Two Counterexamples of Global Differential Geometry for Polyhedra Abdˆenago Barros, Esdras Medeiros and Romildo Silva Departatamento de Matem´ atica Universidade Federal do Cear´ a 60450-520 Fortaleza-Cear´ a, Brazil
Abstract In this article we show, by giving counterexamples, that two classical results of global differential geometry are not valid for polyhedral surfaces. Considering the set of compact smooth surfaces we exhibit counterexamples concerning the Cohn-Vossen theorem and the existence of elliptical points. The key point on the construction of such polyhedral surfaces is the application of a nice discrete version of the Gaussian curvature. Keywords: Gauss-Bonnet, Cohn-Vossen, polyhedra, discrete curvature 2000 MSC: 52B70, 53C45 1. Introduction The study of polyhedral surfaces in the context of differential geometry finds its motivation in both pure and applied mathematics. Consider for example the pioneering work of Alexandrov [2] which investigates the geometry of surfaces by approximating by polyhedral metrics. In other fields such as computer graphics and numerics, we encounter a strong need for a discrete differential geometry of arbitrary meshes (see for instance Bobenko et al. [1] and Meyer et al. [7]). There are many classical results of differential geometry extended to other topological spaces. One fundamental theorem for a given smooth surface S is Email address:
[email protected],
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[email protected] (Abdˆenago Barros, Esdras Medeiros and Romildo Silva)
Preprint submitted to Elsevier
October 19, 2011
the celebrated Gauss-Bonnet theorem which links the topology of S and its total curvature. In the sixties, Banchoff [3] showed that this theorem is also valid for any compact polyhedral surface thanks to the nice discretization of the Gaussian curvature established by Alexandrov [2]. More recently, Forman [5] extended Morse Theory to cell complexes. Such examples of connections between smooth and discrete surfaces motivated us to investigate other similarities of global differential geometry and topology in polyhedral surfaces. Among then we are interested in studying two results intrinsically related to the Gaussian curvature. They are: • A little variation of the Gauus-Bonnet theorem is the Cohn-Vossen theorem that relates the topology of S with its total absolute curvature. We build a simple polyhedral surface topologically equivalent to the torus for which the Cohn-Vossen theorem does not work. This is proved analytically after some elementary computations of the discrete Gaussian curvature. • A fundamental theorem concerning the existence of elliptical points, more specifically, the existence one point with positive Gaussian curvature. By the Gauss-Bonnet theorem, this result is guaranteed for polyhedra with genus zero or one. However we can create a complicated geometric shape topologically equivalent to the 2-torus with no elliptical vertices, more than that, every vertex has strictly negative discrete Gaussian curvature. Our method follows a recent trend of computer assisted proofs [6] which means that the proof is established by numerical analysis tools. The article is organized as follows. The next section gives an explanation on smooth surfaces by giving some basic results. In the section 3 we explore the Gauss-Bonnet theorem for polyhedral surfaces and study the existence of elliptical points in surfaces with genus zero and one. In section 4 we show the Cohn-Vossen theorem counterexample. In section 5 we show the 2-torus counterexample for the non-existence of elliptical points the interval arithmetic method is applied to validate the model. We also give a simple generalization for genus greater than two. In section 6 we finish the article with a discussion.
2
2. Total curvature and Cohn-Vossen theorem The approach given originally by Gauss to define the total curvature of a neighborhood U on a smooth surface M 2 in the Euclidean space R3 is the best way to deal with extrinsic curvature theory. Indeed, according to Gauss the total curvature of a small convex neighborhood U1 on which the Gauss map g is injective and orientation-preserving is defined as the area of e 1 ) = A(g(U1 )), whereas, for non the spherical image g(U1 ) on S2 , i.e., K(U convex regions V on which g is injective, but orientation-reversing, its total curvature is minus the quoted area g(V ), see e.g. Banchoff [3] and Do Carmo [4]. With this approach we may have the Gaussian curvature at a point p ∈ U as the following limit: e i) K(U K(p) = lim , di →0 A(Ui ) where di stands for the diameter of Ui , while Ui is a neighborhood of p. Then we also have Z e K(U) = K(p)dA, U
where dA is the area measure of M 2 . Finally the famous Gauss-Bonnet theorem states that the total curvature of M 2 is given by Z 2 e K(M ) = K(p)dA = 2πχ(M 2 ), M2
