metric has the simplest form. Isothermal coordinates preserves local shapes. It is preferable for visualization and texture mapping purposes.
In the later discussion, we will demonstrate the powerful conformal geometric meth- ods for various engineering applications. A thorough literature review is beyond the scope of this work. Most conformal geometric methods are for planar domains or topological disks genus zero surface with a single bound- ary , whereas our current work focuses on methods for surfaces with complicated topologies.
Therefore, many important works for planar domains or topological disks may be skipped due to the page limit. Planar Domains. Conventional computational complex analysis meth- ods focus on conformal mappings on planar domains. Thorough surveys can be found in [60], [55], [56],[57], [58] and [59]. Recently, a geodesic zipper algorithm based on iterating simple maps has been in- troduced in [61], and a linear conformal mapping algorithm based on hyperbolic geometry can be found in [64].
A robust algorithm based on cross ratio and De- launay triangulation can be found in [62]. Genus Zero Surfaces. Discrete harmonic maps were constructed in [5], where the cotan formula was intro- duced.
Discrete intrinsic parameterization by minimiz- ing Dirichlet energy was introduced by [7]. Mean value coordinates were introduced in [8] to compute generalized harmonic maps. Conformal mappings for topological spheres are discussed in [9] and [10]. High Genus Surfaces. Discrete holomorphic forms are introduced by Gu and Yau [11] to compute global conformal structure for high genus surfaces.
All the computations are carried out on discrete polyhedral surfaces. Another approach of discrete holomorphy was introduced in [12] by discretization of the Cauchy-Riemann equation. The method requires regular connectivity of the mesh. General discrete exterior calculus was presented in [13]. Gortler et al. Tong et al. Surface Ricci Flow. Hamilton in a seminal pa- per [23] for Riemannian manifolds of any dimension. This leads the way for potential applications to computer graphics.
There are many ways to discretize smooth surfaces. The one which is partic- ularly related to a discretization of conformality is the circle packing metric intro- duced by Thurston [25]. Thurston conjectured in [27] that for a discretization of the Jor- dan domain in the plane, the sequence of circle packings converge to the Riemann mapping.
This was proved by Rodin and Sullivan [28]. Another related discretization method is called circle pattern; it considers both the combinatorics and the geometry of the original mesh, and can be regarded as a variant to circle packings.
Circle pattern was proposed by Bowers and Hurdal [33], and has been proven to be a minimizer of a convex energy by Bobenko and Springborn [34]. Yamabe Flow on Surfaces. A comprehensive survey on this topic was given by Lee and Parker in [40]. In a very nice recent work of Springborn et al. They constructed an algorithm based on their explicit formula. It is applied for computing hyperbolic structure and the canonical homotopy class representative in [44]. Any non-orientable surface has a two-fold cover which is orientable.
In the following discussion, by replacing a non-orientable surface by its orientable double cover,we will always assume surfaces are orientable. Algebraic Topology.
Theorem 3. It plays an important role in computational algorithms as well. Hodge Decomposition. The following discussion can be found in [78]. Exact forms are closed. Definition 3. Then Theorem 3. Riemann Surface. Classical textbooks for Riemann surface theory are [72] and [73]. Intuitively, a Riemann surface is a topological surface with an extra structure, which can measure angles, but not the lengths. Manifold and atlas. Because all the local coordinate transitions are holomorphic, the measurements of angles are independent of the choice of coordinates.
The maximal conformal atlas is a conformal structure, Definition 3. Two conformal atlases are equivalent if their union is still a conformal atlas. Each equivalence class of conformal atlases is called a conformal structure. A topological surface with a conformal structure is called a Riemann surface.
A circle domain in a Riemann surface is a domain so that the components of the complement of the domain are closed geodesic disks and points. Here a geodesic disk in a Riemann surface is a topological disk whose lifts in the universal cover is a round desk in E2 or S2 or H2. Harmonic Maps.
The critical points of the harmonic energy in the space of maps are called harmonic maps. For the 2-sphere in the standard metric, we have, Theorem 3. A very useful property of harmonic maps is the following.
