Computing and Visualizing Constant-Curvature Metrics on Hyperbolic ...
Computing and Visualizing Constant-Curvature Metrics on Hyperbolic 3-Manifolds with Boundaries Xiaotian Yin1 , Miao Jin2 , Feng Luo3 , and Xianfeng David Gu1 1
Computer Science Department, State University of New York at Stony Brook {xyin,gu}@cs.sunysb.edu 2 Center for Advanced Computer Studies, University of Louisiana at Lafayette [email protected] 3 Department of Mathematics, Rutgers University, [email protected]
Abstract. Almost all three dimensional manifolds admit canonical metrics with constant sectional curvature. In this paper we proposed a new algorithm pipeline to compute such canonical metrics for hyperbolic 3manifolds with high genus boundary surfaces. The computation is based on the discrete curvature flow for 3-manifolds, where the metric is deformed in an angle-preserving fashion until the curvature becomes uniform inside the volume and vanishes on the boundary. We also proposed algorithms to visualize the canonical metric by realizing the volume in the hyperbolic space H3 , both in single period and in multiple periods. The proposed methods could not only facilitate the theoretical study of 3-manifold topology and geometry using computers, but also have great potentials in volumetric parameterizations, 3D shape comparison, volumetric biomedical image analysis and etc.
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Introduction
Many practical problems in computer graphics and geometric modeling can be reduced to computing special metrics (i.e. edge length) on a given geometric object represented as a mesh. One of the most widely considered problems is parameterization, which maps a given surface or volume mesh to a parameter domain with regular and simple shape; in another word, assigns a new set of edge length to the original mesh. Furthermore, many applications need a metric with special curvature properties. Intuitively, curvature measures how the space is locally distorted with respect to the Euclidean space. In many applications it is natural to ask for a metric with uniformly distributed curvature. We call it a constant curvature metric. Actually the existence of such canonical metrics on surfaces is justified
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by the Uniformization Theorem [1]. Similarly, most 3-manifolds (like those we are studying) admit constant curvature metric. In the surface parameterization literature, there are many works of computing metrics with special curvature distributions; the readers are referred to the parameterization survey papers [2] and [3] for general information. Among all those works, there are several which are able to compute the constant curvature metric for general surfaces, such as [4] and [5]. For volumes (or 3-manifolds), on the other hand, there is less work on computing particular metrics in the computer graphics literature. Most of such work is based on parameterization. To name a few, Wang et al. [6] parameterize brain volumes with solid ball via 3D harmonic mapping. Li et al. [7] computes 3D harmonic deformation between volumes with the same topology. Martin et al. [8] use harmonic functions to parameterize femur volumes, which are topologically balls with special shapes, by extending the boundary surface parameterization into the inside of the volume. All these methods assign a metric to the volume in some sense, but none of them target at uniform curvature distributions. In this work, we investigate the computation of constant curvature metric for 3-manifolds. In particular, we focus our attention on a special class of volumes, called hyperbolic 3-manifolds, whose boundaries are surfaces of genus greater than one, such as the Thurston’s knotted Y shape in figure 4. These volumes can be used to model some human soft tissues or biochemical structures. The computation is based on discrete 3D curvature flows, where the metric is evolving under the driven of the curvature. Luo [9] laid down the theoretical foundations of discrete curvature flows for hyperbolic 3-manifolds with boundaries. Our computational method is directly based on that work. The contribution of this paper can be briefly outlined as follows. 1. We proposed a discrete computational method based on curvature flow for hyperbolic 3-manifolds with boundaries. The resulting canonical metric induces constant curvature inside the volume and zero curvature on the boundary (i.e. geodesic boundary). 2. We proposed an algorithm to realize and visualize such canonical metrics in the hyperbolic space H3 . Both the single period representation (i.e. fundamental domain) and multiple period representation (i.e. universal covering space) can be computed. 3. We analyzed the convergence and stability of the proposed algorithms through experiments, and pinpointed several potential applications in both science and engineering fields. In the rest of the paper, we will introduce some underlying concepts and theories in section 2. Then the algorithm pipeline is presented in section 3, together with detailed discussion for each step. In section 4 we show the convergence and performance of the proposed methods, and pinpoint several potential applications. Finally we conclude the paper in section 5.
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Concepts and Theories
In this section, we provide some background knowledge that is necessary to understand the algorithm pipeline. Only the most related concepts and theories are presented here; for further details, the readers are referred to textbooks like [10], [1] and [11]. Hyperbolic Tetrahedron and Truncated Hyperbolic Tetrahedron A 3manifold can be triangulated using tetrahedra. If one assigns a hyperbolic metric to a tetrahedron, it is called a hyperbolic tetrahedron, such as the one [v1 v2 v3 v4 ] shown in figure 1a, where each face fi is a hyperbolic plane, and each edge eij is a hyperbolic line segment. If a 3-manifold has boundaries, it can also be tessellated using truncated tetrahedron. In this case, the 3-manifold is called hyperideal triangulated. Again, a truncated tetrahedron assigned with hyperbolic metric is called truncated hyperbolic tetrahedron (figure 1b). It can be constructed from a hyperbolic tetrahedron by cutting off each vertex vi with a hyperbolic plane perpendicular to the edges eij , eik , eil . Each truncated hyperbolic tetrahedron has four right-angled hyperbolic hexagon faces (figure 1d), which can be glued to other hexagon faces and therefore become interior faces in a truncated tetrahedron mesh; and also four hyperbolic triangle faces (figure 1c), which will compose the boundary surface of the truncated tetrahedron mesh. y1
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Fig. 1. Hyperbolic tetrahedron (a) and truncated hyperbolic tetrahedron (b) with hyperbolic triangle faces (c) and right-angled hexagons (d).
The geometry (or the edge length) of the truncated tetrahedron can be determined by the six dihedral angles {θ1 , θ2 , · · · , θ6 } (figure 1b). For a hyperbolic triangle (figure 1c) with inner angles {θi , θj , θk }, the edge length xi (the one against θi ) can be determined by the hyperbolic cosine law: cosh xi = (cos θi + cos θj cos θk ) / (sin θj sin θk ). For a right-angled hyperbolic hexagon (1d) with three cutting edge {xi , xj , xk }, its internal edge yi (which is against xi ) can be computed using another hyperbolic cosine law: cosh yi = (cosh xi + cosh xj cosh xk ) / (sinh xj sinh xk ).