where χ(M 2 ) denotes the Euler characteristic of M 2 . The Cohn-Vossen theorem for closed surfaces which deals with the total R absolute curvature, i.e. U | K(p) | dA, is a beautiful remark of the GaussBonnet theory combined with the fact that the Gauss map is onto when restricted to the almost convex part of M 2 . Indeed, if we let M + = {p ∈ M 2 : K(p) ≥ 0} and M − = {p ∈ M 2 : K(p) < 0} we have that the Gauss map restrict to M + is onto S2 . As consequence we have Z K(p)dA ≥ 4π. (1) M+
In order to deduce the above inequality we argue as follows: Given a direction ξ on R3 we move a perpendicular plane to ξ so far from the surface until this touches the surface at the first time at a point p0 . At least at a point p0 we have two nice consequences: The first one tell us that the 3
Gauss map is ±ξ, so we may choice our direction in such way that the Gauss map g(p0 ) = ξ whereas the second one gives us that the Gaussian curvature K(p0 ) > 0, since the surface at p0 is more curved than the Euclidean sphere centered in the origin with radius | p0 |, i.e. we have an elliptic point in M 2 . Next we write Z Z Z K(p)dA = K(p)dA + K(p)dA (2) M2
and
Z
| K(p) | dA = M2
M+
M−
Z
Z
K(p)dA − M+
K(p)dA
(3)
M−
Adding equations (2) and (3) and using Gauss-Bonnet theorem we arrive at Z Z | K(p) | dA = 2 K(p)dA − 2πχ(M 2 ). (4) M2
M+
In order to deduce Cohn-Vossen theorem it is enough to apply inequality (1) to equation (4). Indeed, after that we obtain: Z | K(p) | dA ≥ 2π(4 − χ(M 2 )). (5) M2
3. Polyhedral Surfaces: Discrete curvatures A polyhedral surface M(V, E, F ) is a set V of vertices, E edges and F faces such that each edge is the boundary of exactly two faces. The term polyhedron refers to a closed polyhedral surface. In this setting the polyhedron is closed, but might be not simple, i.e. non homeomorphic to the sphere. Without loss of generality we assume that the polyhedra are simplicial complexes which means that all the faces in F are triangular. The following definition will be useful to the next theorem: Definition 1. The star Str(v) of a vertex v is the union of all the faces and edges that contain this vertex. A polyhedron embedded in Euclidean space R3 may be seen as a piecewise linear surface. As we know, the non differentiability at the vertices restricts the computation of the curvatures by means of the tools from the differential geometry on smooth surfaces. 4
Despite the fact that the Gaussian curvature is not well defined for polyhedra, there is a definition which captures this concept in a natural way (see Banchoff [3]). Based on this work, we may establish a result which mimics the classical Gauss-Bonnet theorem for polyhedra: Theorem 1. Let P be a polyhedron. Then X K(v) = 2πχ(P ) v∈V
P where K(v) = 2π − θ (∗) and θ = αi is the total angle around a vertex v, and αi are those angles of the faces in the Str(v) that are incident to v. The theorem above suggests that K(v) is simply the Gaussian curvature around the vertex v. Indeed, equation (∗) is the definition of discrete Gaussian curvature. It follows thatPthe existence of elliptical points for polyhedra with genus zero holds since v∈V K(v) = 2πχ(P ) = 4π. However, for toroidal polyhedra, we can only affirm that there no exists anyone with negative curvature at all vertices. The discrete Gaussian curvature gave us an important contribution in validating the Gauss-Bonnet theorem for polyhedra. Meanwhile, some results such as the Cohn-Vossen theorem and the existence of elliptical points for polyhedra with genus greater than one remain open. Next, we construct counterexamples for each one. 4. Tori of Small Total Curvature Let ∆′ and ∆ be two concentrical equilateral triangles on the plane ′ ′ ′ Γ = {(x, y, z) ∈ R3 : z = 0 with vertices at z01 , z02 , z03 and z01 , z02 , z03 , ′ respectively, in such way that ∆ ⊂ ∆. Denote Tt : R3 → R3 the vertical translation given by Tt (x, y, z) = (x, y, z + t). Let zλi = Tλ (z0i ) and ′ ′ ′ zλi = Tλ (z0i ), for i = 1, 2, 3. Notice that for each i, zλi and zλi lie in the 3 plane z = λ. Now we consider the polyhedron Tλ ⊂ R whose set of vertices ′ is Vλ = {z±λi , z±λi , i = 1, 2, 3}. We notice that the Gauß curvature at each vertex is given by K(z±λi ) = 2π −
π π π 2π + + = 2 2 3 3
while 5
′ K(z±λi ) = 2π −
Then we have
V X
π π π 2π + + 2π − =− . 2 2 3 3
K(vi ) = 6 ×
i=1
2π 2π +6× − =0 3 3
while V X
|K(vi )| = 6 ×
i=1
2π 2π +6×| − | = 8π. 3 3
Now we consider the polyhedron T⋆λ ⊂ R3 (fig. 1) by degenerating the ′ sides zλi zλi to the vertex z0i , for i = 1, 2, 3. Hence the set of vertices of T⋆λ is ′ given by Vλ⋆ = {z0i , zλi , z−λi , i = 1, 2, 3} whose set of edges is given by ′ ′ ′ ′ ′ E ⋆ = {z01 z02 , z01 zλ1 , z01 z−λ1 , z02 z03 , z02 zλ2 , z02 z−λ2 , z03 z01 , z03 zλ3 , z03 z−λ3 }. ′ , as well as Now let us denote by α the angle between z01 z02 and z01 zλ1 ′ with respect to the other edges z0i z0i+1 and z0i z±λi . In order to compute the Gauß curvature at each vertex we notice that
K(vi ) = 2π − 4α, for vi ∈ {z0i } whereas π π ′ + + π − α + π − α = 2α − π, for vi ∈ {z±λi }. 2 2 Taking into account that α ∈ π6 , π2 we arrive at K(vi ) = 2π −
V X i=1
K(vi ) = 3 × 2π − 4α + 6 × 2α − π = 0
while V X τ T⋆λ = |K(vi )| = 3 × | 2π − 4α | + 6 × | 2α − π | = 12 π − 2α . i=1
From where we deduce τ (α) = τ T⋆λ = 12(π−2α). Therefore limπ τ (α) =
8π while limπ α→ 2
τ (T⋆λ )
α→ 6
= 0.
