SURFACE TRIANGULATION {THE METRIC APPROACH
SURFACE TRIANGULATION { THE METRIC APPROACH
arXiv:cs/0401023v1 [cs.GR] 26 Jan 2004
EMIL SAUCAN Abstract. We embark in a program of studying the problem of better approximating surfaces by triangulations(triangular meshes) by considering the approximating triangulations as nite metric spaces and the target smooth surface as their Haussdor -Gromov limit. This allows us to de ne in a more natural way the relevant elements, constants and invariants s.a. principal directions and principal values, Gaussian and Mean curvature, etc. By a "natural way" we mean an intrinsic, discrete, metric de nitions as opposed to approximating or paraphrasing the di erentiable notions. In this way we hope to circumvent computational errors and, indeed, conceptual ones, that are often inherent to the classical, "numerical" approach. In this rst study we consider the problem of determining the Gaussian curvature of a polyhedral surface, by using the embedding curvature in the sense of Wald (and Menger). We present two modalities of employing these de nitions for the computation of Gaussian curvature.
1. Introduction 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 de ned for smooth surfaces only, and usually attacked with di erential tools, tools that { however ingenious and learned { can hardly represent good approximations for curvature of P L-surfaces, since they are usually just discretizations of formulas developed in the smooth (i.e. of class at least C 2 ) case.1 Moreover, since considering triangulations, one is faced with nite graphs, or, in many cases (when given just the vertices of the triangulation) only with nite {thus discrete { metric spaces. Therefore, the following natural questions arise: (A) Is one fully justi ed in employing discrete metric spaces when evaluating numerical invariants of continuous surfaces? and (B) Can one nd discrete, metric equivalents of the di erentiable notions, notions that are intrinsically more apt to describe the properties of the nite spaces under investigations? One is further motivated to ask the questions above, since the metric method we propose to employ have already Date: 24.01.2004. We would like to thank Prof. Gershon Elber of The Computer Science Department, The Technion, Haifa, who motivated and sustained this project. 1 However one can nd very scienti cally sound discrete versions of Surface Curvature can be found, for instance, in [Ba2], [BCM], [C-SM] . 1
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EMIL SAUCAN
successfully been used in the such diverse elds 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 a rmative, 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 e orts on the theoretical level and study the Lipschitz and Gromov-Hausdor 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-Hausdor limit of a sequence of nite graphs. 2 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: rst 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 rst 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 identi es 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 de nition is based upon a very simple 3 idea: it measures the relative di erence between metrics, more precisely it evaluates their ratio; i.e.: x;f y) 1; The metric spaces (X; dX ), (Y; dY ) are close i 9 f : X ! Y s.t. dYdX(f(x;y) 2 For the relevance of these notions in the study of classical curvatures convergence, see [CMS],
[F] .
3 That is to say: very intuitive, i.e. based upon physical measurements.
SURFACE TRIANGULATION { THE METRIC APPROACH
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8 x; y 2 X.4 Technically, we give the following: De nition 2.1. The map f : (X; dX ) ! (Y; dY ) is bi-Lipschitz i 9 c; C > 0 s.t.: c
d (x; y) X
dY (f x; f y)
C
Xd(x; y) :
Remark 2.2. The same de nition applies for two di erent metrics d 1 ; d2 on the same space X. De nition 2.3. Given a Lipschitz map f : X ! Y , we de ne the dilatation of f by: dY (f x; f y) : dil f = sup x6=y2X dX (x; y) Remark 2.4. The dilatation represents the minimal Lipschitz constant of maps between X and Y . 4
Remark 2.5. If f is not Lipschitz, then dil f = 1. Remark 2.6. (1) f Lipschitz ) f continuous. (2) f bi-Lipschitz ) f homeo. on its image. Remark 2.7. We have the following results: Proposition 2.8. Let f; g : X ! Y be Lipschitz maps. Then: (a) g f is Lipschitz and (b) dil (g f ) dil f dil g Proposition 2.9. The set ff : (X; d) ! R j f Lipschitzg is a vector space. Now we can return to our main interest and de ne the following notion: De nition 2.10. Let (X; dX ), (Y; dY ) be metric spaces. Then the Lipschitz distance between (X; dX ) and (Y; dY ) is de ned as: dL (X; Y ) =
inf
log max (dil f; dil f
1
)
f :X !Y f bi Lip:
Remark 2.11. If 6= f bi-Lipschitz between X and Y , then { remembering Remark 4 2.2 { we put dL (X; Y ) = 1 (i.e. dL is not suited for pairs of spaces that are not bi-Lipschitz equivalent.) having de ned the distance between two metric spaces we now can de ne the convergence in this metric in the following natural way: De nition 2.12. The sequence of metric spaces f(Xn ; dn )g convergence to the metric space f(X; d)g i lim dL (Xn ; X) = 0 (In this case we write: (Xn ; dn )
n !0). L
Example 2.13. Let St be a family of regular surfaces, St = ft (U ); where U is an open set, U = int U R; such that the family fft g of parametrizations is smooth not ! 0. (i.e. F : U R ! R3 2 C 1 ; where F ((u; v); t) = ft (u; v)). Then dn (St ; S0 )t!0 4 Here and in the sequel "f x" etc. ... stands as a short-hand version of "f (x)".
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EMIL SAUCAN
Remark 2.14. If F is not smooth (only continuous) then we do not necessarely have !S . that St t!0 0 We have the following signi cant theorem: Theorem 2.15. The dL satis es the following conditions: (a) dL 0; (b) dL is symmetric; (c) dL satis es the triangle inequality; Moreover, if X; Y are compact, then: (d) dL (X; Y ) = 0 , X = Y (i.e. X is isometric to Y ); that is
dL is a metric on the space of isometry classes of compact metric spaces Remark 2.16. Let us recall the following De nition 2.17. (Xn ; dn )
!(X; d) u
4
= du
! u
sup jdn (x; y)
x;y2X
d as a real function; i.e.
d(x; y)j u!0
(where "u" denotes uniform convergence.) Then Xn u!X ) Xn L!X but Xn indeed Xn u!X , Xn L!X.
!X L
) = Xn
!X. u
However, for nite spaces
2.2. Gromov-Hausdor 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 nite (even for pairs of non-homeomorphic spaces.) )5 We start by rst introducing the classical 2.2.1. Hausdor distance. De nition 2.18. Let A; B (X; d). We de ne the Hausdor distance between A and B as: dH (A; B) = inffr > 0 j A Ur (B); B Ur (A)g 4 S (see Fig. 1); where Ur (A) is the r-neighborhood of A, Ur (A) = a2A Br (a); (here, as usual: Br (a) = fx 2 X j d(a; x) < rg.) Another (equivalent) way of de ning the Hausdor distance is as follows: dH (A; B) = maxfsup d(a; B); sup d(b; A)g : a2A
b2B
(see Fig. 2) We have the following Proposition 2.19. Let (X; d) be a metric space. Then: (a) dH is a semi-metric (on 2X ). (i.e. A = B ) dH (A; B) = 0.) (b) dH (A; A) = 0; 8A X. (c) A = A and B = B ) dH (A; B) = 0 , A = B . i.e. dH is a metric on the set of closed subsets of X. 5The relationship between the Lipschitz and the Hausdor distances is akin to that between the C 0 and C 1 norms in Functional Spaces.
SURFACE TRIANGULATION { THE METRIC APPROACH
U(A) r
5
U(B) r
B
r
A
X
Figure 1. X j A = Ag; dH = 2X =dH .
Notation We put: M(X) = fA
Remark 2.20. (1) if X is compact and if fAn gn subsets of X, then: T (a) An+1 An ) An H! n 1 An . (b) An
An+1 ) An
(2) For general subsets An
! H
S n 1
!A H
1
X is a sequence of compact
An .
2 M(X) , and
(a) A = flimn an j an 2 An ; n
1g .
T S1 (b) A = H! n 1 m=n Am . ! (3) If An H A, and if the sets An are all convex, then A is convex sets. We have the following two important results, which we present without their respective (lengthy) proofs: Proposition 2.21. X complete ) M(X) complete . Theorem 2.22. (Blaschke) X compact ) M(X) compact .
d(a,b)
b
a
A
B
X Figure 2.
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EMIL SAUCAN
2.3. The Gromov-Hausdor Distance. We are now able to de ne the GromovHausdor distance using the following basic guide-lines: we want to get the maximum distance that satis es the following two conditions: (a) dGH (A; B) dH (A; B); 8A; B X (i.e. set that are close as subsets of X will still be close as abstract metric spaces); and (b) X isometric to Y () dGH (X; Y ) = 0. De nition 2.23. Let X; Y be metric spaces. Then the Gromov-Hausdor distance between X and Y is de ned by: 4
dGH (X; Y ) = inf dZ H (f (X); g(Y )) where the in mum is taken over all the isometric embeddings f : X ,! Z; g : Y ,! Z into some metric space Z. (See Fig. 3). Y
X
g
f
f(X)
dHZ
g(Y)
Z
Figure 3. Remark 2.24. If X = S2 , with the spherical metric, and Z = R3 , with the Euclidian metric, then f (X) 6= X (!) Example 2.25. Let Y be an "-net6 in X. Then dGH (X; Y ) Proof Take Z = X = X 0 ; Y = Y 0 .
".
Remark 2.26. It is su cient ‘ to consider embeddings f into the disjoint union of the spaces X and Y , X Y . Remark 2.27.
(1) X; Y bounded =) dGH (X; Y )
(2) If diamX; diamY < 1, then dGH (X; Y ) 6 De nition Let (X; d) be a metric space, and let A
d(x; A)
"; 8x 2 X.
4 7diamX = supx;y2X d(x; y)
1.
1 2 jdiamX
diamY j7.