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By using these hyperbolic cosine laws, all the edge lengths can be computed from the set of dihedral angles. On the other hand, the dihedral angles can also be uniquely determined by the edge lengths using the inverse cosine laws. Discrete Curvature For surfaces represented as triangle meshes, the discrete curvature is represented as the angle deficit around vertices. For aPvertex vi , P the curvature K(vi ) equals 2π − jk αijk for internal vertex or π − jk αijk for boundary vertex, where {αijk } are the surrounding corner angels (figure 2a). For 3-manifolds represented as tetrahedron meshes, there are two types of discrete curvature representations: vertex curvature and edge curvature. For tetrahedron [vi , vj , vk , vl ], let {αijkl , αjkli , αklij , αlijk } denote the solid angles at the verkl tices (figure 2b), βij be the dihedral angle on edge eij (figure 2c). The vertex P curvature K(vi ) is also defined as angle deficit: 4π − jkl αijkl for interior vertex P or 2π − jkl αijkl for boundary vertex. The edge curvature K(eij ) is defined P P kl kl 2π − kl βij if eij is an interior edge, or π − kl βij otherwise. The two types of discrete curvatures are closely related. Actually, the vertex curvature can be completely determined by the edge curvature: K(vi ) = P j K(eij ). vi
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Fig. 2. Discrete curvatures: vertex curvature (a) for 2-manifolds, vertex curvature (b) and edge curvature (c) for 3-manifolds.
Discrete Curvature Flow Curvature flow refers to a flow of the metric driven by the curvature. Given a hyperbolic tetrahedron with edge lengths lij and dihedral angles θij , the volume V of the tetrahedron is a function of the dihedral angles V = V (θ12 , θ13 , θ14 , θ23 , θ24 , θ34 ), and the P Schlaefli formula can be expressed as ∂V /∂θij = −lij /2, and thus dV = −1 ij lij dθij . It can be further 2 proved that the volume of a hyperbolic truncated tetrahedron is a strictly concave function of the dihedral angles ([9]). From this volume function, Luo ([9]) defines another energy function, which is a strictly convex function of the edge length, and the discrete curvature flow is defined as the negative gradient flow of the energy function.
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The curvature flow for 3-manifolds owns some special properties. One of the most prominent properties is the so called Mostow rigidity [12]. It states that the geometry of a finite volume hyperbolic manifold (for dimension greater than two) is determined by the fundamental group. Intuitively, different 3-manifolds have equivalent constant curvature metrics if they have the same topology. As a consequence, the tessellation will not affect the computational results the discrete curvature flow on 3-manifolds. Utilizing this special property, we are allowed to reduce the computational complexity of 3-manifold curvature flow by using the simplest tessellation for a given 3-manifold, as we will see in the algorithm section 3.1. For example, the Thurston’s knotted Y shape can be either represented as a high resolution tetrahedral mesh (figure 4d) or a mesh with only 2 truncated tetrahedra (figure 3a), and the resulting canonical metrics are identical. Hyperbolic Space Model There are several realizations of the 2D and 3D hyperbolic space. In this work, we use the upper half plane model for the 2D hyperbolic space H2 = {(x, y) ∈ R2 |y > 0}, with the Riemannian metric ds2 = dx2 + dy 2 /y 2 . In this model, hyperbolic lines are circular arcs and half lines orthogonal to the x-axis. The rigid motion is given by the so-called M¨ obius transformation, (az + b)/(cz + d), where ac − bd = 1, a, b, c, d ∈ R. Similarly, we use the upper half space model to realize 3D hyperbolic space H3 = {(x, y, z) ∈ R3 |z > 0}, with Riemannian metric ds2 = dx2 + dy 2 + dz 2 /z 2 . In H3 , the hyperbolic planes are hemispheres or vertical planes, whose equators are on the xy-plane. The xy-plane represents all the infinity points in H3 . The rigid motion in H 3 is determined by its restriction on the xy-plane, which is a M¨ obius transformation on the plane, in the form of (az + b)/(cz + d), where ac − bd = 1, and a, b, c, d ∈ C.
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Algorithms
Given a 3-manifold’s boundary surface, represented as a triangular mesh, our algorithm pipeline will go through the following steps: 1. Tessellate the volume with tetrahedra, and simplify the tessellation to minimum number of truncated tetrahedra; (section 3.1) 2. Compute the canonical hyperbolic metric using discrete 3D curvature flow; (section 3.2) 3. Realize and visualize the volume with the computed canonical metric in hyperbolic space H3 . (section 3.3)
3.1
Tessellation and Simplification
Given the boundary surfaces of a 3-manifold, we use the volumetric Delaunay triangulation algorithm introduced in [13] to tessellate the interior of the volume with tetrahedra. Then the tessellation is simplified by the following steps.
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Fig. 3. Simplified tessellation of the Y shape with only two truncated tetrahedra (a) glued together according to the pattern (b). The simplification is carried out through edge collapsing, which turns (c) into (d) by identifying v0 and v6 .
