The paramount importance of triangulations of surfaces and their ubiquity in various implementations (s.a. in numerous algorithms applied in robot (and computer) vision, computer graphics and geometric modelling, with a wide range of applications from industrial ones, to biomedical engineering to cartography and astrography – to number just a few) has hardly to be underlined here. In consequence, determining the intrinsic proprieties of the surfaces under study, and especially computing their Gaussian curvature is essential. However Gaussian curvature is a notion that is defined for smooth surfaces only, and usually attacked with differential tools, tools that – however ingenious and learned – can hardly represent good approximations for curvature of-surfaces, since they are usually just discretizations of formulas developed in the smooth (i.e. of class at least ) case.111 However one can find very scientifically sound discrete versions of Surface Curvature can be found, for instance, in [Ba2], [BCM], [C-SM] .
Moreover, since considering triangulations, one is faced with finite graphs, or, in many cases (when given just the vertices of the triangulation) only with finite –thus discrete – metric spaces. Therefore, the following natural questions arise: (A) Is one fully justified in employing discrete metric spaces when evaluating numerical invariants of continuous surfaces? and (B) Can one find discrete, metric equivalents of the differentiable notions, notions that are intrinsically more apt to describe the properties of the finite spaces under investigations? One is further motivated to ask the questions above, since the metric method we propose to employ have already successfully been used in the such diverse fields as Geometric Group Theory, Geometric Topology and Hyperbolic Manifolds, and Geometric Measure Theory. Their relevance to Computer Graphics in particular and Applied Mathematics in general is made even more poignant by the study of Clouds of Points (see [LWZL], [MD]) and also in applications in Chemistry (see [T]).
We show that the answer to both this questions is affirmative, and we focus our investigations mainly on the study of metric equivalents of the Gauss curvature. Their role is not restricted to that of being yet another discrete version of Gaussian Curvature, but permits us to attach a meaningful notion of curvature to points where the surface fails to be smooth, such as cone points and critical lines. Thus we can employ curvature in reconstructin not only smooth surface, but also surfaces with ”folds”, ”ridges” and ”facets”.
This exposition is organized as follows: in Section 2 we concentrate our efforts on the theoretical level and study the Lipschitz and Gromov-Hausdorff distances between metric spaces, and show that approximating smooth surfaces by nets and triangulations is not only permissible, but is, in a way, the natural thing to do, in particular we show that every compact surface is the Gromov-Hausdorff limit of a sequence of finite graphs.222 For the relevance of these notions in the study of classical curvatures convergence, see [CMS], [F] . In Section 3 we introduce the best candidate for a metric (discrete) version of the classical Gauss curvature of smooth surfaces, that is the Embedding, or Wald curvature. We study its proprieties and investigate the relationship between the Wald and the Gauss curvatures, and show that for smooth surfaces they coincide, so that the Wald curvature represents a legitimate discrete candidate for approximating the Gaussian curvature of triangulated surfaces. Section 4 is dedicated to developing formulas that allow the computation of Wald curvature: first the precise ones, based upon the Cayley-Menger determinants, and then we develop (after Robinson) elementary formulas that approximate well the Embedding curvature. We conclude with three Appendices. In the first Appendix we present three metric analogues for the curvature of curves, namely the Menger, Alt and Haantjes curvatures and study their mutual relationship. Furthermore we show how to relate to these notions as metric analogues of sectional curvature and how to employ them in the evaluation of Gauss curvature of triangulated surfaces. Next we present yet another metric analogue of surfaces curvature, based, in this case, upon a the modern triangle comparison method, namely the Rinow curvature. We investigate its proprieties and show (following Kirk ([K])) that in the case under investigation the Rinow and Wald curvatures coincide (and therefore Rinow curvature also identifies to the Gauss curvature). The third and last Appendix is dedicated to the development of determinant formula for the radius of the circumscribed sphere around a tetrahedron, with a view towards applications.
2. The Haussdorff-Gromov limits
2.1. Lipschitz Distance
This definition is based upon a very simple333 That is to say: very intuitive, i.e. based
upon physical measurements. idea: it measures the relative difference between metrics, more precisely it
evaluates their ratio; i.e.:
The metric spaces , are close iff s.t. .444 Here and in the sequel ”” etc. … stands as a short-hand version of ””.