X. A is called an "-net i
SURFACE TRIANGULATION { THE METRIC APPROACH
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However , the straightforward de nition of d GH may be di cult to implement. Therefore we would like to estimate (compute) dGH by comparing distances in X vs. distances in Y (as done in the cases of uniform and Lipschitz metrics). We start by de ning a correspondence between metric spaces: X ! Y , given by correspondences x $ y between points x 2 X; y 2 Y . Remark 2.28. A correspondence is not necessarely a function, that is to a single x may correspond to several y-’s. We shall prove that ( ) dGH (X; Y ) < r () 9 a correspondenceX 0
!Y
0
0
s:t: (x $ y; x $ y ) =) jdX (x; x )
dY (y; y 0 )j < 2r
Formally, we have: De nition 2.29. Let X; Y denote sets. A correspondence X the Cartesian product of X and Y : R X Y s.t. (i) 8x 2 X; 9y 2 Y; s:t:(x; y) 2 R; and (ii) 8y 2 Y; 9x 2 X; s:t:(x; y) 2 R.
! Y is a subset of
Example 2.30. Any surjective function f : X ! Y represents correspondence R = f(x; f (x))g. Remark 2.31. R is a correspondence () 9 Z and 9 f : Z ! X ; 9g : Z ! Y ; f; g surjective, s.t. R = f(f (z); g(z)) j z 2 Zg. De nition 2.32. Let R be a correspondence between X and Y , where X; Y are metric spaces. We De ne the distortion of R by: dis R = sup jdX (x; x0 )
dY (y; y 0 )j (x; y) ; (x0 ; y 0 ) 2 R :
(See (*) .) Remark 2.33. (1) If R = f(x; f (x))g is a correspondence induced by a surjective function f : X ! Y , then dis R = dis f , where: 4
dis f = sup j dY (f a; f b) a;b2X
dX (a; b) j8 :
(2) If R = f(f (z); g(z))g, where f : X ! Z; g : Y ! Z are surjective functions, then: dis R = sup f j dX (f z; f z 0) dY (gz; gz 0) j g : z;z 0 2Z
(3) R = 0 i R is associated to an isometry. We bring, without proof, the following theorem: Theorem 2.34. Let X; Y be metric spaces. Then: 1 dGH (X; Y ) = inf (dis R) ; 2 R where the in mum is taken over all the correspondences X
R
!Y.
8 Remember that any correspondence can be expressed in this functional manner.
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EMIL SAUCAN
Before bringing the next result (which is very important in determining the topology ) we rst introduce one more notion: De nition 2.35. f : X ! Y is called an "-isometry (" > 0), i (i) dis f ", and (ii) f (x) is an "-net in Y . Remark 2.36. f "-isometry =) = f continuous. Corollary 2.37. Let X; Y be metric spaces and let " > 0. Then: (i) dGH (X; Y ) < " =) 9 2" isometry f : X ! Y . (ii) 9 " isometry f : X ! Y =) dGH (X; Y ) < 2". R
Proof (i) Let X ! Y s.t. dis R < 2". For any x 2 X and f (x) 2 Y , choose y = f (x) s.t. (x; f (x)) 2 R. Then x 7! f (x) de nes a map f : X ! Y . Moreover: dil f dil R < ". We shall prove that f (X) is a 2"-net in Y . Indeed, let x 2 X and y 2 Y s.t. (x; y) 2 R. Then d(y; f x) d(x; x) + dis R < 2r, thence d(y; f (X)) < 2r. Let f be an 2"-isometry. De ne R X Y; R = f(x; y) j d(y; f x) Then, since f (X) is an "-net it follows that R is a correspondence. Then 8 (x; y); (x0 ; y 0 ) 2 R we have: j dY (y; y 0 ) dX (x; x0 ) j
j d(f x; f x0 ) d(x; x0 ) j+d(y; f x)+d(y 0 ; f x0 )
=) dis R
3" =) dGH (x; y)
"g.
dis f +"+"
3r=2 < 2r :
The next result is of great importance (in particular so in our context): Theorem 2.38. dGH is a ( nite) metric on the set of isometry classes of compact metric spaces. Proof It su ces to prove that d GH (X; Y ) = 0 =) X Y .9 Indeed, let X; Y be compact spaces s.t. dGH = 0. Then it follows from the previous Corollary (for " = 1=n) that 9 (fn )n 1 ; fn : X ! Y s.t. dis fn n!0. let S X; S ; jSj = @0 . Using a Cantor-diagonal argument one easily shows that (fn )n 1 s.t. (fnk ) 1 converges in Y; 8x 2 S. Without restricting the 9 (fnk )k 1 generality we may assume that this happens for (fn )n 1 itself. Thus we can de ne a function f : X ! Y by putting: f (x) = limn fn (x). But j d(fn x; fn y) d(x; y) j dis fn n!0 =) d(f x; f y) = lim d(fn x; fn y). In other words f jS is an isometry. But S = S, therefore this isometry can be extended to an isometry f~ from X to Y . In a analogous manner one shows the existence of an ~ isometry f~ : X ! Y . Remark 2.39. Xn
!X L
=) Xn
! u X
! X. =) Xn GH
In fact, the following relationship exists between " Theorem 2.40. Xn 9We shall write: X
!X GH
() "-nets in Xn
Y if X is isometric to Y .
!" L
!"-nets L
! ": and "GH
in X.
3" :
SURFACE TRIANGULATION { THE METRIC APPROACH
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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: De nition 2.41. Let X; Y be compact metric spaces, and let "; > 0. X; Y are called "- -approximations (of each-other) i : 9 fxi gN X, 9 fyi gN Y s.t. i=1 i=1 N N (i) fxi gi=1 is an "-net in X and fyi gi=1 is an "-net in Y ; (ii) j dX (xi ; xj ) d( yi ; yj ) j < 8 i; j 2 f1; :::; N g. An ("; ")-approximation is called, for short: an "-approximation. The relationship between this last de nition and the Gromov-Hausdor distance is rst revealed in Proposition 2.42. Let X; Y be compact metric spaces. Then: (1) If Y is a ("; )-approximation of X, then dGH (X; Y ) < 2" + . (2) dGH (X; Y ) < " =) Y is a 5"-approximation of X. Proof (1) Condition (ii) of Def. 2.41. is equivalent to dis RX0 Y0 < , where N X0 = fxi gN =) dGH (X0 ; Y0 ) < =2. Now, since i=1 ; fyi gi=1 . But dis RX0 Y0 < X0 and Y0 are "-nets in X, resp. Y , it follows that dGH (X; X0 ) "; dGH (Y; Y0 ) < ". From here and from the dGH (X0 ; Y0 ) < =2 follows, by means of he triangle inequality, that dGH (X; Y ) < 2" + . (2) By Cor. 2.37., there exists a 2"-isometrie f : X ! Y . Let X0 = fxi gN i=1 be an "-net, and let yi = f (xi ). Then j d(xi ; xj ) d( yi ; yj ) j < 2" < 5". Therefore su ce to prove that Y 0 = fyi gN i=1 is a 5"-net in Y . Indeed, if y 2 Y , then, since f (X) is an 2"-net in Y , 9 x 2 X s.t. d(y; f (x)). Now, since X0 is an "-net in X, 9xi 2 X0 , s.t. d(x; xi ) ". Therefore: d(y; yi ) = d(y; f (xi )) d(y; f (x)) + d(f (x); f (xi )) 2" + d(x; xi ) + dis f 2" + " + 2" 5". Remark 2.43. Prop. 2.42. () (Xn
! X) GH
, (8 " > 0; Xn is an "-approximation, 8 n large enough.)
More precisely we have the following Proposition: Proposition 2.44. Let X; fXn g1 1 compact metric spaces. Then: Xn !X () 8" > 0; 9 a nite "-net S X and 9 a nite "-net Sn GH ! Sn S and, moreover, jSn j = jSj, for large enough n.
Xn , s.t.
GH
2:42:
Proof ((=) If S; Sn exist, then =) Xn is an "-approximation of X =) Xn
!X GH
8 n.
(=)) Let S be an nite "=2-net in X. We construct in Xn corresponding nets Sn (to be more precise, we de ne: S n = fn (X), where fn is an "n -approximation, fn : X ! Xn ; "n ! 0.) Then Sn !S GH and, in addition, Sn is an "-net in S (for n large enough). We make the following extremely important Remark:
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EMIL SAUCAN
Remark 2.45. Let M(n; k; D) be an n-dimensional manifold, of (sectional, Ricci) curvature k, and s.t. diam M D. Then (M; dGH ) is compact. However, it should be noted that this result doesn’t hold for curvature < k. (only for V ol(M) V0 ) and injectivity radius r0 . Note With the notations of the precedent Proposition, the distances in Sn con! X, therefore The Geometric Proprieties verge to the distances in S, as Xn GH of Sn will converge to those of S. Thus we can use the Gromov-Hausdor each and every time The Geometric Proprieties of Xn can be expressed in term of a nite number of points, and, by passing to the limit, automatically obtain proprieties of X. 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 R3 .) The induced (intrisic) metric
The Euclidian metric
c
S
3
R
Figure 4. On a more formal note, we have the following characterization of intrinsic metrics: Theorem 2.46. Let (X; d) be a complete metric space. (1) If 8 x; y 2 X; 9 21 xy, then d is strictly intrinsic. (2) If 8 x; y 2 X and 8" > 0; 9 the "-middle of xy, then d is intrinsic. Where we used the following de nitions and notations: De nition 2.47. (1) Given x; y points in (X; d), the middle (or midpoint) of the segment xy (more correctly: ’a midpoint between "x" and "y" ’) is de ned as: 1 xy = z; d(x; z) = d(z; y) : 2 (2) d is called strictly intrinsic i the length structure is associated with is complete. (3) Let d be an intrinsic metric. z is an "-middle (or an "-midpoit) for xy i : j 2d(x; z) d(x; y) j " and j 2d(y; z) d(x; y) j ". Remark 2.48. The converse of Thm. 2.46. holds in any metric space, more precisely we have: Proposition 2.49. If d is an intrinsic metric, then
1 2 xy
exists, 8x; y.