Denote the boundary surface of a 3-manifold M as ∂M = {S1 , S2 , · · · , Sn }. First, create a cone vertex vi for each boundary component Si ; for each triangle face fj ∈ Si , create a new tetrahedron Tji , whose vertex set consists of vi and the vertices of fj . In this way, M is augmented with a set of cone vertices and a set of new tetrahedra. Second, take the edge collapsing operation (shown in figure 3c-d) iteratively, until all the vertices are removed except for those cone vertices {v1 , v2 , · · · , vn } created in the previous step. Denote the resulting tetrahedral ˜ . Finally, for each tetrahedron T˜i ∈ M ˜ , trim off its vertices (which are mesh as M cone vertices) by the original boundary surface, and thus make it a truncated tetrahedron, denoted as Ti . As an example, the Y shape can be simplified to two truncated tetrahedra {T1 , T2 } (figure 3a), together with the gluing pattern (figure 3b) among their hexagon faces. For each Ti , let {Ai , Bi , Ci , Di } be its four hexagon faces, let {ai , bi , ci , di } be the truncated vertices. Its gluing pattern is given as follows, where the arrow → means to identify the former and the later: A1 B1 C1 D1
→ B2 → A2 → C2 → D2
{b1 → c2 , d1 → a2 , c1 → d2 } {c1 → b2 , d1 → c2 , a1 → d2 } {a1 → a2 , d1 → b2 , b1 → d2 } {a1 → a2 , b1 → c2 , c1 → b2 }
Please note that the edge collapsing operation does not change the fundamental group of the volume, and the simplified 3-manifold is topologically equivalent to the original 3-manifolds. From the discussion in section 2, it guarantees the later computation can be carried out on the simplified tessellation instead of the original one. 3.2
Metric Computation via Curvature Flow
Given an hyperideal triangulated 3-manifold, define the discrete metric function as x : E → R+ , where E is the set of edges in the triangulation. The discrete curvature function can be defined as K : E → R. Given an edge eij ∈ E, its
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edge length and edge curvature can be represented as xij and Kij respectively. From the discussion in section 2, the edge curvature can be determined by the dihedral angles, which in turn is a function of edge length. Therefore, the set {Kij } can be calculated from {xij }. Then the discrete curvature flow is then defined as dxij = Kij , (1) dt From this differential equation, the deformation of the metric is driven by the edge curvature, and the whole process is like a heat diffusion. Any numerical method for solving the discrete heat diffusion problem can be applied to solve the curvature flow equation. And in practice, we set the initial edge length to be xij = 1. P 2 During the flow, the total edge curvature ij Kij is strictly decreasing. When the flow reaches the equilibrium state, both the edge curvature and the vertex curvature vanish. The boundary surface will become a hyperbolic geodesic, while all the curvature (which is negative) is uniformly distributed within each truncated hyperbolic tetrahedron. Due to the fact the total curvature is negative, the resulting metric is a hyperbolic one. 3.3
Realization and Visualization
Once the canonical hyperbolic metric is computed, one is ready to realize it in the hyperbolic space H3 . There are two ways to realize the metric. The first one is a single period representation (figure 6), which is a union of multiple truncated hyperbolic tetrahedra. The second is a multiple period representation (figure 7), which is composed by multiple copies of the single period representation glued together nicely. As explained in section 2, we will use the upper half space and upper half plane as the model for H3 and H2 respectively. Realization of a Single Truncated Hyperbolic Tetrahedron Given the edge length of a truncated hyperbolic tetrahedron, its dihedral angles will be uniquely determined and the truncated tetrahedron can be realized in H3 uniquely up to rigid motions. Actually, its embedding is determined by the position of its four right-angled hexagon faces f1 , f2 , f3 , f4 and that of its four triangle faces v1 , v2 , v3 , v4 . Each of these faces now is a hyperbolic plane (i.e. semi-sphere, see figure 5b), separating H3 into two half spaces. By choosing the right half space for each face and taking the intersection of all these half spaces, one will get the realization of the truncated hyperbolic tetrahedron (figure 5c). To compute the position of each hyperbolic plane, let’s consider its intersection with the infinity plane z = 0, which is an Euclidean circle (figure 5a). Here we reuse the symbol fi and vj to represent the intersection circle by the hyperbolic plane fi and vj respectively. As shown in figure 5a, all the circles can be computed explicitly, such that circle fi and circle fj intersect at the given dihedral angle θk , while circle vi is orthogonal to circles fj , fk , fl . In order to remove the ambiguity caused by rigid motion, we fix circle f1 to be line y = 0, f2 to be line y = tan θ1 x, and normalize the circle f3 to have radius 1.
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Once the intersection circles are settled, we can directly construct hemispheres (i.e. hyperbolic planes) whose equators are those circles. By choosing the right half space for each hemisphere and using CSG operations to compute the intersection of these half spaces, we can get a visualization of a single truncated hyperbolic tetrahedron as shown in figure 5c.
Realization of a Single Period A single period representation of the whole 3-manifold with canonical hyperbolic metric is a union of all its constituting truncated hyperbolic tetrahedra. It can be constructed as the following. First, realize one truncated hyperbolic tetrahedron T0 as explained above. Then, pick another not-embedded truncated hyperbolic tetrahedron T1 , which is neighboring to T0 through hexagon faces f1 ∈ T1 with f0 ∈ T0 . Compute a M¨obius transformation in H3 , which rigidly move T1 to such a position that f1 ∈ T1 can be perfectly glued to f0 ∈ T0 . Now we get a partially embedded volume. Repeat the process of picking, moving and gluing a neighboring truncated hyperbolic tetrahedron, until the whole volume is embedded. The above algorithm is essentially a bread-first-search (BFS) in the given 3manifold; it results in a tree spanning all the truncated hyperbolic tetrahedra in the volume. Due to the nature of the constant curvature hyperbolic metric, such gluing (or spanning) operation can be carried out seamlessly, until finally all the truncated tetrahedra are glued together nicely into a simply connected volume, which is a topological ball. Such a single period representation is usually called the fundamental domain for the original volume (see [11]). Figure 6 visualizes fundamental domain of the Y shape embedded in H3 . Realization of Multiple Periods A multiple period representation of the canonical hyperbolic metric is a union of multiple copies of the fundamental domain, and is usually called the universal covering space (UCS) of the original 3-manifolds (see [11]), which is also a simply connected topological ball. Similar to the realization of the fundamental domain, the UCS representation can also be constructed through a sequence of gluing operations; the difference is, the primitive construction block is the embedded fundamental domain rather than the truncated hyperbolic tetrahedron. The gluing operation here can be explained as the following. Recall the algorithm for realizing a fundamental domain, any two truncated hyperbolic tetrahedra are glued through a pair of hexagon faces. After the algorithm is complete, there will be some hexagon faces left open, without gluing to any neighboring face. This is natural because otherwise the fundamental domain will not be simply connected. All the open hexagon faces are grouped into several connected components, each component constitutes a gluable face for the whole fundamental domain. All the gluable faces can be coupled nicely; that is, for each gluable face, there exist another unique gluable face in the same fundamental domain such that they are able to attach to each other nicely. Actually, the fundamental domain can be viewed as a result of cutting the original volume open through
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the coupling gluable faces. And two copies of the fundamental domain can be glued together through one pair of the gluable faces. Different to the construction of one fundamental domain, the gluing operation among fundamental domains can be repeated infinitely many times, getting infinitely many copies involved in the UCS. In practice, we can only afford to realize a finite portion of the UCS, as the one visualized in figure 7.