Technically, we give the following:
The map is bi-Lipschitz iff s.t.:
The same definition applies for two different metrics on the same space .
Given a Lipschitz map , we define the dilatation of f by:
The dilatation represents the minimal Lipschitz constant of maps between and .
If is not Lipschitz, then .
bi-Lipschitz homeo. on its image.
We have the following results:
Let be Lipschitz maps. Then:
(a) is Lipschitz
is a vector space.
is a vector space.
Now we can return to our main interest and define the following notion:
Let , be metric spaces. Then the Lipschitz distance between and is defined as:
If bi-Lipschitz between and , then – remembering Remark 2.2 – we put (i.e. is not suited for pairs of spaces that are not bi-Lipschitz equivalent.)
having defined the distance between two metric spaces we now can define the convergence in this metric in the following natural way:
The sequence of metric spaces convergence to the metric space iff
(In this case we write: ).
Let be a family of regular surfaces, ; where is an open set, ; such that the family of parametrizations is smooth (i.e. ; where ). Then .
If is not smooth (only continuous) then we do not necessarely have that .
We have the following significant theorem:
The satisfies the following conditions:
(b) is symmetric;
(c) satisfies the triangle inequality;
Moreover, if are compact, then:
(d) (i.e. is isometric to );
is a metric on the space of isometry classes of compact metric spaces
Let us recall the following
as a real function; i.e.
(where ”” denotes uniform convergence.)
Then but . However, for finite spaces indeed .
2.2. Gromov-Hausdorff distance
This is also a distance between compact metric spaces ((distinguished) up to isometry!). However it gives a weaker topology (In particular: it is always finite (even for pairs of non-homeomorphic
spaces.) )555The relationship between the Lipschitz and the Hausdorff distances is akin to that between
the and norms in Functional Spaces.
We start by first introducing the classical
2.2.1. Hausdorff distance
Let . We define the Hausdorff distance between and as:
(see Fig. 1);
where is the -neighborhood of A, ; (here, as usual: .)
Another (equivalent) way of defining the Hausdorff distance is as follows:
(see Fig. 2)
We have the following
Let be a metric space. Then:
(a) is a semi-metric (on ). (i.e. .)
i.e. is a metric on the set of closed subsets of X.
Notation We put: .
if is compact and if is a sequence of compact subsets of , then:
For general subsets , and
If , and if the sets are all convex, then is convex sets.
We have the following two important results, which we present without their respective (lengthy) proofs:
complete complete .
(Blaschke) compact compact .
2.3. The Gromov-Hausdorff Distance
We are now able to define the Gromov-Hausdorff distance using the
following basic guide-lines: we want to get the maximum distance that satisfies the following two
(a) (i.e. set that are close as subsets of will still be close as abstract metric spaces);
(b) isometric to .
Let be metric spaces. Then the Gromov-Hausdorff distance between and is defined by:
where the infimum is taken over all the isometric embeddings into some metric space Z. (See Fig. 3).
If , with the spherical metric, and , with the Euclidian metric, then (!)
Let be an -net666 Definition
Let be a metric space, and let . is called an -net iff
. in . Then .
Proof Take .
It is sufficient to consider embeddings into the disjoint union of the spaces and , .
If , then 777.
However , the straightforward definition of
may be difficult to implement. Therefore we would like to estimate (compute)by comparing distances in vs. distances in (as done in the cases of uniform and Lipschitz metrics). We start by defining a correspondence between metric spaces: , given by correspondences between points .
A correspondence is not necessarely a function, that is to a single may correspond to several -’s.
We shall prove that
Formally, we have:
Let denote sets. A correspondence is a subset of the Cartesian product of
and : s.t.
Any surjective function represents correspondence
is a correspondence and ; surjective, s.t. .
Let be a correspondence between and , where are metric spaces. We Define the distortion of by:
(See (*) .)
If is a correspondence induced by a surjective function , then , where:
If , where are surjective functions, then:
iff is associated to an isometry.