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The following Theorem shows that length spaces are closed in the GH-topology : Theorem 2.50. Let fXn g be length spaces and let X be a complete metric space s.t. Xn !X. GH Then X is a length space. Proof We have already presented the idea of the proof: it is su cient to show that for every x; y there exist an "-midpoit (8" > 0). " . Then, from the a preceding result, it follows Indeed, let n be such that dGH < 10 that there exist a correspondence Xn R!X s.t. dis R < 5" . R R Let x; y 2 Xn , x$ x, y $ y. Since Xn is a length space, =) 9z 2 Xn s.t. z = 5" R midpoint of xn yn . Consider z 2 X; z $ z. Then: 1 " 2" 1 jxyj jxzj jxyj + 2dis R < + < ": jxzj 2 2 5 5 (Here we write jxyj instead of d(x; y), etc.) In a similar manner we show that: jyzj 21 jxyj < "; i.e. "-midpoit of xy. The next Theorem and its Corollary are of paramount importance: Theorem 2.51. Any compact length space is the GH-limit of a sequence of nite graphs. Proof Let "; ( ") small enough, and let S be a -net in X. Let G = (V; E) be the graph with V = S and E = f(x; y) j d(x; y) < "g. we shall prove that G is an "-approximation of X, for small enough (i.e. for < "2 4 diam(X)). (See Fig. 5.)
Sn
G
X
Figure 5. But, since S is an "-net both in X and in G, and since dG (x; y) dX (x; y), it is su cient to prove that: dG (x; y) dX (x; y) + " : Let be the shortest path between x and y, and let x1 ; :::; xn 2 s.t. n length( )=" (and jxi ; xi+1 )j "=2). Since 8 xi 9yi 2 S s.t. jxi ; yi j , it follows that jyi yi+1 j jxi xi+1 j + 2 < ": (See Fig. 6)
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EMIL SAUCAN
Therefore, (for < "=4) 9 an edge e 2 G; e = yi yi+1 . From this we get the following upper bound for dG (x; y): n 0 jyi yi+1 j
dG (x; y) But n < 2length( )="
+2 n
2diam(X)="; therefore:
dG (x; y) (because
n 0 jxi xi+1 j
jxyj +
4diam(X) < jxyj + " : "
< "2 =4diam(X)).
y=x
xn-1 x i+1
.
xi
. . . .. x y
3
x
2
n
yn
...
S
yi+1
yn-1
y
i
G
y2
x=x 1 y1
x Figure 6. So, for any " > 0; 9 G = G" an "-approximation of X. Then, Gn ! " X. Corollary 2.52. Let X be a compact length space. Then X is the GromovHausdor limit of a sequence fG n gn 1 of nite graphs, isometrically embedded in X. Remark 2.53.
(1) If Gn ! " X, Gn = (Vn ; En ). If 9N0 2 N s.t. (?) jEn j
N0 ; 8n 2 N ;
then X is a nite graph. (2) If condition (?) is replaced by: (??) jVn j
N0 ; 8n 2 N ;
then X will still be always a graph, but not necessarily nite(!)
SURFACE TRIANGULATION { THE METRIC APPROACH
13
3. The Embedding Curvature 3.1. Theoretical Setting. This is basically a comparison-curvature (as is the more "modern" CAT 10 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 classically11 the "minimal" geometric gures that allow the di erentiation 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 R3 .12 De nition 3.1. Let (M; d) be a metric space, and let Q = fp1 ; :::; p4 g M, together with the mutual distances: dij = dji = d(pi ; pj ); 1 i; j 4. The set Q together with the set of distances fdij g1 i;j 4 is called a metric quadruple. Remark 3.2. One can de ne metric quadruples in slightly more abstract manner, without the aid of the ambient space: a metric quadruple being a 4 point metric space; i.e. Q = fp1 ; :::; p4 g; fdij g , where the distances dij verify the axioms for a metric. Before we proceed to the next de nition, let us introduce the following Notation S denotes the complete, simply connected surface of constant curvature , if < 0. Here H2p , i.e. S R2 , if = 0; S S2p , if > 0; and S p 2 2 p p stands for the S S denotes the Sphere of radius R = 1= , and S H p Hyperbolic Plane of curvature , as represented by the Poincare Model of the p plane disk of radius R = 1= De nition 3.3. The embedding curvature (Q) of the metric quadruple Q is dened be the curvature of S into which Q can be isometrically embedded. (See Figures 7 and 8 for embeddings of the metric quadruple in S and H , respectively.) We can now de ne the embedding curvature at a point in a natural way by passing to the limit (but without neglecting the existence conditions), more precisely: De nition 3.4. Let (M; d) be a metric space, and let p 2 M be an accumulation point. Then p is said to have Wald curvature W (p) i (i) =9 N 2 N (p); N linear13 ; (ii) 8 " > 0; 9 > 0 s.t. Q = fp1 ; :::; p4 g M ; and s.t. d(p; pi ) < (i = 1; :::; 4) =) j (Q) (p)j < ". W Remark 3.5. (1) If one uses the second (abstract) de nition of the metric curvature of quadruples, then the very existence of (Q) is not assured, as it is shown by the following Counterexample 3.6. The metric quadruple of lengths d12 = d13 = d14 = 1; d23 = d24 = d34 = 2 10 i.e. Cartan-Alexandrov-Topogonov 11 as illustrated by the time-honored principles of Projective Geometry... 12 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.. 13 The neighborhood N of p is called linear i N is contained in a geodesic.
14
EMIL SAUCAN
X
p3
d 34 d12 23
d24
d14
p1 d12
d
p4
p2
f d34 d12
d23 R=
1
d14
d24
d12 k
2
Sk
Figure 7. admits no embedding curvature. (2) Even if a quadruple has an embedding curvature, it still may be not unique (even if Q is not liniar), indeed, one can study the following examples: Example 3.7. (a) The quadruple Q of distances dij = =2; 1 i<j is isometrically embeddable both in S0 = R2 and in S1 = S2 .
X
p3
d 34 d12 d
23
p1 d12
p4 d24
d14
p2
d23
f
d13 d12
d34
R=
d14
d24
2
Hk
Figure 8.
1 -k
4
SURFACE TRIANGULATION { THE METRIC APPROACH
15
(b) The quadruple Q of distances d13 = d14 = d23 = d24 = ; d 12 = d34 = 3 =2 admits exactly two embedding curvatures: 1 2 (1:5; 2) and 2 = 3. (See [BM].) However, for "good" metric spaces14 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): Proposition 3.8. Let 4(p1 ; q1 ; r1 ) S 1 and 4(p2 ; q2 ; r2 ) p1 q1 = p2 q2 ; p1 r1 = p2 r2 and \(q1 ; p1 ; r1 ) = \(q2 ; p2 ; r2 ). Then: 1 < 2 =) q1 r1 > q2 r2 .
S 2 , s.t.
Proposition 3.9. Let p1 ; q1 ; r1 2 S 1 ; p2 ; q2 ; r2 2 S 2 two isometric triples of points, s.t. the triple p1 ; q1 ; r1 is not linear. Then: \(q1 ; p1 ; r1 ) < \(q2 ; p2 ; r2 ); \(p1 ; q1 ; r1 ) < \(p2 ; q2 ; r2 ); \(q1 ; r1 ; p1 ) < \(q2 ; r2 ; p2 ). Proposition 3.10. Let Q1 = fp1 ; q1 ; r1 ; s1 g; Q2 = fp2 ; q2 ; r2 ; s2 g be non-linear and non-degenerate quadruples in S 1 ; S 2 , respectively. If 4(p1 ; q1 ; r1 ) = 4(p2 ; q2 ; r2 ) and 1 < 2 , then: (1) p1 s1 = p2 s2 ; q1 s1 = q2 s2 =) r1 s1 > r2 s2 ; (2) r1 s1 = r2 s2 ; q1 s1 = q2 s2 =) p1 s1 > p2 s2 ; (3) p1 s1 = p2 s2 ; r1 s1 = r2 s2 =) q1 s1 < q2 s2 . In order that we fully exploit the results above we need the following de nition: De nition 3.11. A metric quadruple Q = Q(p1 ; p2 ; p3 ; p4 ), of distances dij = dist(pi ; pj ); i = 1; :::; 4, is called semi-dependent (a sd-quad, for brevity), i 3 of its points are on a common geodesic, i.e. there exist 3 indices { e.g. 1,2,3 { s.t.: d12 + d23 = d13 . Now we can easily formulate the following immediate consequence of Prop. 3.10. : Corollary 3.12. A sd-quad admits at most one embedding curvature. Unfortunately { as we have already noticed { in the general case the uniqueness of the embedding curvature is not guaranteed. However we can be a bit more explicit using the following de nition: De nition 3.13. Let Q = fp; q; r; sg be a non-linear and non-degenerate quadruple. Q is called planar i \(q; p; r) + \(q; p; s) + \(s; p; r) = 2 . Then we have Proposition 3.14. Let Q = fp; q; r; sg be a a non-linear and non-degenerate quadruple in S . Then (1) If Q is planar, then it admits no isometric embedding in S 1 ; 1 > . 14 i.e. spaces that are locally "plane like"
16
EMIL SAUCAN
(2) If Q is not planar, then it admits no isometric embedding in S 2 ;
2
< .