6
w Figure 1: Tλ (left) and T⋆λ (right)
5. 2-Torus with Strictly Negative Curvature Let us consider a simplicial complex that looks like a block with two holes, i.e., the 2-torus B = (V, E, F ) as showed on the left of figure 2 such that |V | = 24 and |F | = 52. The “inner” vertices, have negative curvature − π2 whereas the “outer” vertices have positive curvature π2 . P In this basic model the reader can easily verify the Gauss-Bonnet equation K(v) = −4π. Analogously to the previous section we transform the geometry of B by pushing up or down the inner vertices in the vertical direction. Let us again call such transformation as Ta (x, y, z) = (x, y, z + a), ∀a ∈ R. During this process, the curvatures of the outer vertices decrease whereas the inner vertices increase. Indeed, by the Gauss-Bonnet theorem the total curvature is invariant. A computer graphics system was implemented to solve a task too arduous or too tiring for any human. It allowed us to observe in real time the curvature changes of the vertices as they are moving through the transformation Ta . After some attempts, we were able to find out the polyhedron with strictly negative curvatures at all vertices, here denoted as B⋆ . In figure 2 we illustrate this transformation process and the resulting polyhedron B⋆ . The figure 3 represents the wired and the unfolded versions of B⋆ . In the appendix at the end of the article we have table 1 and table 2 that present the geometry and combinatorial topology of B⋆ respectively. Surprisingly B⋆ is also another counterexample of the P Cohn-Vossen theo⋆ rem. Since, using the computer, we obtained τ B = |K(vi )| = 2. Numerical Validation We computed numerically the curvature by using the dot product between two triangle edges and the arc cosine function (acos). All variables have 7
w Figure 2: B (left) and B⋆ (right)
(a)
(b)
Figure 3: wired (a) and unfolded (b) B⋆
double precision and, consequently, the results are accurate. However, the main concern for the 2-torus B⋆ relies on the difference of the “true” value x of the corresponding quantity and the computed value xˆ. To clarify it, in ˆ table 1, let us look at the smallest value (in modulo) of the curvatures K(p) and notice that it is reasonably near to zero. Now, consider the following ˆ question: how far from K(p) is its true value K(p)? If such distance is greater ˆ than k(p) then the true value can be positive, invalidating our example. One of the simplest and most efficient models to validate numerical computations is interval arithmetic [8, 9]. In such model an interval is a pair of numbers which represents all the numbers between these two. The fundamental property of interval arithmetic is: “if f is a function on a set of numbers, f can be extended to a new function defined on intervals. This new function f takes one interval argument and returns an interval such
8
Figure 4: Building a new handle.
