Computational Hyperbolic Geometry
Hyperbolic geometry:
Hyperbolic geometry is a non-Euclidean geometry, created in the first half of the nineteenth century in the midst of attempts to understand Euclid's axiomatic basis for geometry. The parallel postulate in Euclidean geometry, which states that in two dimensional space, given a line and a point not on it, there is exactly one line going through that point that is parallel to the given line, is replaced in hyperbolic geometry. In hyperbolic geometry there is more than one line going through that point that is parallel to the given line. Unusual consequences of this change came to be recognized as fundamental and surprising properties of non-Euclidean geometry: a straight line was in fact not straight but curved; angle sum in a triangle was not equal to PI, and so forth.
Connections with other areas of mathematics:
Hyperbolic geometry and its geometric insights have applications in diverse areas of mathematics. The below figure (image courtesy of Hyperbolic Geometry by James Cannon et al.) only shows its connections to the three areas: conformal mappings and complex variables, topology, and group theory.
Models of the hyperbolic plane:
There are a number of different models for the hyperbolic geometry.Each model has its own metric, geodesics, isometries, and so on. They are, of course, all equivalent, but provide different views. Each view supplies its own natural intuitions. Here is the Poincare disk model, also called conformal disk model (some cool video). In this model, all the points belong to the open unit disk of the complex plane. The geodesics are the circular arcs orthogonal to the unit circle. The adjective conformal stems from the fact that the angles in the Poincare disk model are the same as the angles in the Euclidean plane.
In the left Poincare disk, the straight line passing through points p and r, and the straight line passing through points q and r, are both parallel to the line l. The angle sum of the hyperbolic triangle with vertices p, q, and r, is 75.49. The right one is a print by Escher that illustrates the fact that the metric in the hyperbolic plane is different from the Euclidean metric, so the pattern is much denser in the boundary than in the center of the disk, although Escher himself did not have mathematical training.
Rigid motion in hyperbolic plane:
According to Felix Klein'sErlanger program, a geometry is the study of properties of a space X invariant under a group G of transformations of X. For hyperbolic geometry,if X is the hyperbolic plane H2 and we use the Poincare disk model, the rigid motion G in H2 is the Mobius transformation group ( complex linear rational maps).
Here is an example to visualize the Mobius transformation. From left to right, the first one is a scanned 3D human face with face number 50k. The triangulated 3D human face is conformally mapped to a planar unit disk, as shown in the third figure. Conformal map is also called angle preserved map, which can be visualized by the second figure, by pulling back the planar checkboard texture to the 3D face according to the inverse map of the face to the unit disk, and all the right angles of the planar checkboard are arepreserved in the 3D face. From the third figure to the forth figure, they only differ a Mobius transformation. Since Mobius transformation is a rigid motion in Poincare disk, which means both the hyperbolic metrics and the angles are preserved, we will find that the pulled back right angled texture are still preserved based on the new inverse map from the disk to the 3D face, as shown in the last figure.
From left to right, datas of this example can be found here: face.m, face.m, face_disk.m, face_Mobius_disk.m, face_Mobius.m.
viewer: g3dogl.exe; usage of g3dogl: g3dogl.txt; checkboard: check.bmp.
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Thurston Uniformization theorem:
The Uniformization theorem says that any surfaces (2D manifold) can conformally embedded to one of the three basic canonical spaces: the sphere, the Euclidean plane, and the hyperbolic plane. The new metric is angle preserved with original one and has constant Gaussian curvature. As shown in the figure with three triangulated surfaces, the genus zero bima model (with positive Euler number) is conformally deforms to a unit sphere (genus is the surface handle). The genus one kitten model (with zero Euler number) is conformally deforms to the Euclidean plane. The genus three David model (with negative Euler number) is conformally deforms to the Poincare disk with infinite tiling. Details can be found in paper: Discrete Surface Ricci Flow.
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Universal cover and Deck transformation:
Here is one example to visualize universal cover and deck transformation of negative Euler number surfaces, whose universal cover can be embedded in hyperbolic plane (here we use Poincare disk model), and the deck transformations are Mobius transformations. From left to right, the amphora model with face number 15k, as shown in the first figure, is cut open along one set of canonical fundamental basis (which is marked on the amphora.m and can be seen with the given viewer). Both disrete hyperbolic Ricci flow and discrete hyperbolic Yamabe flow can compute the embedding of the fundamental domain of the amphora surface to hyperbolic plane, as shown in the second figure with embedding in Poincare disk. Since Poincare disk model is also conformal model, we can use the Euclidean coordinates of each vertex embedded in Poincare disk as their uv coordinates, and the conformal texture mapping is visualized in the third figure. The forth figure shows the basis of the deck transformations of this model. The fifth figure is a portion of the universal cover of the model: finite tiling of the fundamental domain in Poincare disk.
From left to right, datas of this example can be found here: amphora.m, fundamental_domain.m, amphora_uv.m, deck_transformation.m, universal_cover.m.
viewer: g3dogl.exe; usage of g3dogl: g3dogl.txt; checkboard: check.bmp.