Corollary 3.15. Let Q = fp; q; r; sg be a a non-linear and non-degenerate quadruple. Then Q has at most two di erent embedding curvatures. In fact we can state a much stronger assertion, of which Example 3.7.(a) is just a very particular case: Proposition 3.16. 8 p 2 S , and 8 > 0; 9 U 2 N (p) s.t. 9 a nonlinear, non-degenerate quadruple Q U of embedding curvature 0. Proof. Let 1 ; 2 2 U , two great-circle arcs s.t. 1 \ 2 = p. p Let q1 ; q2 2 1 s.t. pq1 = pq2 6= 0 and let q 2 2 s.t. pq < =2 . 15 Consider 4(q10 q20 ; q 0 ) R2 ; 4(q10 q20 ; q 0 ) = 4(q1 q2 ; q), let p0 = 12 q10 q20 , and let h = q 0 p0 .
x0
q1
U
γ
q2
p
1
γ
2
q
Sκ 2
q'' h x'0
q'1
p'
q'2
h
R
2
q'
Figure 9. Then since 0 < , Proposition 3.10.(3) applied to the quadruples fq; q 1 ; q2 ; pg and fq 0 ; q10 ; q20 ; p0 g implies that h < pq. Now let x 2 2 , x between p and q, and let x0 2 R2 s.t. 4(q10 q20 ; x0 ) = 4(q1 q2 ; x) ! s.t. x and q 0 are on di erent sides of the line q1 q2 . Then, x = p ) xq > x0 q 0 , and x = q ) xq = 0 < x0 q 0 = 2h, where, in this case x0 = q 00 . (See Figure 9.) 15 i.e a quarter of the length of a great circle in S
SURFACE TRIANGULATION { THE METRIC APPROACH
17
Then it follows from a continuity argument that 9 x0 2 2 ; x0 between p and q, s.t. x0 q = x00 q 0 , thus implying that fq1 ; q2 ; q; xg = fq10 ; q20 ; q 0 ; x0 g. Remark 3.17. fq1 ; q2 ; q; xg is planar, while fq10 ; q20 ; q 0 ; x0 g is not planar. 3.2. The Wald Curvature vs. Gauss Curvature. The discussion above would be nothing more than a nice intellectual exercise where it not for the fact that the metric (Wald) and the classical (Gauss) curvatures coincide whenever the second notion makes sense, that is for smooth (i.e. of class C2 ) surfaces in R3 . More precisely the following theorem holds: Theorem 3.18. (Wald) Let S R3 ; S 2 C m ; m 2 be a smooth surface. Then W (p) exists, for all p 2 S, and W (p) = G (p); 8 p 2 M . Moreover, Wald also proved that a partial reciprocal theorem holds, more precisely he proved the following: Theorem 3.19. Let M be a compact and convex metric space. If W (p) exists, for all p 2 M , then M is a smooth surface and W (p) =
G (p); 8 p
2 M.
Remark 3.20. I one tries to restrict oneself, in the building of De nition 3.4. only to sd-quads, then Theorem 3.19. holds only if the following presumption is added: Condition 3.21. M is locally homeomorphic to R2 . However the proof of this facts is involved and, as such, beyond the scope of this presentation. Therefore we shall restrict ourselves to a succinct description of the principal steps towards the proofs. The basic idea is to show that if a metric M space admits a Wald curvature at any point, than M is locally homeomorphic to R2 , thus any metric proprieties of R2 can be translated to M , (in particular the rst fundamental form). The rst of these partial results is: Theorem 3.22. Let M be a convex metric space. Then M admits at most one Wald curvature W (p); 8 p 2 M . Proof By Corollary 3.12. it su ces to prove that any disk neighborhood B(p ; ) 2 N (p) contains a non degenerate sd-quad. Without loss of generality one can assume that B(p ; ) contains three points p 1 ; p2 ; p3 s.t. d(p; pi ) < =2; i = 1; 2; 3.16 Then, by the convexity of M it follows that 9 q 2 M s.t. p 6= p2 ; p3 and p2 q + p3 q = p2 p3 . But p2 p3 pp2 + pp3 < =) (pq < =2) _ (pp2 < =2). In the rst inequality holds, then pq pp2 + p2 q < , i.e q 2 B(p ; ); and if the second one holds, then pd pp3 + p3 q < , i.e. q 2 B(p ; ). But p 6= q, therefore p; p2 ; p3 ; q are not linear. Our next step will be to analyze those neighborhoods that display "a normal behavior", both metrically and curvature-wise: that is precisely those disk neighborhoods in which the Wald curvature is de ned and ranges over a small, bounded set of values prescribed by the very radius of the disk: De nition 3.23. A disk neighborhood B(p ; ); > 0 is called regular i 8 nondegenerate quadruple Q B(p ; ) , W (Q) exists and j W (Q)j < 2 =16 2 . 16 See [B]
18
EMIL SAUCAN
Remark 3.24. If regular.
W (p)
exists, then for any su ciently small , B(p ; ) will be
It turns out that regular neighborhoods, in compact, convex spaces have the following "nice" (i.e. Real Plane like) proprieties: Proposition 3.25. Let M be a compact, convex metric space and let B(p ; ) M be a regular neighborhood. Then if a non-degenerate quadruple Q B(p ; ) contains two linear triples of points, then Q is linear. Proposition 3.26. Let M be a compact, convex metric space. Then: 8 p 2 M and 8 B(p ; ) regular, 9 q; r 2 B(p ; ) s.t. p; q; r are not linear. Proposition 3.27. Any regular neighborhood B(p ; ) of a compact, convex metric space is strictly convex, i.e. q; r 2 B(p ; ) =) int(qr) B(p ; ). While the proof of this last Proposition is lengthy, that of the following important Corollary is not: Corollary 3.28. Let B(p ; ) be a regular neighborhood. Then, 8 q; r 2 B(p ; ); 9! qr and int(qr) B(p ; ). Proof. By the convexity of B(p ; ) it follows the existence of at least one geodesic qr; 8 q; r 2 B(p ; ). If s 2 int(q)r, then by the proposition above we have that s 2 B(p ; ). It follows that B(p ; ) contains all the geodesics with end points q; r. Hence, by Proposition 3.25. , the geodesic segment qr is unique. We can now begin to prove that a compact, convex metric space locally mimics R2 . We start by showing that the sinus function is de ned on M , thus allowing for angle measure (hence for the de nition of Polar Coordinates on regular neighbourhoods17). First, let M be as before, and let p 2 M s.t. W p exists. Let q; r 2 B(p ; ); q 6= p 6= r, where B(p ; ) is a regular neighborhood of p. Then, 8 x 2 [0; minfpq; prg), de ne q(x) 2 pq; r(x) 2 pr by: d(p; q(x)) = x = d(p; r(x)), and let d(x) = d(; q(x); r(x)) (see Figure 10 bellow). Proposition 3.29. The following limit exists: d(x) : lim x!0 x We omit the proof since it is rather involved (but canonical for any axiomatic approach to Euclidian Geometry { see, for instance, [B], [RR].) Now we can de ne the measure of angles at p : De nition 3.30. The measure of the angle \(q; p; r)) is given by: 1 d(x) : lim 2 x!0 x Remark 3.31. The De nition above enables us to de ne Polar Coordinates on regular neighborhoods18 in the following manner: Let p1 ; p2 2 B(p ; ) s.t. p; p1 ; p2 are not collinear. (Such points exist by Proposition 3.26.). To every point q 2 B(p ; ) we associate the following pair of real 4
m(\(q; p; r)) = 2arcsin
17 In the same way geodesic polar coordinates are used on classical surfaces. 18 ... and once Coordinates (be they Polar or Cartesian) are introduced, the (local) homomor-
phism with R2 is immediate.
SURFACE TRIANGULATION { THE METRIC APPROACH
B(p;ρ)
19
q
q(x)
d(x)
x x
p
r(x)
r
M
Figure 10. numbers (de ning the Polar Coordinates of q relative to the frame determined by p; p1 ; p2 ): (r(q); (q)), where 4
r(q) = d(p; q) and 4
(q)) =
2
m(\(q; p; p1 )) if jm(\(p2 ; p; p1 )) m(\(q; p; p1 )) if jm(\(p2 ; p; p1 ))
m(\(q; p; p1 ))j = m(\(q; p; p1 )) ; m(\(q; p; p1 ))j 6= m(\(q; p; p1 )) .
We can now safely state the foretold homomorphism result: Proposition 3.32. Any convex, compact metric space is locally homeomorphic to the real plane. 4. Computing Embedding Curvature In this section we develop formulas for the computation of Embedding Curvature of Quadruples. First we follow the classical approach of Wald-Blumenthal that employs the so-called Cayley-Menger determinants (see bellow). Unfortunately, the formulas obtained, albeit precise are transcendental. Therefore we present, in the next subsection, the approximate formulas developed by C.V. Robinson. 4.1. Embedding Curvature { The Determinant Approach. Given a general metric quadruple Q = Q(p1 ; p2 ; p3 ; p4 ), of distances dij = dist(pi ; pj ); i = 1; :::; 4, we denote by D(Q) = D(p1 ; p2 ; p3 ; p4 ) the following determinant:
(4.1)
D(p1 ; p2 ; p3 ; p4 ) =
0 1 1 0 1 d212 1 d213 1 d214
1 d212 0 d223 d224
1 d213 d223 0 d234
1 d214 d224 d234 0
20
EMIL SAUCAN
Then the embedding curvature (Q) of Q is given { depending upon the embedding space (i.e. upon the sign of the curvature) { by the following formulae:
(4.2)
(Q) =
8 > > < > > :
; ;
0 if D(Q) = 0p; < 0 if det(coshp 0; ijd) =p d d > 0 if det(cos ) and ij ij and all the principal minors of order 3 are
0.
The determinant D(Q) = D(p1 ; p2 ; p3 ; p4 ) is called the Cayley-Menger determinant (of the points p1 ; :::p4 )19 and, in order to prove (4.2) we need rst to investigate some of its properties. We start with the following Lemma 4.1. Let: p1 ; :::; p4 be points in R3 . Then: ! ! 1 (4.3) Gram(p1 p! 2 ; p1 p3 ; p1 p4 ) = D(p1 ; p2 ; p3 ; p4 ) ; 8 where ! ! def ! ! Gram(p1 p! 2 ; p1 p3 ; p1 p4 ) = det(p1 pi p1 pj )i;j=2;3;4 3 (Here " " denotes the standard scalar (dot) product in R .)