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Experiments and Applications
We tested our algorithms on about 120 hyperbolic 3-manifolds, including the Thurston’s Y shape, tessellated with truncated hyperbolic tetrahedra. All the experiments converge, and the resulting numerical metrics are perfectly consistent with the results computed using algebraic methods, by a difference less than 1e − 8. From the experiments, we also notice that the stability of the algorithms depends on two issues: the initial metric and the tessellation. Due to numerical errors, for certain initial assignments of edge lengths, the angle computed using hyperbolic cosine law is too close to zeros and thus leads to instability. In our experiments, an initial metric with all edge lengths being 1.0 will always lead to the desired solution. On the other hand, the curvature flow may fail on certain tetrahedralization for the given 3-manifolds; it is probably because the critical point of the volume energy is too close to the boundary of the metric space. It remains an open problem to find the conditions under which the tessellation will guarantee the convergence to the desired metric. Regarding applications, the canonical metrics computed on surfaces have brought lights to huge amount of applications in geometric modeling, computer graphics, computer vision, visualization and biomedical data analysis. We believe that our methods for 3-manifolds will also play fundamental roles in many science and engineering fields in the future. And here we only address some of the potential applications in very brief words, while leaving the detailed discussions of particular applications to future exploration, which is far beyond the scope of this paper. Firstly, the theoretical study of the 3-manifold topology and geometry has been a hot area in the mathematical society for many years; but most 3-manifolds cannot be realized in the Euclidean space and even hard to imagine. we hope that with our work the researchers could gain more insights into the structure of 3-manifolds and therefore facilitate their study, at least in a discrete sense. Secondly, The realization of the canonical metric actually gives a canonical domain for the original 3-manifolds. This domain might be used for the purpose of volumetric texture mapping, volume discretization and remeshing, volume spline construction, volume registration and comparison, and etc, just like how the canonical parameter domains of surfaces help tackle the 2D problems. Thirdly, we noticed that the volumes we studied here could be good models for some volumetric biomedical data, such as some soft tissues in the human body with the penetrating vessels removed, or the complementary space of some
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protein structures. It will be a very interesting research topic to apply our computational methods to these biomedical applications.
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Conclusion
In this work we proposed a new computational method to compute constant curvature metrics for hyperbolic 3-manifolds with high genus boundary surfaces. The algorithm is directly based on discrete volumetric curvature flow. Experiments show the convergence and stability of the algorithm. In order to visualize the metric, we proposed algorithms to realize the metric in the upper half space model of the hyperbolic space. The whole pipeline can not only facilitate the theoretical study of 3-manifold topology and geometry, but also have potential applications in many engineering fields, like volume parameterization, biomedical data analysis and etc. How to apply our methods to those specific applications would be a very interesting future research direction. Meanwhile, it is also a challenging research topic to extend the current framework to other type of 3-manifolds.
References 1. Petersen, P.: Riemannian Geometry. Springer (1997) 2. Sheffer, A., Praun, E., Rose, K.: Mesh parameterization methods and their applications. Foundations and Trendsr in Computer Graphics and Vision 2 (2006) 3. Floater, M.S., Hormann, K.: Surface parameterization: a tutorial and survey. In: Advances in Multiresolution for Geometric Modelling. Springer (2005) 157–186 4. Jin, M., Kim, J., Luo, F., Gu, X.: Discrete surface ricci flow. IEEE Transaction on Visualization and Computer Graphics (2008) 5. Springborn, B., Schr¨ oder, P., Pinkall, U.: Conformal equivalence of triangle meshes. SIGGRAPH 2008 (2008) 6. Wang, Y., Gu, X., Thompson, P.M., Yau, S.T.: 3d harmonic mapping and tetrahedral meshing of brain imaging data. In: Proceeding of Medical Imaging Computing and Computer Assisted Intervention (MICCAI), St. Malo, France. (2004) 7. Li, X., Guo, X., Wang, H., He, Y., Gu, X., Qin, H.: Harmonic volumetric mapping for solid modeling applications. In: Proceeding of Symposium on Solid and Physical Modeling. (2007) 109–120 8. Martin, T., Cohen, E., Kirby, M.: Volumetric parameterization and trivariate bspline fitting using harmonic functions. In: Proceeding of Symposium on Solid and Physical Modeling. (2008) 9. Luo, F.: A combinatorial curvature flow for compact 3-manifolds with boundary. Electron. Res. Announc. Amer. Math. Soc. 11 (2005) 12–20 10. Guggenheimer, H.W.: Differential Geometry. Dover Publications (1977) 11. Gu, X., Yau, S.T.: Computational Conformal Geometry. Higher Education Press, China (2007) 12. Mostow, G.D.: Quasi-conformal mappings in n-space and the rigidity of the hyperbolic space forms. Publ.Math.IHES 34 (1968) 53–104 13. Si, H.: Tetgen: A quality tetrahedral mesh generator and three-dimensional delaunay triangulator. (http://tetgen.berlios.de/)
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Fig. 4. An example of hyperbolic 3-manifold, Thurston’s knotted Y shape, constructed from a solid ball with three entangled tunnels removed. (a) and (b) show the boundary surface, (c) and (d) show the internal tessellation with tetrahedra. v4 f2 v3
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Fig. 5. Realization of a truncated hyperbolic tetrahedron (c) in H3 by taking CSG among hemispheres (b) based on their intersection circles with the infinity plane z = 0 (a).