Locally, isothermal coordinates always exist [82]. An atlas with all local co- ordinates being isothermal is a conformal atlas. Therefore a Riemannian metric uniquely determines a conformal structure, namely Theorem 3. All oriented metric surfaces are Riemann surfaces. The Gaussian curvature of the surface is given by 3. Although the Gaussian curvature is intrinsic to the Riemannian metric, the total Gaussian curvature is a topological invariant: Theorem 3.
Spherical Euclidean Hyperbolic Figure 4. Surface uniformization theorem. Quasi-Conformal Maps. A generalization of a conformal map is called the quasi-conformal map, which is an orientation-preserving homeomorphism be- tween Riemann surfaces with bounded conformality distortion.
In particular, a conformal homeomorphism is quasi-conformal. Conformal and quasi-conformal mappings for a topo- logical disk. If equation 3. From the texture mappings in frames c and d , we can see the conformality is maintained well around the nose tip circles to circles , while it is changed a lot along the boundary area circles to quasi-ellipses.
The moduli space, intro- duced and studied by Riemann, is of fundamental importance for theoretical study on Riemann surfaces.
However, the moduli space is very complicated geometrically and topologically. We call it the uniformization metric. Fur- thermore, if two hyperbolic pants are to be glued isometrically along boundary components, the gluing map is not unique.
It depends on how much twist one must use to glue. Thus there is an extra twisting parameter for each gluing boundary. The twisting angle and length of each cutting loop give the Fenchel- Nielsen coordinates in the shape space. Detailed proofs can be found in [77] and [85]. Figure 8. Harmonic map from a genus zero closed surface to the unit sphere.
According to theorem 3. Definition 4. For a genus zero surface with a single boundary, we can convert it to a sym- metric closed surface by doubling, which means we glue the surface with a copy of itself with the reversed orientation along their common boundaries, to form a closed symmetric surface.
Then by mapping the doubled surface to the unit sphere, we can compute the conformal mapping of the original surface. Discrete Harmonic Map. Figure Computing homology group basis. Homology Basis.
We use two approaches for it, the algebraic approach and the combinatorial approach. In the algebraic approach, we triangulate the surface to a simplcial complex a triangular mesh , and build chain complexes. The 0, 1 and 2 dimensional simplexes are vertices, edges and faces. Figure 10 shows the homology group generators of a genus two surface. Cohomology Basis.
There are two approaches to calculate the cohomology group basis as well, the algebraic approach and the combinatorial approach. Computing harmonic 1-form group basis. Harmonic 1-form Basis. According to Hodge theory 3. Figure 11 shows the harmonic group generators of a genus two surface. Computing holomorphic 1-form group basis. Holomorphic 1-form Basis. The conjugate of a discrete harmonic 1-form 4. So the linear system in Equation 4.
Therefore, each discrete harmonic 1-form has a unique conjugate discrete harmonic 1-form. Figure 12 shows the holomorphic 1-form group basis for the genus two surface. Algorithmic details for computing the basis for the holomorphic 1-form group can be found in [11] and [48]. Given a smooth Riemann surface, we can approximate it by a sequence of discrete surfaces [52], such that the Riemannian metrics of the discrete surfaces converge to the smooth metric.
Discrete intrinsic parameterization by minimizing Dirichlet energy was introduced by [7]. Mean value coordinates were introduced in [8] to compute generalized harmonic maps; Discrete spherical conformal mappings are used in [9] and [10]. Discrete holomorphic forms are introduced by Gu and Yau [11] to compute global conformal surface parameterizations for high genus surfaces.
Another approach of discrete holomorphy was introduced in [12] using discrete exterior calculus [13]. The problem of computing optimal holomorphic 1-forms to reduce area distortion was con- sidered in [14]. Gortler et al. Tong et al. Discrete one-forms have been applied for meshing point clouds in [17], surface tiling [18], surface quadrangulation [19].
Holomorphic 1-form method has been applied for virtual colonoscopy [20]. The colon surface is reconstructed from MRI images, and conformally mapped to the planar rectangle. This improves the efficiency and accuracy for detecting polyps. Conformal mapping is used for brain cortex surface morphology study in [10].