Proof. Use expansion and manipulation of determinants. Since is a known fact that: ! ! Gram(p1 p! 2 ; p1 p3 ; p1 p4 ) = V ol(p1 ; p2 ; p3 ; p4 )
2
;
where V ol(p1 ; p2 ; p3 ; p4 ) denotes the (un-oriented) volume of the parallelepiped de! ! termined by the vertices p1 ; :::; p4 (and with edges p1 p! 2 ; p1 p3 ; p1 p4 ); formula (2.3) shows that: (4.4)
D(p1 ; p2 ; p3 ; p4 ) = 8 V ol(p1 ; p2 ; p3 ; p4 )
2
:
Therefore the following assertion is immediate: Proposition 4.2. The points p1 ; :::; p4 are the vertices of a simplex in R3 i D(p1 ; p2 ; p3 ; p4 ) 6= 0_. However, we can prove the much strong result bellow: Theorem 4.3. Let dij > 0 ; 1 4 ; i 6= j. Then there exists a simplex T = T (p1 ; :::; p4 ) D(pi ; pj ) < 0 ; (8) fi; jg f1; :::; 4g and D(pi ; pj ; pk ) > 0 ; (8) fi; j; kg f1; :::; 4g; where, for instance, 0 1 D(p1 ; p2 ) = 1 0 1 d212 and 0 1 1 0 D(p1 ; p2 ; p3 ) = 1 d212 1 d213
R3 s.t. dist(xi ; xj ) = dij ; i 6= j; i
1 d212 0 1 d212 0 d223
1 d213 d223 0
;
19 This de nition readily generalizes to any dimension, as do the results bellow.
SURFACE TRIANGULATION { THE METRIC APPROACH
21
etc... In fact, the necessary and su cient condition above can be relaxed, indeed one can also show that the following holds20: Proposition 4.4.21 Let dij > 0 ; 1 4 ; i 6= j. Then there exists a simplex T = T (p1 ; :::; p4 ) R3 s.t. dist(xi ; xj ) = dij ; i 6= j; i D(p1 ; p2 ; p3 ; p4 ) 6= 0 and sign D(p1 ; p2 ; p3 ; p4 ) = +1 . Proof.(Sketch) Su cient to show (by using standard operations on determinants) that: 2 D(p1 ; p2 ; p3 )D(p1 ; :::; p^i ; :::; p4 ) = Mi4 + D(p1 ; :::; p^i ; :::; p4 )D(p1 ; p2 ; p3 ; p4 ) ;
where Mi4 is the cofactor (in D) of d2i4 , and were we used the notation: fp1 ; :::; p^i ; :::; p4 g = fp1 ; p2 ; p3 ; p4 g n fpi g. Proving the formula for the spherical and hyperbolical cases would prove to be to technical for this limited exposition; su ce to say that they essentially reproduce the proof given in the Euclidian case, and tacking into account the fact that performing computations in the spherical (resp. hyperbolic) metric one has to replace the distances dij by cos dij (resp. cosh dij )22. 4.2. Embedding Curvature { Approximate Formulas. The formulas we just developed in are not only transcendental, but also the computed curvature may fail to be unique (see preceding section). However, uniqueness is guaranteed for sd-quads. Moreover, the relatively simple geometric setting of sd-quads allows for the development of simple (i.e. rational) formulas for the approximation of the Embedding Curvature. Proposition 4.5. Given the metric quadruple Q = Q(p1 ; p2 ; p3 ; p4 ), of distances dij = dist(pi ; pj ); i = 1; :::; 4, the embedding curvature (Q) is well approximated by: 6(cos \0 2 + cos \0 20 ) (4.5) K(Q) = d24 (d12 sin2 (\0 2) + d23 sin2 (\0 20 )) where: \0 2 = \(p1 p2 p4 ) ; \0 20 = \(p3 p2 p4 ) represent the angles of the Euclidian triangles of sides d12 ; d14 ; d24 and d23 ; d24 ; d34 , respectively. The error R can be estimated by using the following inequality: (4.6)
jRj = jR(Q)j = j (Q)
K(Q)j < 4 2 (Q)diam2 (Q)= (Q)
where we put: (Q) = d 24 (d12 sin \0 2+d23 sin \0 20 )=S 2 , and where S = M axfp; p0 g; 2p = d12 + d14 + d24 ; 2p0 = d32 + d34 + d24 . Proof The basic idea of the proof is to recreate, in a general metric setting, the Gauss Map { in this case one measures the curvature by the amount of "bending" one has to apply to a general planar quadruple so that it may be "straightened" (i.e. isometrically embedded as a sd-quad) in some S . Consider two plane23 triangles 4p1 p2 p4 and 4p2 p3 p4 , and denote by 4p1;k p2;k p4;k 20 For the direct proof of Theorem 4.3., see [Be] or, alternatively [B] 21 We formulate this result { for convenience and practicality { for the case n = 3, only.
However it is readily generalized to any dimension. 22 See [B] for the full details 23i.e. embedded in R2 S0
22
EMIL SAUCAN
and 4p2;k p3;k p4;k their respective isometric embeddings into Sk . Then pi;k pj;k will denote the geodesic (of Sk ) through pi;k and pj;k . Also, let \k 2 and \k 20 denote, respectively, the following angles of 4p1;k p2;k p4;k and 4p2;k p3;k p4;k : \k 2 = \p1;k p2;k p4;k and \k 20 = \p2;k p3;k p4;k . (See Fig. 11) d
4
d 34
d
14
d 24 p3
p
1
d
2
2'
12
p
d 23
2
Figure 11. But \k 2 and \k 20 are strictly increasing as functions of k. Therefore the equation \ k 2 + \ k 20 =
(4.7)
has at most one solution k , i.e. k represents the unique value for which the points p1 ; p2 ; p4 are on a geodesic in Sk (for instance on p1 p4 ). But that means that k is precisely the Embedding Curvature, i.e. k = (Q) , where Q = Q(p1 ; p2 ; p3 ; p4 ). Equation (4:7) is equivalent to \k 20 \k 2 + cos2 = 1 2 2 The basic idea being the comparison between metric triangles with equal sides, embedded in S0 and Sk , respectively, it is natural to consider instead of the previous equation, the following: cos2
(4.8)
(k; 2)
2 \0 2 cos + 2
(k; 20 )
2 cos
\0 20 = 1 2
where we denote: (k; 2) :=
\k 2 2 cos2 \20 2
cos2
;
0
(k; 2 ) :=
\k 20 2 cos2 \20 2
cos2
:
Since we want to approximate (Q) by K(Q) we shall resort { naturally { to expansion into MacLaurin series. We are able to do this because of the existence of the following classical formulas: p p sin(p k) sin(d k) 2 \k 2 p ; k > 0; p = cos 2 sin(d12 k) sin(d 24 k)
SURFACE TRIANGULATION { THE METRIC APPROACH
p \k 2 sinh(p k) p cos = 2 sinh(d12 k) 2
23
p sinh(d k) p ; k < 0; sinh(d 24 k)
and, of course cos2
pd \0 2 = ; 2 d12 d24
were: d = p d14 = (d12 + d24 d14 )=2 :24 By using the development into series of f1 (x) = (easily) gets the desired expansion for (k; 2):
p sin p x x
1 (k; 2) = 1 + kd12 d24 cos(\0 2) 6
(4.9)
and f2 (x) =
p sinh p x; x
one
1 +r;
where: jrj < 83 k 2 p4 , for jkp2 j < 1=16 . By applying (4.9.) to (4.8), we receive: 1 [1 + k d12 d24 cos(\0 2) 6
(4.10)
1 + r] cos2
\0 2 + 2
1 \0 20 [1 + k d23 d24 cos(\0 20 ) 1 + r0 ] cos2 = 1; 6 2 for: jrj + jr0 j < 34 (k )2 (M axfp; p0 g)4 = 34 (k )2 S 4 . By solving linear equation (in variable k ) (4.10) and using some elementary trigonometric transformation one has: k =
6(cos \0 2 + cos \0 20 ) +R d24 (d12 sin2 (\0 2) + d23 sin2 (\0 20 ))
where: jRj
0 and, moreover, (Q) ! 0 ) Q ! linearity.] (b) Since (Q) 6= 0 it follows that: K(Q) 2 R for any quadrangle Q. In addition: sign(k(Q)) = sign(K(Q)). (c) If Q is any sd-quad, then 2 (Q)diam2 (Q)= (Q) < 1. Moreover, 25 if (Q) = 0, then 2 (Q)diam2 (Q)= (Q) = 0 i.e. jRj is small if Q is not close to linearity. In this case jR(Q)j diam2 (Q) (for any given Q). 24and the analogous formulas for cos 2 \k0 2 . 2 25i.e. for not very small values of (Q)
24
EMIL SAUCAN
Since the Gaussian curvature kG (p) at a point p is given by: kG (p) = lim (Q n ) ; n!0
where Qn ! Q = p1 pp3 p4 ; diam(Qn ) ! 0,from Remark 4.6.(c) we immediately infer that the following holds26: Theorem 4.7. Let S be a di erentiable surface. Then, for any point p 2 S: kG (p) = lim K(Qn ) ; n!0
for any sequence fQn g of sd-quads that satisfy the following condition: Qn ! Q =
p1 pp3 p4 ; diam(Qn ) ! 0 :
Remark 4.8. In the following special cases even "nicer" formulas are obtained: (1) If d12 = d32 , then cos \0 2 + cos \0 20 ; d13 24 d sin2 \0 2 + sin2 \0 20 (here we have of course: d13 = 2d12 = 2d32 ); or, expressed as a function of distances alone: 2d212 + 2d224 d214 d213 (4.12) K(Q) = 12 2 2 2 8d12 d24 (d12 + d224 d214 )2 (d212 + d224 d234 )2
(4.11)
K(Q) =
12
(2) If d12 = d32 = d24 and if the following condition also holds: (3) \0 20 = =2; i.e. if d 234 = d212 + d224 or, considering (2), also: d234 = 2d212 then 2d212 d214 6 cos \0 2 = 4 (4.13) K(Q) = : 2 4d12 + 4d214 d212 d414 d12 (1 + sin \0 2) 5. Appeendix 1 { The Menger and Haantjes Curvatures Better known than the Wald Curvature, the Menger Curvature is a metric definition of curvature of curves, as is the Haantjes Curvature. As such they can be employed as sectional curvatures to approximate curvature of triangulated surfaces. We begin by introducing the the Menger Curvature: this is a metric expression for the circum-radius of a triangle27, based upon elementary high-school formulas: De nition 5.1. Let (M; d) be a metric space, and let p; q; r 2 M be three distinct points. Then: p (pq + qr + rp)(pq + qr rp)(pq qr + rp)( pq + qr + rp) ; KM (p; q; r) = pq qr rp is called the Menger Curvature of the points p; q; r. We can now de ne the Menger Curvature at a given point by passing to the limit: De nition 5.2. Let (M,d) be a metric space and let p 2 M be an accumulation point. Then M has at p Menger Curvature M (p) i 8 " > 0; 9 > 0 s.t. d(p; pi ) < ; i = 1; 2; 3 =) jK(Q) M (p)j < ". 26and gives theoretical justi cation to the algorithm
27 thus giving in the limit a metric de nition of the Osculatory Circle
SURFACE TRIANGULATION { THE METRIC APPROACH
25
Remark 5.3. The apparent equivalent notion of Alt Curvature, in which one uses only two points converging to the third, is in fact more general, where we de ne the Arp curvature by: De nition 5.4. Let (M,d) be a metric space and let P 2 M be an accumulation point. Then M has at p Alt Curvature A (p) i the following limit exists A (p)
4
= lim K(p; q; r) : q;r!p
However, both M (p) and A (p) su er from the same imperfection: since they are both modelled closely after the Euclidian Plane, they convey this Euclidian type of curvature upon the space they are de ned on. However, the next de nition doesn’t mimic closely R2 so it better tted for generalizations: De nition 5.5. Let (M,d) be a metric space and let c : I = [0; 1] ! M be a homeomorphism, and let p; q; r 2 c(I); q; r 6= p. Denote by qr b the arc of c(I) between q and r, and by qr segment from q to r. (See Figure 12 bellow.)