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Teichmuller Space:
Related Publications
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Computing Teichmuller Shape SpaceM. Jin, W. Zeng, F. Luo and X. Gu Shape indexing, classification, and retrieval are fundamental problems in computer graphics. This work introduces a novel method for surface indexing and classification based on Teichm¨ uller theory. Two surfaces are conformal equivalent, if there exists a bijective angle-preserving map between them. The Teichm¨ uller space for surfaces with the same topology is a finite dimensional manifold, where each point represents a conformal equivalence class, and a curve in the Teichm¨ uller space represents a deformation process from one class to the other. In this work, we apply Teichm¨ uller space coordinates as shape descriptors, which are succinct, discriminating and intrinsic, invariant under the rigid motions and scalings, insensitive to resolutions. |
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Discrete Curvature Flows for Surfaces and 3-ManifoldsX. Yin, M. Jin, F. Luo, X. Gu Intrinsic curvature flows can be used to design Riemannian metrics by prescribed curvatures. This chapter presents three discrete curvature flow methods that are recently introduced into the engineering fields: the discrete Ricci flow and discrete Yamabe flow for surfaces with various topology, and the discrete curvature flow for hyperbolic 3-manifolds with boundaries. For each flow, we introduce its theories in both the smooth setting and the discrete setting, plus the numerical algorithms to compute it. We also provide a brief survey on their history and their link to some of the engineering applications in computer graphics, computer vision, medical imaging, computer aided design and others. |
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Computing Fenchel-Nielsen Coordinates in Teichm¨uller Shape SpaceM. Jin, W. Zeng, N. Ding, X. Gu This work focuses on the computation of the coordinates of high genus surfaces in the Teichm¨ uller space. The coordinates are called as Fenchel-Nielsen coordinates. The main idea is to decompose the surface to pairs of hyperbolic pants. Each pair of pants is a genus zero surface with three boundaries, equipped with hyperbolic metric. Furthermore, all the boundaries are geodesics. Each pair of hyperbolic pants can be uniquely described by the lengths of its boundaries. The way of gluing different pairs of pants can be represented by the twisting angles between two adjacent pairs of pants which share a common boundary. |
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Canonical Homotopy Class Representative Using Hyperbolic StructureW. Zeng, M. Jin, F. Luo, X. Gu Homotopy group plays a role in computational topology with a fundamental importance. Each homotopy equivalence class contains an infinite number of loops. This work introduces a rigorous and practical method to compute a unique representative for each homotopy class based on hyperbolic structure. we apply hyperbolic Yamabe curvature flow to compute the unique Riemannian metric, which has constant negative one curvature everywhere and is conformal to the original metric. Then we compute the Fuchsian group generators of the surface on the hyperbolic space. For a given loop on the surface, we lift it to the universal covering space, to obtain the Fuchsian transformation corresponding to the homotopy class of the loop. The unique closed geodesic inside the homotopy class is the axis of the Fuchsian transformation, which is the canonical representative. |
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Discrete Surface Ricci FlowM. Jin, J. Kim, F. Luo, X. Gu This work introduces a unified framework for discrete surface Ricci flow algorithms, including spherical, Euclidean, and hyperbolic Ricci flows, which can design Riemannian metrics on surfaces with arbitrary topologies by user-defined Gaussian curvatures. Furthermore, the target metrics are conformal (angle-preserving) to the original metrics. Ricci flow conformally deforms the Riemannian metric on a surface according to its induced curvature, such that the curvature evolves like a heat diffusion process. Eventually, the curvature becomes the user defined curvature. Discrete Ricci flow algorithms are based on a variational framework. Given a mesh, all possible metrics form a linear space, and all possible curvatures form a convex polytope. The Ricci energy is defined on the metric space, which reaches its minimum at the desired metric. The Ricci flow is the negative gradient flow of the Ricci energy. Furthermore, the Ricci energy can be optimized using Newton’s method more efficiently. |
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Discrete Curvature Flow for Hyperbolic 3-Manifold with Complete Geodesic BoundariesX. Yin, M. Jin, F. Luo, X. Gu Three dimensional manifolds admit canonical metrics, which induce constant sectional curvature. Canonical metrics on 3- manifolds are valuable for the study of 3D topology and have the potential for volumetric parameterization and shape matching. This paper generalizes discrete curvature flow from surfaces to hyperbolic 3-manifolds with complete geodesic boundaries. The metric deforms according to the curvature, until the curvature is constant everywhere. The theoretical results are introduced, the algorithm is explained in details, and thorough experiments are carried out to demonstrate the effectiveness and efficiency of discrete 3-manifold curvature flow. |
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Computing General Geometric Structures on Surfaces Using Ricci FlowM. Jin, F. Luo, and X. Gu Surfaces have natural geometric structures, such as spherical structure, affine structure, projective structure, hyperbolic structure and conformal structure. Geometric structures allow different geometries to be defined on various manifolds, such that the algorithms designed for planar domains can be systematically and straightforwardly generalized to the surface domains. In this paper, we thoroughly explain both theoretic and practical aspects of the computational methodology for geometric structures based on Ricci flow, and demonstrate several important applications of Geometric Structures. |
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Computing Geodesic Spectra of SurfacesM. Jin, F. Luo, S.-T. Yau, X. Gu Three dimensional manifolds admit canonical metrics, which induce constant sectional curvature. Canonical metrics on 3- manifolds are valuable for the study of 3D topology and have the potential for volumetric parameterization and shape matching. This paper generalizes discrete curvature flow from surfaces to hyperbolic 3-manifolds with complete geodesic boundaries. The metric deforms according to the curvature, until the curvature is constant everywhere. The theoretical results are introduced, the algorithm is explained in details, and thorough experiments are carried out to demonstrate the effectiveness and efficiency of discrete 3-manifold curvature flow. |
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Computing Surface Hyperbolic Structure and Real Projective StructureM. Jin, F. Luo, X. Gu Surfaces with negative Euler characteristic numbers admit hyperbolic structures and allow hyperbolic geometry. This paper introduces theoretically rigorous and practically simple algorithms to compute hyperbolic structures and real projective structures for general surfaces. The algorithms have been verified on real surfaces scanned from sculptures. The method is efficient and robust in practice. To the best of our knowledge, this is the first work of introducing algorithms based on Ricci flow to compute hyperbolic structure and real projective structure. |