Fig. 6. Fundamental domain of the Y shape: the single period realization of the canonical metric in H3 from various views.
Fig. 7. Universal Covering Space of the Y shape: the multiple period realization of the canonical metric in H3 from various views.
Computer Science Department, State University of New York at Stony Brook {xyin,gu}@cs.sunysb.edu 2 Center for Advanced Computer Studies, University of Louisiana at Lafayette [email protected] 3 Department of Mathematics, Rutgers University, [email protected]
Abstract. Almost all three dimensional manifolds admit canonical metrics with constant sectional curvature. In this paper we proposed a new algorithm pipeline to compute such canonical metrics for hyperbolic 3manifolds with high genus boundary surfaces. The computation is based on the discrete curvature flow for 3-manifolds, where the metric is deformed in an angle-preserving fashion until the curvature becomes uniform inside the volume and vanishes on the boundary. We also proposed algorithms to visualize the canonical metric by realizing the volume in the hyperbolic space H3 , both in single period and in multiple periods. The proposed methods could not only facilitate the theoretical study of 3-manifold topology and geometry using computers, but also have great potentials in volumetric parameterizations, 3D shape comparison, volumetric biomedical image analysis and etc.
1
Introduction
Many practical problems in computer graphics and geometric modeling can be reduced to computing special metrics (i.e. edge length) on a given geometric object represented as a mesh. One of the most widely considered problems is parameterization, which maps a given surface or volume mesh to a parameter domain with regular and simple shape; in another word, assigns a new set of edge length to the original mesh. Furthermore, many applications need a metric with special curvature properties. Intuitively, curvature measures how the space is locally distorted with respect to the Euclidean space. In many applications it is natural to ask for a metric with uniformly distributed curvature. We call it a constant curvature metric. Actually the existence of such canonical metrics on surfaces is justified
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by the Uniformization Theorem [1]. Similarly, most 3-manifolds (like those we are studying) admit constant curvature metric. In the surface parameterization literature, there are many works of computing metrics with special curvature distributions; the readers are referred to the parameterization survey papers [2] and [3] for general information. Among all those works, there are several which are able to compute the constant curvature metric for general surfaces, such as [4] and [5]. For volumes (or 3-manifolds), on the other hand, there is less work on computing particular metrics in the computer graphics literature. Most of such work is based on parameterization. To name a few, Wang et al. [6] parameterize brain volumes with solid ball via 3D harmonic mapping. Li et al. [7] computes 3D harmonic deformation between volumes with the same topology. Martin et al. [8] use harmonic functions to parameterize femur volumes, which are topologically balls with special shapes, by extending the boundary surface parameterization into the inside of the volume. All these methods assign a metric to the volume in some sense, but none of them target at uniform curvature distributions. In this work, we investigate the computation of constant curvature metric for 3-manifolds. In particular, we focus our attention on a special class of volumes, called hyperbolic 3-manifolds, whose boundaries are surfaces of genus greater than one, such as the Thurston’s knotted Y shape in figure 4. These volumes can be used to model some human soft tissues or biochemical structures. The computation is based on discrete 3D curvature flows, where the metric is evolving under the driven of the curvature. Luo [9] laid down the theoretical foundations of discrete curvature flows for hyperbolic 3-manifolds with boundaries. Our computational method is directly based on that work. The contribution of this paper can be briefly outlined as follows. 1. We proposed a discrete computational method based on curvature flow for hyperbolic 3-manifolds with boundaries. The resulting canonical metric induces constant curvature inside the volume and zero curvature on the boundary (i.e. geodesic boundary). 2. We proposed an algorithm to realize and visualize such canonical metrics in the hyperbolic space H3 . Both the single period representation (i.e. fundamental domain) and multiple period representation (i.e. universal covering space) can be computed. 3. We analyzed the convergence and stability of the proposed algorithms through experiments, and pinpointed several potential applications in both science and engineering fields. In the rest of the paper, we will introduce some underlying concepts and theories in section 2. Then the algorithm pipeline is presented in section 3, together with detailed discussion for each step. In section 4 we show the convergence and performance of the proposed methods, and pinpoint several potential applications. Finally we conclude the paper in section 5.
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Concepts and Theories
In this section, we provide some background knowledge that is necessary to understand the algorithm pipeline. Only the most related concepts and theories are presented here; for further details, the readers are referred to textbooks like [10], [1] and [11]. Hyperbolic Tetrahedron and Truncated Hyperbolic Tetrahedron A 3manifold can be triangulated using tetrahedra. If one assigns a hyperbolic metric to a tetrahedron, it is called a hyperbolic tetrahedron, such as the one [v1 v2 v3 v4 ] shown in figure 1a, where each face fi is a hyperbolic plane, and each edge eij is a hyperbolic line segment. If a 3-manifold has boundaries, it can also be tessellated using truncated tetrahedron. In this case, the 3-manifold is called hyperideal triangulated. Again, a truncated tetrahedron assigned with hyperbolic metric is called truncated hyperbolic tetrahedron (figure 1b). It can be constructed from a hyperbolic tetrahedron by cutting off each vertex vi with a hyperbolic plane perpendicular to the edges eij , eik , eil . Each truncated hyperbolic tetrahedron has four right-angled hyperbolic hexagon faces (figure 1d), which can be glued to other hexagon faces and therefore become interior faces in a truncated tetrahedron mesh; and also four hyperbolic triangle faces (figure 1c), which will compose the boundary surface of the truncated tetrahedron mesh. y1
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Fig. 1. Hyperbolic tetrahedron (a) and truncated hyperbolic tetrahedron (b) with hyperbolic triangle faces (c) and right-angled hexagons (d).
The geometry (or the edge length) of the truncated tetrahedron can be determined by the six dihedral angles {θ1 , θ2 , · · · , θ6 } (figure 1b). For a hyperbolic triangle (figure 1c) with inner angles {θi , θj , θk }, the edge length xi (the one against θi ) can be determined by the hyperbolic cosine law: cosh xi = (cos θi + cos θj cos θk ) / (sin θj sin θk ). For a right-angled hyperbolic hexagon (1d) with three cutting edge {xi , xj , xk }, its internal edge yi (which is against xi ) can be computed using another hyperbolic cosine law: cosh yi = (cosh xi + cosh xj cosh xk ) / (sinh xj sinh xk ).