C
r qr qr
p
q
Figure 12. Then c has Haantjes Curvature 2 H (p)
H (p)
at the point p i :
= 24 lim
l(qr) b
q;r!p
d(q; r)
l(qr) b
3
;
where "l(qr)" b denotes the length28 of qr. b Remark 5.6. H exists only for recti able curves, but if of c, then c is recti able.
M
exists at any point p
Remark 5.7. Evidently we have the following relationship between curvatures: 9
M
=) 9
9
A
=) = 9
A
:
while However, we can prove the following theorem: 28 given by the intrinsic metric induced by d
M
:
26
EMIL SAUCAN
Theorem 5.8. Let c : I ! M be a recti able curve, and let p 2 M . If A (or M ) exists, then H (p) exists and A
=
H (p) :
Remark 5.9. This last result and the Remark preceding it allow as to employ any of the curvatures above in estimating curvatures of smooth curves on triangulated surfaces. 6. Appendix 2 { The Rinow Curvature The curvatures introduced before may seem a bit archaic in comparison to the more fashionable approach of comparison triangles, with their ar reaching applications. We present here one of these comparison criteria and show its equivalence with the Wald curvature. We start with the following de nition: De nition 6.1. Let (M; d) be a metric space, together with the intrinsic metric induced by d. Let R = int(R) M be a region of M . We say that R is a region of curvature ( 2 R) i (1) 8p; q 2 R ; 9 a geodesic segment pq R; (2) 8 T (p; q; r) R is isometrically embeddable in S ; (3) If T (p; q; r) R and x 2 pq; y 2 pr, and if the points p ; q ; r ; x ; y 2 S satisfy the following conditions: (a) T (p; q; r) = T (p ; q ; r ); (b) T (p; q; x) = T (p ; q ; x ); (c) T (p; r; y) = T (p ; r ; y ); then xy xy . By replacing the condition: "xy x y " with: "xy x y ", we obtain the de nition of a region of curvature . (See Fig. 13.)
xκ
qκ
yκ
pκ
2
Sκ
rκ
q x
R
p
M
y
r
Figure 13.
(κ > 0)
SURFACE TRIANGULATION { THE METRIC APPROACH
27
We now pass to the localization of the De nition above: De nition 6.2. Let (M; d) be a metric space, together with the intrinsic metric induced by d, and let p 2 M be an accumulation point. Then M has at p Rinow Curvature R (p) i (i) =9 N 2 N (p); N linear; (ii) 8 " > 0; 9 :0, s.t.B(p; ) is (a) a region of Rinow curvature R (p) + " and (b) a region of Rinow curvature (p) ". R While its greater generality endows the Rinow curvature with more exibility in applications and makes it easier in generalization, it is even more di cult to compute than Wald Curvature. However this quandary was has an almost ideal solution, due to Kirk (see [K]), solution which we brie y expose here: De nition 6.3. Let M be a compact, convex metric space, and let p 2 M . If W (p) exists, then R (p) exists, and R (p) = W (p). Unfortunately, since R (p) makes no presumption of dimensionality, the existence of R (p) does not imply the existence of W (p). Counterexample 6.4. Let M R3 . Then R (p) 0 but W (p) does not exist at any point, since every neighborhood contains linear quadruples. The solution (due to Kirk) of this problem is to consider the Modi ed Wald curvature W K , de ned as follows: De nition 6.5. Let (M; d) be a metric space, together with the intrinsic metric induced by d, and let p 2 M . Then M has at p Modi ed Wald curvature Curvature W K (p) i (i) =9 N 2 N (p); N linear; (ii) 8 " > 0; 9 :0, s.t. if Q B(p; ) is a non-degenerate sd-quad, then W (Q) exists and j W K (p) W (Q)j < ". Remark 6.6. 9
W (p)
=) 9
W K (p)
but 9
KW (p)
=) =
W (p).
This modi ed curvature indeed represents the wished for solution, as proved by the following to Theorems: Theorem 6.7. Let (M; d) be a metric space. Then: 9 R (p) =) 9 W K (p) and R (p) = W K (p). Theorem 6.8. Let (M; d) be a metric space together with the associated intrinsic metric, and let p 2 M . Then, if (i) W K (p); and if (ii) 9 B(p; ) 2 N (p), s.t. qr B(p; ); 8q; r 2 B(p; ); then R (p) exists and R (p) = W K (p). 7. Appendix 3 { The Radius Formula The Cayley-Menger determinant allows one to express not only the volume and area29 of simplices in Rn but (as expected) it may be used to compute the radius 29The 2-dimensional analogue of Formula (4.4) for the area of the triangle T (p ; p ; p ) being: 1 2 3
Area (p1 ; p2 ; p3 )
2
=
D(p1 ; p2 ; p3 ) :
28
EMIL SAUCAN
of the circumscribed sphere around an Euclidian simplex. To be more precise, we have the following result30: Theorem 7.1. (1) The radius R = R(p1 ; p2 ; p3 ; p4 ) of the sphere circumscribed around the tetrahedron T (p1 ; p2 ; p3 ; p4 ) 2 R3 is given by: R2 =
1 (p 1 ; p2 ; p3 ; p4 ) 2 D(p1 ; p2 ; p3 ; p4 )
where:
0 d212 d213 d214
(p 1 ; p2 ; p3 ; p4 ) =
d212 0 d223 d224
d213 d223 0 d234
d214 d224 d234 0
(2) The points p1 ; p2 ; p3 ; p4 ; p5 2 R3 are coplanar or co-spherical i (p 1 ; p2 ; p3 ; p4 ; p5 ) = 0 : Proof (1) If p0 2 R3 is s.t. d0i = R; i = 1; :::; 5, then by direct computation we obtain: (p 0 ; :::; p5 ) =
2R2 (p 1 ; :::; p5 )
(p 1 ; p2 ; p3 ; p4 ; p5 ) ;
from which the desired formula follows immediately if we chose p0 as the center of the sphere circumscribed around the points p1 ; p2 ; p3 ; p4 ; p5 . (2) (=)) Let fx1 ; x2 ; x3 g be any orthonormal coordinate frame for R3 , and let pji ; i = 1; :::; 5; j = 1; 2; 3; represent the coordinates of the points p1 ; p2 ; p3 ; p4 ; p5 relative to this coordinate system. Then p1 ; p2 ; p3 ; p4 ; p5 belong to the same sphere or plane i 9 (a; b; c j ) 6= (0; 0; 0); j = 1; 2; 3; s.t. ajjpi jj2 + b +
3 X
cj pji ; i = 1; :::; 5:
j=1
Then
1
=
2
= 0, where:
1 (p1 ; p2 ; p3 ; p4 ; p5 )
and 2 (p1 ; p2 ; p3 ; p4 ; p5 )
Therefore proven.