We bring, without proof, the following theorem:
Let be metric spaces. Then:
where the infimum is taken over all the correspondences .
Before bringing the next result (which is very important in determining the topology ) we first introduce one more notion:
is called an -isometry (), iff
(ii) is an -net in .
Let be metric spaces and let . Then:
Proof (i) Let s.t. .
For any and , choose s.t. . Then defines a map . Moreover: .
We shall prove that is a -net in .
Indeed, let and s.t. . Then , thence .
Let be an -isometry. Define .
Then, since is an -net it follows that is a correspondence.
Then we have:
The next result is of great importance (in particular so in our context):
is a (finite) metric on the set of isometry classes of compact metric spaces.
It suffices to prove that .999We shall write:
if is isometric to .
Indeed, let be compact spaces s.t. . Then it follows from the previous Corollary (for ) that s.t. .
let . Using a Cantor-diagonal argument one easily shows that s.t. converges in . Without restricting the generality we may assume that this happens for itself. Thus we can define a function by putting: .
But . In other words is an isometry. But , therefore this isometry can be extended to an isometry from to . In a analogous manner one shows the existence of an isometry .
In fact, the following relationship exists between ”” and ””:
-nets in -nets in .
One can formulate this assertion in a more formal manner and it directly (see [G+], pg. 73). However we shall proceed in more ”delicate” manner, starting with:
Let be compact metric spaces, and let . are called
--approximations (of each-other) iff: ,
(i) is an -net in and is an -net in ;
An -approximation is called, for short: an -approximation.
The relationship between this last definition and the Gromov-Hausdorff distance is first revealed in
Let be compact metric spaces. Then:
If is a -approximation of , then .
is a -approximation of .
Proof (1) Condition (ii) of Def. 2.41. is equivalent to , where
. But . Now, since and are -nets in , resp. , it follows that . From here and from the follows, by means of he triangle inequality, that .
(2) By Cor. 2.37., there exists a -isometrie . Let be an -net, and let .
Then . Therefore suffice to prove that is a -net in .
Indeed, if , then, since is an -net in , s.t. . Now, since is an -net in , , s.t. .
Prop. 2.42. ( is an -approximation, large enough.)
More precisely we have the following Proposition:
Let compact metric spaces. Then:
a finite -net and a finite -net , s.t. and, moreover, , for large enough .
() If exist, then is an -approximation of
() Let be an finite -net in .
We construct in corresponding nets (to be more precise, we define: , where is an -approximation, .) Then and, in addition, is an -net in (for large enough).
We make the following extremely important Remark:
Let be an -dimensional manifold, of (sectional, Ricci) curvature , and s.t. . Then is compact. However, it should be noted that this result doesn’t hold for curvature . (only for and injectivity radius .
Note With the notations of the precedent Proposition, the distances in converge to the distances in
, as , therefore The Geometric Proprieties of will converge to those of . Thus
we can use the Gromov-Hausdorff each and every time The Geometric Proprieties of can be expressed in
term of a finite number of points, and, by passing to the limit, automatically obtain proprieties of .
A typical example is that of the intrinsic metric i.e. the metric induced by a length structure (i.e. path length) by a metric on a subset (of a given metric space). (See Fig. 4 for the classical example of surfaces in .)
On a more formal note, we have the following characterization of intrinsic metrics:
Let be a complete metric space.
If , then is strictly intrinsic.
If and the -middle of , then is intrinsic.
Where we used the following definitions and notations:
Given points in , the middle (or midpoint) of the segment (more correctly: ’a midpoint between ”” and ”” ’) is defined as:
is called strictly intrinsic iff the length structure is associated with is complete.
Let be an intrinsic metric. is an -middle (or an -midpoit) for iff:
The converse of Thm. 2.46. holds in any metric space, more precisely we have:
If is an intrinsic metric, then exists, .
The following Theorem shows that length spaces are closed in the GH-topology :
Let be length spaces and let be a complete metric space s.t. .
Then is a length space.
Proof We have already presented the idea of the proof: it is sufficient to show that for every there
exist an -midpoit ().