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By using these hyperbolic cosine laws, all the edge lengths can be computed from the set of dihedral angles. On the other hand, the dihedral angles can also be uniquely determined by the edge lengths using the inverse cosine laws. Discrete Curvature For surfaces represented as triangle meshes, the discrete curvature is represented as the angle deficit around vertices. For aPvertex vi , P the curvature K(vi ) equals 2π − jk αijk for internal vertex or π − jk αijk for boundary vertex, where {αijk } are the surrounding corner angels (figure 2a). For 3-manifolds represented as tetrahedron meshes, there are two types of discrete curvature representations: vertex curvature and edge curvature. For tetrahedron [vi , vj , vk , vl ], let {αijkl , αjkli , αklij , αlijk } denote the solid angles at the verkl tices (figure 2b), βij be the dihedral angle on edge eij (figure 2c). The vertex P curvature K(vi ) is also defined as angle deficit: 4π − jkl αijkl for interior vertex P or 2π − jkl αijkl for boundary vertex. The edge curvature K(eij ) is defined P P kl kl 2π − kl βij if eij is an interior edge, or π − kl βij otherwise. The two types of discrete curvatures are closely related. Actually, the vertex curvature can be completely determined by the edge curvature: K(vi ) = P j K(eij ). vi
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Discrete Curvature Flow Curvature flow refers to a flow of the metric driven by the curvature. Given a hyperbolic tetrahedron with edge lengths lij and dihedral angles θij , the volume V of the tetrahedron is a function of the dihedral angles V = V (θ12 , θ13 , θ14 , θ23 , θ24 , θ34 ), and the P Schlaefli formula can be expressed as ∂V /∂θij = −lij /2, and thus dV = −1 ij lij dθij . It can be further 2 proved that the volume of a hyperbolic truncated tetrahedron is a strictly concave function of the dihedral angles ([9]). From this volume function, Luo ([9]) defines another energy function, which is a strictly convex function of the edge length, and the discrete curvature flow is defined as the negative gradient flow of the energy function.
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The curvature flow for 3-manifolds owns some special properties. One of the most prominent properties is the so called Mostow rigidity [12]. It states that the geometry of a finite volume hyperbolic manifold (for dimension greater than two) is determined by the fundamental group. Intuitively, different 3-manifolds have equivalent constant curvature metrics if they have the same topology. As a consequence, the tessellation will not affect the computational results the discrete curvature flow on 3-manifolds. Utilizing this special property, we are allowed to reduce the computational complexity of 3-manifold curvature flow by using the simplest tessellation for a given 3-manifold, as we will see in the algorithm section 3.1. For example, the Thurston’s knotted Y shape can be either represented as a high resolution tetrahedral mesh (figure 4d) or a mesh with only 2 truncated tetrahedra (figure 3a), and the resulting canonical metrics are identical. Hyperbolic Space Model There are several realizations of the 2D and 3D hyperbolic space. In this work, we use the upper half plane model for the 2D hyperbolic space H2 = {(x, y) ∈ R2 |y > 0}, with the Riemannian metric ds2 = dx2 + dy 2 /y 2 . In this model, hyperbolic lines are circular arcs and half lines orthogonal to the x-axis. The rigid motion is given by the so-called M¨ obius transformation, (az + b)/(cz + d), where ac − bd = 1, a, b, c, d ∈ R. Similarly, we use the upper half space model to realize 3D hyperbolic space H3 = {(x, y, z) ∈ R3 |z > 0}, with Riemannian metric ds2 = dx2 + dy 2 + dz 2 /z 2 . In H3 , the hyperbolic planes are hemispheres or vertical planes, whose equators are on the xy-plane. The xy-plane represents all the infinity points in H3 . The rigid motion in H 3 is determined by its restriction on the xy-plane, which is a M¨ obius transformation on the plane, in the form of (az + b)/(cz + d), where ac − bd = 1, and a, b, c, d ∈ C.
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Given a 3-manifold’s boundary surface, represented as a triangular mesh, our algorithm pipeline will go through the following steps: 1. Tessellate the volume with tetrahedra, and simplify the tessellation to minimum number of truncated tetrahedra; (section 3.1) 2. Compute the canonical hyperbolic metric using discrete 3D curvature flow; (section 3.2) 3. Realize and visualize the volume with the computed canonical metric in hyperbolic space H3 . (section 3.3)
3.1
Tessellation and Simplification
Given the boundary surfaces of a 3-manifold, we use the volumetric Delaunay triangulation algorithm introduced in [13] to tessellate the interior of the volume with tetrahedra. Then the tessellation is simplified by the following steps.
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Fig. 3. Simplified tessellation of the Y shape with only two truncated tetrahedra (a) glued together according to the pattern (b). The simplification is carried out through edge collapsing, which turns (c) into (d) by identifying v0 and v6 .