1
t 2
= 0. But
1
=
=
1 1 1 1 1 t 2
jjp1 jj2 jjp2 jj2 jjp3 jj2 jjp4 jj2 jjp5 jj2 jjp1 jj2 jjp2 jj2 jjp2 jj2 jjp2 jj2 jjp2 jj2
1 1 1 1 1
p11 p12 p13 p14 p15 2p11 2p12 2p13 2p14 2p15
p21 p22 p23 p24 p25 2p21 2p22 2p23 2p24 2p25
p31 p32 p33 p34 p35 2p31 2p32 2p33 2p34 2p35
= (p 1 ; p2 ; p3 ; p4 ; p5 ), so this implication is
30that can be readily generalized to higher dimensions
SURFACE TRIANGULATION { THE METRIC APPROACH
((=) (p 1 ; p2 ; p3 ; p4 ; p5 ) = 0 =) (0; 0; 0); j = 1; 2; 3; s.t. ajjpi jj2 + b +
1
3 X
29
= 0 and there exist numbers (a; b; cj ) 6=
cj pji = 0 ; i = 1; :::; 5;
j=1
i.e. p1 ; p2 ; p3 ; p4 ; p5 belong to the plane or the sphere given by the equation ajjXjj2 + b + c
3 X
cj X = 0 ; X = (x1 ; x2 ; x3 ) :
j=1
References [Ba1] Bancho , T.A. { Critical points and curvature for embedded polyhedra, J. Di erential Geometry, 1 (1967), 257-268. [Ba2] Bancho , T.A. { Critical Points and Curvature for Embedded Polyhedral Surfaces, Amer. Math. Monthly, 77 (1970), 475-485. [Be] Berger, M. { Geometry I, Universitext, Spinger-Verlag, 1987. [B] Blumenthal, L. M. Distance Geometry { Theory and Applications, Claredon, 1953. [BM] Blumenthal, L. M. and Menger, K. { Studies in Geometry, Freeman and Co., 1970. [BH] Bridson, M. R. and Hae iger, A. { Metric spaces of non-positive curvature , Grundlehren der mathematischen Wissenschaften, Springer-Verlag, Berlin, 1999. [BBI] Burago, D. , Burago, Y. and Ivanov, S. { A Course in Metric Geometry, GSM, AMS, RI, 2000. [BCM] Borrelli, V. Cazals, F. and Morvan, J.-M. { On the angular defect of triangulations and the poitwise approximation of Curvatures, Computer Aided Geometric Designs, 20, pp. 319341, 2003. [CMS] Cheeger, J. , Muller, W. , and Schrader, R. { On the Curvature of Piecewise Flat Spaces, Comm. Math. Phys. , 92, 1984, 405-454. [C-SM] Cohen-Steiner, D. and Morvan, J.-M. { Restricted Delaunay triangulations and normal cycle, preprint, 2003. [F] Fu, J. H. G. { Convergence of Curvatures in Secant Approximation, J. Di erential Geometry, 37, 1993, 177-190. [G+] Mikhail Gromov { Metric structures for Riemannian and non-Riemannian spaces, Progress in Mathematics 152, Birkhauser, Boston, 1999. [K] Kirk, W. A. { On Curvature of a Metric Space at a Point, Paci c J. Math. 14: 195-198, 1964. [LWZL] Liu, G.H. , Wong, Y.S. , Zhang, Y.F. and Loh, H.T. { Adaptive fairing of digitized point data with discrete curvature, Comuter Aided Design, vol. 34(4), 309-320, 2002. [MD] Maltret, J.-L. and Daniel, M. { Discrete curvatures and applications: a survey, preprint, 2003. [P] Pajot, H. { Analytic Capacity, Recti cabilility, Menger Curvature and the Cauchy Integral, LNM 1799, Springer, Berlin, 2002. [RR] Ramsay, A. and Richtmayer, R.D. { Introduction to Hyperbolic Geometry, Universitext, Spinger-Verlag, 1991. [Rat] Ratcli e, J.C. : Foundations of Hyperbolic Manifolds, GTM 194, Springer Verlag, N.Y., 1994. [R] Robinson, C.V. { A Simple Way of Computing the Gauss Curvature of a Surface, Reports of a Mathematical Colloquium, Second Series, Issue 5-6, 16-24, 1944. [SMSER] Surazhsky, T. , Magid, E. , Soldea, O. , Elber, G. and Rivlin, E. { A Comparison of Gaussian and Mean Curvatures Estimation Methods on TRiangular Meshes, preprint, 2003. [T] Troyanov, M. { Tangent Spaces to Metric Spaces: Overview and Motivations, preprint, 2003.
30
EMIL SAUCAN
Department of Mathematics,Technion & Department of Software Engineering, Ort Braude College, Karmiel E-mail address: [email protected]
arXiv:cs/0401023v1 [cs.GR] 26 Jan 2004
EMIL SAUCAN Abstract. We embark in a program of studying the problem of better approximating surfaces by triangulations(triangular meshes) by considering the approximating triangulations as nite metric spaces and the target smooth surface as their Haussdor -Gromov limit. This allows us to de ne in a more natural way the relevant elements, constants and invariants s.a. principal directions and principal values, Gaussian and Mean curvature, etc. By a "natural way" we mean an intrinsic, discrete, metric de nitions as opposed to approximating or paraphrasing the di erentiable notions. In this way we hope to circumvent computational errors and, indeed, conceptual ones, that are often inherent to the classical, "numerical" approach. In this rst study we consider the problem of determining the Gaussian curvature of a polyhedral surface, by using the embedding curvature in the sense of Wald (and Menger). We present two modalities of employing these de nitions for the computation of Gaussian curvature.
1. Introduction 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 de ned for smooth surfaces only, and usually attacked with di erential tools, tools that { however ingenious and learned { can hardly represent good approximations for curvature of P L-surfaces, since they are usually just discretizations of formulas developed in the smooth (i.e. of class at least C 2 ) case.1 Moreover, since considering triangulations, one is faced with nite graphs, or, in many cases (when given just the vertices of the triangulation) only with nite {thus discrete { metric spaces. Therefore, the following natural questions arise: (A) Is one fully justi ed in employing discrete metric spaces when evaluating numerical invariants of continuous surfaces? and (B) Can one nd discrete, metric equivalents of the di erentiable notions, notions that are intrinsically more apt to describe the properties of the nite spaces under investigations? One is further motivated to ask the questions above, since the metric method we propose to employ have already Date: 24.01.2004. We would like to thank Prof. Gershon Elber of The Computer Science Department, The Technion, Haifa, who motivated and sustained this project. 1 However one can nd very scienti cally sound discrete versions of Surface Curvature can be found, for instance, in [Ba2], [BCM], [C-SM] . 1
2
EMIL SAUCAN
successfully been used in the such diverse elds 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 a rmative, 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 e orts on the theoretical level and study the Lipschitz and Gromov-Hausdor 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-Hausdor limit of a sequence of nite graphs. 2 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: rst 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 rst 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 identi es 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 de nition is based upon a very simple 3 idea: it measures the relative di erence between metrics, more precisely it evaluates their ratio; i.e.: x;f y) 1; The metric spaces (X; dX ), (Y; dY ) are close i 9 f : X ! Y s.t. dYdX(f(x;y) 2 For the relevance of these notions in the study of classical curvatures convergence, see [CMS],
[F] .
3 That is to say: very intuitive, i.e. based upon physical measurements.
SURFACE TRIANGULATION { THE METRIC APPROACH
3
8 x; y 2 X.4 Technically, we give the following: De nition 2.1. The map f : (X; dX ) ! (Y; dY ) is bi-Lipschitz i 9 c; C > 0 s.t.: c
d (x; y) X
dY (f x; f y)
C
Xd(x; y) :
Remark 2.2. The same de nition applies for two di erent metrics d 1 ; d2 on the same space X. De nition 2.3. Given a Lipschitz map f : X ! Y , we de ne the dilatation of f by: dY (f x; f y) : dil f = sup x6=y2X dX (x; y) Remark 2.4. The dilatation represents the minimal Lipschitz constant of maps between X and Y . 4
Remark 2.5. If f is not Lipschitz, then dil f = 1. Remark 2.6. (1) f Lipschitz ) f continuous. (2) f bi-Lipschitz ) f homeo. on its image. Remark 2.7. We have the following results: Proposition 2.8. Let f; g : X ! Y be Lipschitz maps. Then: (a) g f is Lipschitz and (b) dil (g f ) dil f dil g Proposition 2.9. The set ff : (X; d) ! R j f Lipschitzg is a vector space. Now we can return to our main interest and de ne the following notion: De nition 2.10. Let (X; dX ), (Y; dY ) be metric spaces. Then the Lipschitz distance between (X; dX ) and (Y; dY ) is de ned as: dL (X; Y ) =
inf
log max (dil f; dil f
1
)
f :X !Y f bi Lip:
Remark 2.11. If 6= f bi-Lipschitz between X and Y , then { remembering Remark 4 2.2 { we put dL (X; Y ) = 1 (i.e. dL is not suited for pairs of spaces that are not bi-Lipschitz equivalent.) having de ned the distance between two metric spaces we now can de ne the convergence in this metric in the following natural way: De nition 2.12. The sequence of metric spaces f(Xn ; dn )g convergence to the metric space f(X; d)g i lim dL (Xn ; X) = 0 (In this case we write: (Xn ; dn )
n !0). L
Example 2.13. Let St be a family of regular surfaces, St = ft (U ); where U is an open set, U = int U R; such that the family fft g of parametrizations is smooth not ! 0. (i.e. F : U R ! R3 2 C 1 ; where F ((u; v); t) = ft (u; v)). Then dn (St ; S0 )t!0 4 Here and in the sequel "f x" etc. ... stands as a short-hand version of "f (x)".
4
EMIL SAUCAN
Remark 2.14. If F is not smooth (only continuous) then we do not necessarely have !S . that St t!0 0 We have the following signi cant theorem: Theorem 2.15. The dL satis es the following conditions: (a) dL 0; (b) dL is symmetric; (c) dL satis es the triangle inequality; Moreover, if X; Y are compact, then: (d) dL (X; Y ) = 0 , X = Y (i.e. X is isometric to Y ); that is
dL is a metric on the space of isometry classes of compact metric spaces Remark 2.16. Let us recall the following De nition 2.17. (Xn ; dn )
!(X; d) u
4
= du
! u
sup jdn (x; y)
x;y2X
d as a real function; i.e.
d(x; y)j u!0
(where "u" denotes uniform convergence.) Then Xn u!X ) Xn L!X but Xn indeed Xn u!X , Xn L!X.
!X L
) = Xn
!X. u
However, for nite spaces
2.2. Gromov-Hausdor 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 nite (even for pairs of non-homeomorphic spaces.) )5 We start by rst introducing the classical 2.2.1. Hausdor distance. De nition 2.18. Let A; B (X; d). We de ne the Hausdor distance between A and B as: dH (A; B) = inffr > 0 j A Ur (B); B Ur (A)g 4 S (see Fig. 1); where Ur (A) is the r-neighborhood of A, Ur (A) = a2A Br (a); (here, as usual: Br (a) = fx 2 X j d(a; x) < rg.) Another (equivalent) way of de ning the Hausdor distance is as follows: dH (A; B) = maxfsup d(a; B); sup d(b; A)g : a2A
b2B
(see Fig. 2) We have the following Proposition 2.19. Let (X; d) be a metric space. Then: (a) dH is a semi-metric (on 2X ). (i.e. A = B ) dH (A; B) = 0.) (b) dH (A; A) = 0; 8A X. (c) A = A and B = B ) dH (A; B) = 0 , A = B . i.e. dH is a metric on the set of closed subsets of X. 5The relationship between the Lipschitz and the Hausdor distances is akin to that between the C 0 and C 1 norms in Functional Spaces.