Indeed, let be such that . Then, from the a preceding result, it follows that there exist a correspondence s.t. .
Let , , . Since is a length space, s.t. -midpoint of . Consider . Then:
(Here we write instead of , etc.)
In a similar manner we show that: ; i.e. -midpoit of .
The next Theorem and its Corollary are of paramount importance:
Any compact length space is the GH-limit of a sequence of finite graphs.
Proof Let small enough, and let be a -net in
Let be the graph with and . we shall prove that is an -approximation of , for small enough (i.e. for ). (See Fig. 5.)
But, since is an -net both in and in , and since , it is sufficient to prove that:
Let be the shortest path between and , and let
s.t. (and . Since
s.t. , it follows that (See Fig. 6)
Therefore, (for ) an edge . From this we get the following upper bound for :
But ; therefore:
So, for any an -approximation of . Then, .
Let be a compact length space. Then is the Gromov-Hausdorff limit of a sequence of finite graphs, isometrically embedded in .
If , . If s.t.
then is a finite graph.
If condition is replaced by:
then will still be always a graph, but not necessarily finite(!)
3. The Embedding Curvature
3.1. Theoretical Setting
This is basically a comparison-curvature (as is the more ”modern” 101010 i.e. Cartan-Alexandrov-Topogonov approach). This is done with quadruples instead of triangles (like in the Alexandrov-Topogonov method). It is in a sense a more natural idea, since quadruples are classically111111 as illustrated by the time-honored principles of Projective Geometry… the ”minimal” geometric figures that allow the differentiation between metric spaces. This allows for a much more easier and rapid development of the theory than the triangle-based comparison. Moreover we shall show that the two Theories coincide on those metric space on which both can be applied, i.e. metric spaces that are (a) ”planar” and (b) ”rich enough” i.e. contain quadrangles, s.a. classical (PL-smooth) surfaces in .121212 In this sense CAT spaces are more ”potent”: they can be employed in studying mathematical objects that not (neccessarilly) contain quadrangles, e.g. trees, Cayley graphs, etc..
Let be a metric space, and let , together with the mutual distances: . The set together with the set of distances is called a metric quadruple.
One can define metric quadruples in slightly more abstract manner, without the aid of the ambient space: a metric quadruple being a point metric space; i.e. , where the distances verify the axioms for a metric.
Before we proceed to the next definition, let us introduce the following
Notation denotes the complete, simply connected surface of constant curvature , i.e. , if ; , if ; and , if . Here denotes the Sphere of radius , and stands for the Hyperbolic Plane of curvature , as represented by the Poincare Model of the plane disk of radius
The embedding curvature of the metric quadruple is defined be the curvature of into which can be isometrically embedded. (See Figures 7 and 8 for embeddings of the metric quadruple in and , respectively.)
We can now define the embedding curvature at a point in a natural way by passing to the limit (but without neglecting the existence conditions), more precisely:
Let be a metric space, and let be an accumulation point. Then is said to have Wald
(i) linear131313 The neighborhood of is called linear iff is contained in a geodesic. ;
(ii) s.t. and s.t.
If one uses the second (abstract) definition of the metric curvature of quadruples, then the very existence of is not assured, as it is shown by the following
The metric quadruple of lengths
admits no embedding curvature.
Even if a quadruple has an embedding curvature, it still may be not unique (even if is not liniar), indeed, one can study the following examples:
The quadruple of distances is isometrically embeddable both in and in .
The quadruple of distances admits exactly two embedding curvatures: and . (See [BM].)
However, for ”good” metric spaces141414 i.e. spaces that are locally ”plane like” the embedding curvature
exists and it is unique. And, what is even more relevant for us, this embedding curvature coincides with the
classical Gaussian curvature. The proof of this result is rather long and tedious, therefore we shall present here
only a brief sketch of it. (This will prove to be somewhat redundant anyhow, in view of the more general results
presented in the previous section, a fact but we shall emphasize later in our presentation).)
The Main ingredient for this proof, and for the analysis of yet another another approach to curvature (the CAT one) is provided by the following string of propositions (which are just generalizations of the well known high-school triangle inequalities):
Let and , s.t.