Denote the boundary surface of a 3-manifold M as ∂M = {S1 , S2 , · · · , Sn }. First, create a cone vertex vi for each boundary component Si ; for each triangle face fj ∈ Si , create a new tetrahedron Tji , whose vertex set consists of vi and the vertices of fj . In this way, M is augmented with a set of cone vertices and a set of new tetrahedra. Second, take the edge collapsing operation (shown in figure 3c-d) iteratively, until all the vertices are removed except for those cone vertices {v1 , v2 , · · · , vn } created in the previous step. Denote the resulting tetrahedral ˜ . Finally, for each tetrahedron T˜i ∈ M ˜ , trim off its vertices (which are mesh as M cone vertices) by the original boundary surface, and thus make it a truncated tetrahedron, denoted as Ti . As an example, the Y shape can be simplified to two truncated tetrahedra {T1 , T2 } (figure 3a), together with the gluing pattern (figure 3b) among their hexagon faces. For each Ti , let {Ai , Bi , Ci , Di } be its four hexagon faces, let {ai , bi , ci , di } be the truncated vertices. Its gluing pattern is given as follows, where the arrow → means to identify the former and the later: A1 B1 C1 D1
→ B2 → A2 → C2 → D2
{b1 → c2 , d1 → a2 , c1 → d2 } {c1 → b2 , d1 → c2 , a1 → d2 } {a1 → a2 , d1 → b2 , b1 → d2 } {a1 → a2 , b1 → c2 , c1 → b2 }
Please note that the edge collapsing operation does not change the fundamental group of the volume, and the simplified 3-manifold is topologically equivalent to the original 3-manifolds. From the discussion in section 2, it guarantees the later computation can be carried out on the simplified tessellation instead of the original one. 3.2
Metric Computation via Curvature Flow
Given an hyperideal triangulated 3-manifold, define the discrete metric function as x : E → R+ , where E is the set of edges in the triangulation. The discrete curvature function can be defined as K : E → R. Given an edge eij ∈ E, its
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edge length and edge curvature can be represented as xij and Kij respectively. From the discussion in section 2, the edge curvature can be determined by the dihedral angles, which in turn is a function of edge length. Therefore, the set {Kij } can be calculated from {xij }. Then the discrete curvature flow is then defined as dxij = Kij , (1) dt From this differential equation, the deformation of the metric is driven by the edge curvature, and the whole process is like a heat diffusion. Any numerical method for solving the discrete heat diffusion problem can be applied to solve the curvature flow equation. And in practice, we set the initial edge length to be xij = 1. P 2 During the flow, the total edge curvature ij Kij is strictly decreasing. When the flow reaches the equilibrium state, both the edge curvature and the vertex curvature vanish. The boundary surface will become a hyperbolic geodesic, while all the curvature (which is negative) is uniformly distributed within each truncated hyperbolic tetrahedron. Due to the fact the total curvature is negative, the resulting metric is a hyperbolic one. 3.3
Realization and Visualization
Once the canonical hyperbolic metric is computed, one is ready to realize it in the hyperbolic space H3 . There are two ways to realize the metric. The first one is a single period representation (figure 6), which is a union of multiple truncated hyperbolic tetrahedra. The second is a multiple period representation (figure 7), which is composed by multiple copies of the single period representation glued together nicely. As explained in section 2, we will use the upper half space and upper half plane as the model for H3 and H2 respectively. Realization of a Single Truncated Hyperbolic Tetrahedron Given the edge length of a truncated hyperbolic tetrahedron, its dihedral angles will be uniquely determined and the truncated tetrahedron can be realized in H3 uniquely up to rigid motions. Actually, its embedding is determined by the position of its four right-angled hexagon faces f1 , f2 , f3 , f4 and that of its four triangle faces v1 , v2 , v3 , v4 . Each of these faces now is a hyperbolic plane (i.e. semi-sphere, see figure 5b), separating H3 into two half spaces. By choosing the right half space for each face and taking the intersection of all these half spaces, one will get the realization of the truncated hyperbolic tetrahedron (figure 5c). To compute the position of each hyperbolic plane, let’s consider its intersection with the infinity plane z = 0, which is an Euclidean circle (figure 5a). Here we reuse the symbol fi and vj to represent the intersection circle by the hyperbolic plane fi and vj respectively. As shown in figure 5a, all the circles can be computed explicitly, such that circle fi and circle fj intersect at the given dihedral angle θk , while circle vi is orthogonal to circles fj , fk , fl . In order to remove the ambiguity caused by rigid motion, we fix circle f1 to be line y = 0, f2 to be line y = tan θ1 x, and normalize the circle f3 to have radius 1.
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Once the intersection circles are settled, we can directly construct hemispheres (i.e. hyperbolic planes) whose equators are those circles. By choosing the right half space for each hemisphere and using CSG operations to compute the intersection of these half spaces, we can get a visualization of a single truncated hyperbolic tetrahedron as shown in figure 5c.
Realization of a Single Period A single period representation of the whole 3-manifold with canonical hyperbolic metric is a union of all its constituting truncated hyperbolic tetrahedra. It can be constructed as the following. First, realize one truncated hyperbolic tetrahedron T0 as explained above. Then, pick another not-embedded truncated hyperbolic tetrahedron T1 , which is neighboring to T0 through hexagon faces f1 ∈ T1 with f0 ∈ T0 . Compute a M¨obius transformation in H3 , which rigidly move T1 to such a position that f1 ∈ T1 can be perfectly glued to f0 ∈ T0 . Now we get a partially embedded volume. Repeat the process of picking, moving and gluing a neighboring truncated hyperbolic tetrahedron, until the whole volume is embedded. The above algorithm is essentially a bread-first-search (BFS) in the given 3manifold; it results in a tree spanning all the truncated hyperbolic tetrahedra in the volume. Due to the nature of the constant curvature hyperbolic metric, such gluing (or spanning) operation can be carried out seamlessly, until finally all the truncated tetrahedra are glued together nicely into a simply connected volume, which is a topological ball. Such a single period representation is usually called the fundamental domain for the original volume (see [11]). Figure 6 visualizes fundamental domain of the Y shape embedded in H3 . Realization of Multiple Periods A multiple period representation of the canonical hyperbolic metric is a union of multiple copies of the fundamental domain, and is usually called the universal covering space (UCS) of the original 3-manifolds (see [11]), which is also a simply connected topological ball. Similar to the realization of the fundamental domain, the UCS representation can also be constructed through a sequence of gluing operations; the difference is, the primitive construction block is the embedded fundamental domain rather than the truncated hyperbolic tetrahedron. The gluing operation here can be explained as the following. Recall the algorithm for realizing a fundamental domain, any two truncated hyperbolic tetrahedra are glued through a pair of hexagon faces. After the algorithm is complete, there will be some hexagon faces left open, without gluing to any neighboring face. This is natural because otherwise the fundamental domain will not be simply connected. All the open hexagon faces are grouped into several connected components, each component constitutes a gluable face for the whole fundamental domain. All the gluable faces can be coupled nicely; that is, for each gluable face, there exist another unique gluable face in the same fundamental domain such that they are able to attach to each other nicely. Actually, the fundamental domain can be viewed as a result of cutting the original volume open through
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the coupling gluable faces. And two copies of the fundamental domain can be glued together through one pair of the gluable faces. Different to the construction of one fundamental domain, the gluing operation among fundamental domains can be repeated infinitely many times, getting infinitely many copies involved in the UCS. In practice, we can only afford to realize a finite portion of the UCS, as the one visualized in figure 7.