Holomorphic 1-form method has also been applied in computer vision [21, 22] for 3D shape matching, recognition and stitching. In geometric modeling field, constructing splines on general surfaces is one of the most fundamental problems. It is proven in [23] that if the surface has an affine structure, then spline can be generalized to it directly. Holomorphic 1-forms can be applied for computing the affine structures of general surfaces.
Curvature Flow. The Ricci Flow on Surfaces. The Ricci flow was introduced by R. Hamilton in a seminal paper [24] for Riemannian manifolds of any dimension. The Ricci flow has revolutionized the study of geometry of surfaces and 3-manifolds and has inspired huge research activities in geometry.
In the paper [25], Hamilton used the 2- dimensional Ricci flow to give a proof of the uniformization theorem for surfaces of positive genus. This leads a way for potential applications to computer graphics. There are many ways to discretize smooth surfaces.
The one which is particularly related to a discretization of conformality is the circle packing metric introduced by Thurston [26]. The notion of circle packing has appeared in the work of Koebe [27]. Thurston conjectured in [28] that for a discretization of the Jordan domain in the plane, the sequence of circle packings converge to the Riemann mapping. This was proved by Rodin and Sullivan [29]. This paved a way for a fast algorithmic implementation of finding the circle packing metrics, such as the one by Collins and Stephenson [31].
They proved a general existence and convergence theorem for the discrete Ricci flow and proved that the Ricci energy is convex. The algorithmic implementation of the discrete Ricci flow was carried out by Jin et al.
Another related discretization method is called circle pattern; it considers both the combinatorics and the geometry of the original mesh, and can be looked as a variant to circle packings. Circle pattern was proposed by Bowers and Hurdal [34], and has been proven to be a minimizer of a convex energy by Bobenko and Springborn [35].
An efficient circle pattern algorithm was developed by Kharevych et al. The Yamabe Flow on Surfaces. The Yamabe problem aims at finding a conformal metric with constant scalar curvature for compact Riemannian manifolds. The first proof with flaws was given by Yamabe [37], which was corrected and extended to a complete proof by several researchers including Trudinger [38], Aubin [39] and Schoen [40]. In [42] Luo studied the discrete Yamabe flow on surfaces. He introduced a notion of discrete conformal change of polyhedral metric, which plays a key role in developing the discrete Yamabe flow and the associated variational principle in the field.
Based on the discrete conformal class and geometric consideration, Luo gave the discrete Yamabe energy as an integration of a differential 1-form and proved that this energy is a locally convex function. He also deduced from it that the curvature evolution of the Yamabe flow is a heat equation. In a very nice recent work of Springborn et al. They constructed an algorithm based on their explicit formula. Another recent work by Gu et al. In addition, discrete hyperbolic Yamabe Flow was presented in [45] for computing hyperbolic structure and the canonical homotopy class representative.
Theoretic Background. This section review the preliminary theoretic back- ground for conformal geometry. Harmonic Maps. A harmonic function is a critical point of the harmonic energy. Figure 3 shows one such kind of map. Conformal Mappings. In details, as shown in Fig.
A harmonic map from a face surface to a planar convex domain. Conformal mappings preserve angles. Locally, conformal mapping is a scaling transformation, it preserves local shapes. For example, it maps infinitesimal circles to infinitesimal circles. As shown in Fig. A circle packing is defined on the plane, and pulled back onto the bunny surface, and all small circles are preserved. If we put a checkerboard on the plane, then on the bunny surface, all the right angles of the checkers are well preserved, as illustrated in the same figure frame a.
Conformal texture mapping. Conformal mappings have deep relation with complex analysis. Conformal map- pings between two planar domains can be represented as holomorphic functions. Conformal Structure. Conformal structure. An atlas is a conformal atlas if all its transition functions are biholomorphic. Each equivalence class of conformal atlases is called a conformal structure of the surface.
A surface with a conformal structure is called a Riemann surface. We also call such local complex param- eter as isothermal coordinates. Therefore conformal geometric concepts and methods are general to all surfaces in real life. Figure 7 shows the isothermal coordinates on surfaces from real life.
Conformal structures isothermal coordinates for surfaces in real life. A conformal map preserves angles. Differential forms play important role in con- formal geometry. The exterior differential operator d is the generalization of conventional grad, curl and divergence operators. The exterior differentiation of a 0-form is its gradient.