that: ∀x ∈ [a, b], f (x) ∈ f ([a, b])”. Based on this simple numerical model, we evaluated the curvature intervals at all vertices. In our experimental results, the magnitude of the larger interval is 10−13 which is highly accurate considering that the magnitude of the smallest curvature is only 10−2 . Such analysis proves, in fact, that all the vertices of B⋆ have negative curvature. Generalization for Genus ≥ 3 Now we construct more examples of polyhedra without elliptical points by adding handles as much as desired. This can be achieved by perforating B assuming that it is not hollow. For instance, we can perform a CSG operation by subtracting from B a rectangular parallelepiped between the two genus and, consequently, generating a new one (see figure 4). Such operation is enough because we can take advantage of two properties. First, the perforation does not affect the transformation Ta , i. e., there is no selfintersections during the translation process of the vertices. Second, we may fine tune the size of the parallelepipeds to fit them appropriately between the two original genus. 6. Discussion In this paper, we showed two counterexamples of classical theorems in differential geometry for polyhedra. The first one was constructed analytically and contradicts the Cohn-Vossen theorem. On the other hand, the construction of a polyhedra without elliptical points (as conjectured by us initially) was a great challenge since all attempts to generate it analytically have failed. Indeed, the calculation of the curvature at each vertex demands much time. We even suspect that this conjecture was false, and we almost moved on to the proof of the existence of an elliptic point. Fortunately the 9
computer sped up the calculations of the discrete curvatures, which assisted us to find successfully the counterexample. In table 1 we observe that the ratio ρ between the maximal and minimal values of the curvatures at the vertices is of the order 102 . Hence, to advance in this work, we study the existence of a polyhedra with constant negative curvature or, more generally, look for the best ρ such that it is possible to generate a polyhedron with strictly negative curvature. Acknowledgments We would like to thank L. Jorge, J. Lira and C. Mota for discussions and suggestions to this paper. We also thank FUNCAP and CNPq for sponsoring our research. [1] John M. Sullivan Gnter M. Ziegler Alexander I. Bobenko, Peter Schrder. Discrete differential geometry. Birkhauser Verlag AG, 2008. [2] A. D. Alexandrov and V. A. Zalgaller. Intrinsic geometry of surfaces. Translation of Mathematical Monographs (AMS), 15, 1967. [3] T. F. Banchoff. Critical points and curvature for embedded polyhedral surfaces. The American Mathematical Monthly, 77(5):475–485, 1970. [4] M. P. Do Carmo. Differential Geometry of curves and surfaces. PrenticeHall, 1989. [5] R. Forman. Morse theory for cell complexes. Advances in Mathematics, 134:90–145, 1998. [6] Michael Joswig and Konrad Polthier. Digital models and computer assisted proofs. Newsletter of the European Mathematical Society (EMS), 2001. [7] Mark Meyer, Mathieu Desbrun, Peter Schr¨oder, and Alan H. Barr. Discrete differential-geometry operators for triangulated 2-manifolds. In Hans-Christian Hege and Konrad Polthier, editors, Visualization and Mathematics III, pages 35–57, Heidelberg, 2003. Springer-Verlag. [8] R. E. Moore. Interval Analysis. Prentice-Hall, 1966. [9] Ramon E. Moore and Fritz Bierbaum. Methods and applications of interval analysis (siam studies in applied and numerical mathematics). 1979. 10
Appendix The following tables describes the geometry and combinatorial topology of the 2-torus counterexample polyhedron B⋆ . index 6 3 11 4 19 8 1 5 9 2 7 10
x -0.5 2.5 0.5 -1.6 -1.6 0.5 -2.5 -0.5 1.6 2.5 -1.6 1.6
y 0.5 1.5 0.5 -0.5 0.5 -0.5 -1.5 -0.5 -0.5 -1.5 0.5 0.5
z 3.1 1.0 3.1 0.1 -0.1 3.1 1.0 3.1 0.1 1.0 0.1 0.1
curv. -0.012 -0.010 -0.012 -1.548 -1.548 -0.012 -0.010 -0.012 -1.548 -0.010 -1.548 -1.548
index 0 16 12 18 13 14 17 20 21 15 22 23
x -2.5 -1.6 -2.5 -0.5 -2.5 2.5 -0.5 0.5 1.6 2.5 1.6 0.5
y 1.5 -0.5 1.5 0.5 -1.5 -1.5 -0.5 -0.5 -0.5 1.5 0.5 0.5
z 1.0 -0.1 -1.0 -3.1 -1.0 -1.0 -3.1 -3.1 -0.1 -1.0 -0.1 -3.1
curv. -0.010 -1.548 -0.010 -0.012 -0.010 -0.010 -0.012 -0.012 -1.548 -0.010 -1.548 -0.012
Table 1: Vertices and curvatures
0 11 4 6 8 6 1 3 8 8 3 2 16
6 6 1 0 2 11 4 11 6 5 10 8 12
3 8 5 7 9 3 0 10 5 1 9 1 19
12 7 3 19 18 13 13 13 14 14 15 15 20
16 0 9 12 12 16 17 20 20 21 21 22 17
13 4 2 18 15 17 20 14 21 15 22 23 18
18 20 12 0 2 3 13 3 0 15 4 4 5
Table 2: Faces.
11
15 18 13 13 1 2 14 14 3 12 5 17 6
23 23 0 1 14 14 1 15 12 3 17 16 17
18 19 6 7 16 8 8 9 22 23 10 11 720
17 18 7 4 19 9 21 10 21 22 11 8 23
6 6 19 19 4 21 20 21 10 10 23 23 8