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Experiments and Applications
We tested our algorithms on about 120 hyperbolic 3-manifolds, including the Thurston’s Y shape, tessellated with truncated hyperbolic tetrahedra. All the experiments converge, and the resulting numerical metrics are perfectly consistent with the results computed using algebraic methods, by a difference less than 1e − 8. From the experiments, we also notice that the stability of the algorithms depends on two issues: the initial metric and the tessellation. Due to numerical errors, for certain initial assignments of edge lengths, the angle computed using hyperbolic cosine law is too close to zeros and thus leads to instability. In our experiments, an initial metric with all edge lengths being 1.0 will always lead to the desired solution. On the other hand, the curvature flow may fail on certain tetrahedralization for the given 3-manifolds; it is probably because the critical point of the volume energy is too close to the boundary of the metric space. It remains an open problem to find the conditions under which the tessellation will guarantee the convergence to the desired metric. Regarding applications, the canonical metrics computed on surfaces have brought lights to huge amount of applications in geometric modeling, computer graphics, computer vision, visualization and biomedical data analysis. We believe that our methods for 3-manifolds will also play fundamental roles in many science and engineering fields in the future. And here we only address some of the potential applications in very brief words, while leaving the detailed discussions of particular applications to future exploration, which is far beyond the scope of this paper. Firstly, the theoretical study of the 3-manifold topology and geometry has been a hot area in the mathematical society for many years; but most 3-manifolds cannot be realized in the Euclidean space and even hard to imagine. we hope that with our work the researchers could gain more insights into the structure of 3-manifolds and therefore facilitate their study, at least in a discrete sense. Secondly, The realization of the canonical metric actually gives a canonical domain for the original 3-manifolds. This domain might be used for the purpose of volumetric texture mapping, volume discretization and remeshing, volume spline construction, volume registration and comparison, and etc, just like how the canonical parameter domains of surfaces help tackle the 2D problems. Thirdly, we noticed that the volumes we studied here could be good models for some volumetric biomedical data, such as some soft tissues in the human body with the penetrating vessels removed, or the complementary space of some
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protein structures. It will be a very interesting research topic to apply our computational methods to these biomedical applications.
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Conclusion
In this work we proposed a new computational method to compute constant curvature metrics for hyperbolic 3-manifolds with high genus boundary surfaces. The algorithm is directly based on discrete volumetric curvature flow. Experiments show the convergence and stability of the algorithm. In order to visualize the metric, we proposed algorithms to realize the metric in the upper half space model of the hyperbolic space. The whole pipeline can not only facilitate the theoretical study of 3-manifold topology and geometry, but also have potential applications in many engineering fields, like volume parameterization, biomedical data analysis and etc. How to apply our methods to those specific applications would be a very interesting future research direction. Meanwhile, it is also a challenging research topic to extend the current framework to other type of 3-manifolds.
References 1. Petersen, P.: Riemannian Geometry. Springer (1997) 2. Sheffer, A., Praun, E., Rose, K.: Mesh parameterization methods and their applications. Foundations and Trendsr in Computer Graphics and Vision 2 (2006) 3. Floater, M.S., Hormann, K.: Surface parameterization: a tutorial and survey. In: Advances in Multiresolution for Geometric Modelling. Springer (2005) 157–186 4. Jin, M., Kim, J., Luo, F., Gu, X.: Discrete surface ricci flow. IEEE Transaction on Visualization and Computer Graphics (2008) 5. Springborn, B., Schr¨ oder, P., Pinkall, U.: Conformal equivalence of triangle meshes. SIGGRAPH 2008 (2008) 6. Wang, Y., Gu, X., Thompson, P.M., Yau, S.T.: 3d harmonic mapping and tetrahedral meshing of brain imaging data. In: Proceeding of Medical Imaging Computing and Computer Assisted Intervention (MICCAI), St. Malo, France. (2004) 7. Li, X., Guo, X., Wang, H., He, Y., Gu, X., Qin, H.: Harmonic volumetric mapping for solid modeling applications. In: Proceeding of Symposium on Solid and Physical Modeling. (2007) 109–120 8. Martin, T., Cohen, E., Kirby, M.: Volumetric parameterization and trivariate bspline fitting using harmonic functions. In: Proceeding of Symposium on Solid and Physical Modeling. (2008) 9. Luo, F.: A combinatorial curvature flow for compact 3-manifolds with boundary. Electron. Res. Announc. Amer. Math. Soc. 11 (2005) 12–20 10. Guggenheimer, H.W.: Differential Geometry. Dover Publications (1977) 11. Gu, X., Yau, S.T.: Computational Conformal Geometry. Higher Education Press, China (2007) 12. Mostow, G.D.: Quasi-conformal mappings in n-space and the rigidity of the hyperbolic space forms. Publ.Math.IHES 34 (1968) 53–104 13. Si, H.: Tetgen: A quality tetrahedral mesh generator and three-dimensional delaunay triangulator. (http://tetgen.berlios.de/)
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Fig. 4. An example of hyperbolic 3-manifold, Thurston’s knotted Y shape, constructed from a solid ball with three entangled tunnels removed. (a) and (b) show the boundary surface, (c) and (d) show the internal tessellation with tetrahedra. v4 f2 v3
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Fig. 5. Realization of a truncated hyperbolic tetrahedron (c) in H3 by taking CSG among hemispheres (b) based on their intersection circles with the infinity plane z = 0 (a).
Fig. 6. Fundamental domain of the Y shape: the single period realization of the canonical metric in H3 from various views.
Fig. 7. Universal Covering Space of the Y shape: the multiple period realization of the canonical metric in H3 from various views.