The difference between exact forms and closed forms conveys the topological information of the surface. Therefore the group of all harmonic 1-forms is isomorphic to the first coho- mology group H 1 S, R. Also, the group of all holomorphic 1-forms is isomorphic to H 1 S, R. Surface Ricci Flow. Let S be a topological surface.
Consider all Rieman- nian metrics on S. Each conformal equivalence class of Riemannian metrics is a conformal structure of S. Hamilton and Chow together proved that sur- face Ricci flow converges to a special metric, whose Gaussian curvature is constant everywhere.
Surface uniformization theorem. Quasi-Conformal Maps. A generalization of conformal map is called the quasi-conformal map which is an orientation-preserving homeomorphism between Rie- mann surfaces with bounded conformality distortion, in the sense that the first order approximation of the quasi-conformal homeomorphism takes small circles to small el- lipses of bounded eccentricity. Thus, a conformal homeomorphism that maps a small circle to a small circle can also be regarded as quasi-conformal.
Conformal and quasi-conformal mappings for a topological disk. The twisting angle and length of each cutting loop give the Fenchel- Nielsen coordinates in the shape space. From the texture mappings in frames c and d , we can see the conformality is kept well around nose tip circles to circles , while it is changed a lot along boundary area circles to quasi-ellipses. In general, this space is complicated to compute. Assume S is with a hyperbolic metric, then its Fenchel-Nielsen coordinates in Tg can be constructed as follows.
Figure 11 illustrates one example. Computational Methods. Harmonic map from a genus zero closed surface to the unit sphere. Harmonic maps in R3 can be computed using heat flow method. We can initialize the map by the canonical Gauss map, then minimize the harmonic energy by the heat flow. Then we compute the tangential component of the Laplacian. S For genus zero closed surface, harmonic maps are conformal.
Figure 15 shows one example computed using this method. For genus zero surface with a single boundary, we can convert it to a symmetric closed surface by double covering. Then by mapping the doubled surface to the unit sphere, we can compute the conformal mapping of the original surface.
The mapping is not unique. Computing homology group basis. Homology Basis. Given a surface S embedded in R3 , we first compute its fundamental group generators. Then the fundamental group of S1 has the same generators as the fundamental group of S. These loops also form a basis of the first homology basis H1 S, Z. Figure 14 shows the homology group generators of a genus two surface.
Cohomology Basis. Then dhk is an exact 1-form on Sk. Because of the consistency along the boundaries, dhk is also a closed 1-form on S. Computing harmonic 1-form group basis. Harmonic 1-form Basis. Figure 15 shows the harmonic group generators of a genus two surface. Computing holomorphic 1-form group basis. Holomorphic 1-form Basis. Figure 16 shows the holomorphic 1-form group basis for the genus two surface.
For surfaces with boundaries, we can convert the surface to a symmetric closed surface by the double covering technique. Discrete Surface Ricci Flow. In engineering field, smooth surfaces are often approximated by simplicial complexes triangle meshes.
Major concepts, such as metric, curvature, and conformal deformation in the continuous setting can be generalized to the discrete setting. Background Geometry. In this case, we say the mesh is with Euclidean background geometry.
The angles and edge lengths of each face satisfy the Euclidean cosine law. Similarly, we can assume that a mesh is embedded in the three dimensional sphere S3 or hyperbolic space H3 , then each face is a spherical or a hyperbolic triangle.
We say the mesh is with spherical or hyperbolic background geometry. The angles and the edge lengths of each face satisfy the spherical or hyperbolic cosine law. Discrete Riemannian Metric. A metric on a mesh with Euclidean metric is a discrete Euclidean metric with cone singularities.
Each vertex is a cone singularity. Similarly, a metric on a mesh with spherical background geometry is a discrete spherical metric with cone singularities; a metric on a mesh with hyperbolic background geometry is a discrete hyperbolic metric with cone sin- gularities. Circle Packing Metric. Discrete Gaussian Curvature. The discrete Gaussian curvatures are deter- mined by the discrete metrics. Discrete Gauss-Bonnet Theorem.
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