SURFACE TRIANGULATION { THE METRIC APPROACH
U(A) r
5
U(B) r
B
r
A
X
Figure 1. X j A = Ag; dH = 2X =dH .
Notation We put: M(X) = fA
Remark 2.20. (1) if X is compact and if fAn gn subsets of X, then: T (a) An+1 An ) An H! n 1 An . (b) An
An+1 ) An
(2) For general subsets An
! H
S n 1
!A H
1
X is a sequence of compact
An .
2 M(X) , and
(a) A = flimn an j an 2 An ; n
1g .
T S1 (b) A = H! n 1 m=n Am . ! (3) If An H A, and if the sets An are all convex, then A is convex sets. We have the following two important results, which we present without their respective (lengthy) proofs: Proposition 2.21. X complete ) M(X) complete . Theorem 2.22. (Blaschke) X compact ) M(X) compact .
d(a,b)
b
a
A
B
X Figure 2.
6
EMIL SAUCAN
2.3. The Gromov-Hausdor Distance. We are now able to de ne the GromovHausdor distance using the following basic guide-lines: we want to get the maximum distance that satis es the following two conditions: (a) dGH (A; B) dH (A; B); 8A; B X (i.e. set that are close as subsets of X will still be close as abstract metric spaces); and (b) X isometric to Y () dGH (X; Y ) = 0. De nition 2.23. Let X; Y be metric spaces. Then the Gromov-Hausdor distance between X and Y is de ned by: 4
dGH (X; Y ) = inf dZ H (f (X); g(Y )) where the in mum is taken over all the isometric embeddings f : X ,! Z; g : Y ,! Z into some metric space Z. (See Fig. 3). Y
X
g
f
f(X)
dHZ
g(Y)
Z
Figure 3. Remark 2.24. If X = S2 , with the spherical metric, and Z = R3 , with the Euclidian metric, then f (X) 6= X (!) Example 2.25. Let Y be an "-net6 in X. Then dGH (X; Y ) Proof Take Z = X = X 0 ; Y = Y 0 .
".
Remark 2.26. It is su cient ‘ to consider embeddings f into the disjoint union of the spaces X and Y , X Y . Remark 2.27.
(1) X; Y bounded =) dGH (X; Y )
(2) If diamX; diamY < 1, then dGH (X; Y ) 6 De nition Let (X; d) be a metric space, and let A
d(x; A)
"; 8x 2 X.
4 7diamX = supx;y2X d(x; y)
1.
1 2 jdiamX
diamY j7.
X. A is called an "-net i
SURFACE TRIANGULATION { THE METRIC APPROACH
7
However , the straightforward de nition of d GH may be di cult to implement. Therefore we would like to estimate (compute) dGH by comparing distances in X vs. distances in Y (as done in the cases of uniform and Lipschitz metrics). We start by de ning a correspondence between metric spaces: X ! Y , given by correspondences x $ y between points x 2 X; y 2 Y . Remark 2.28. A correspondence is not necessarely a function, that is to a single x may correspond to several y-’s. We shall prove that ( ) dGH (X; Y ) < r () 9 a correspondenceX 0
!Y
0
0
s:t: (x $ y; x $ y ) =) jdX (x; x )
dY (y; y 0 )j < 2r
Formally, we have: De nition 2.29. Let X; Y denote sets. A correspondence X the Cartesian product of X and Y : R X Y s.t. (i) 8x 2 X; 9y 2 Y; s:t:(x; y) 2 R; and (ii) 8y 2 Y; 9x 2 X; s:t:(x; y) 2 R.
! Y is a subset of
Example 2.30. Any surjective function f : X ! Y represents correspondence R = f(x; f (x))g. Remark 2.31. R is a correspondence () 9 Z and 9 f : Z ! X ; 9g : Z ! Y ; f; g surjective, s.t. R = f(f (z); g(z)) j z 2 Zg. De nition 2.32. Let R be a correspondence between X and Y , where X; Y are metric spaces. We De ne the distortion of R by: dis R = sup jdX (x; x0 )
dY (y; y 0 )j (x; y) ; (x0 ; y 0 ) 2 R :
(See (*) .) Remark 2.33. (1) If R = f(x; f (x))g is a correspondence induced by a surjective function f : X ! Y , then dis R = dis f , where: 4
dis f = sup j dY (f a; f b) a;b2X
dX (a; b) j8 :
(2) If R = f(f (z); g(z))g, where f : X ! Z; g : Y ! Z are surjective functions, then: dis R = sup f j dX (f z; f z 0) dY (gz; gz 0) j g : z;z 0 2Z
(3) R = 0 i R is associated to an isometry. We bring, without proof, the following theorem: Theorem 2.34. Let X; Y be metric spaces. Then: 1 dGH (X; Y ) = inf (dis R) ; 2 R where the in mum is taken over all the correspondences X
R
!Y.
8 Remember that any correspondence can be expressed in this functional manner.
8
EMIL SAUCAN
Before bringing the next result (which is very important in determining the topology ) we rst introduce one more notion: De nition 2.35. f : X ! Y is called an "-isometry (" > 0), i (i) dis f ", and (ii) f (x) is an "-net in Y . Remark 2.36. f "-isometry =) = f continuous. Corollary 2.37. Let X; Y be metric spaces and let " > 0. Then: (i) dGH (X; Y ) < " =) 9 2" isometry f : X ! Y . (ii) 9 " isometry f : X ! Y =) dGH (X; Y ) < 2". R
Proof (i) Let X ! Y s.t. dis R < 2". For any x 2 X and f (x) 2 Y , choose y = f (x) s.t. (x; f (x)) 2 R. Then x 7! f (x) de nes a map f : X ! Y . Moreover: dil f dil R < ". We shall prove that f (X) is a 2"-net in Y . Indeed, let x 2 X and y 2 Y s.t. (x; y) 2 R. Then d(y; f x) d(x; x) + dis R < 2r, thence d(y; f (X)) < 2r. Let f be an 2"-isometry. De ne R X Y; R = f(x; y) j d(y; f x) Then, since f (X) is an "-net it follows that R is a correspondence. Then 8 (x; y); (x0 ; y 0 ) 2 R we have: j dY (y; y 0 ) dX (x; x0 ) j
j d(f x; f x0 ) d(x; x0 ) j+d(y; f x)+d(y 0 ; f x0 )
=) dis R
3" =) dGH (x; y)
"g.
dis f +"+"
3r=2 < 2r :
The next result is of great importance (in particular so in our context): Theorem 2.38. dGH is a ( nite) metric on the set of isometry classes of compact metric spaces. Proof It su ces to prove that d GH (X; Y ) = 0 =) X Y .9 Indeed, let X; Y be compact spaces s.t. dGH = 0. Then it follows from the previous Corollary (for " = 1=n) that 9 (fn )n 1 ; fn : X ! Y s.t. dis fn n!0. let S X; S ; jSj = @0 . Using a Cantor-diagonal argument one easily shows that (fn )n 1 s.t. (fnk ) 1 converges in Y; 8x 2 S. Without restricting the 9 (fnk )k 1 generality we may assume that this happens for (fn )n 1 itself. Thus we can de ne a function f : X ! Y by putting: f (x) = limn fn (x). But j d(fn x; fn y) d(x; y) j dis fn n!0 =) d(f x; f y) = lim d(fn x; fn y). In other words f jS is an isometry. But S = S, therefore this isometry can be extended to an isometry f~ from X to Y . In a analogous manner one shows the existence of an ~ isometry f~ : X ! Y . Remark 2.39. Xn
!X L
=) Xn
! u X
! X. =) Xn GH
In fact, the following relationship exists between " Theorem 2.40. Xn 9We shall write: X
!X GH
() "-nets in Xn
Y if X is isometric to Y .
!" L
!"-nets L
! ": and "GH
in X.
3" :
SURFACE TRIANGULATION { THE METRIC APPROACH
9
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: De nition 2.41. Let X; Y be compact metric spaces, and let "; > 0. X; Y are called "- -approximations (of each-other) i : 9 fxi gN X, 9 fyi gN Y s.t. i=1 i=1 N N (i) fxi gi=1 is an "-net in X and fyi gi=1 is an "-net in Y ; (ii) j dX (xi ; xj ) d( yi ; yj ) j < 8 i; j 2 f1; :::; N g. An ("; ")-approximation is called, for short: an "-approximation. The relationship between this last de nition and the Gromov-Hausdor distance is rst revealed in Proposition 2.42. Let X; Y be compact metric spaces. Then: (1) If Y is a ("; )-approximation of X, then dGH (X; Y ) < 2" + . (2) dGH (X; Y ) < " =) Y is a 5"-approximation of X. Proof (1) Condition (ii) of Def. 2.41. is equivalent to dis RX0 Y0 < , where N X0 = fxi gN =) dGH (X0 ; Y0 ) < =2. Now, since i=1 ; fyi gi=1 . But dis RX0 Y0 < X0 and Y0 are "-nets in X, resp. Y , it follows that dGH (X; X0 ) "; dGH (Y; Y0 ) < ". From here and from the dGH (X0 ; Y0 ) < =2 follows, by means of he triangle inequality, that dGH (X; Y ) < 2" + . (2) By Cor. 2.37., there exists a 2"-isometrie f : X ! Y . Let X0 = fxi gN i=1 be an "-net, and let yi = f (xi ). Then j d(xi ; xj ) d( yi ; yj ) j < 2" < 5". Therefore su ce to prove that Y 0 = fyi gN i=1 is a 5"-net in Y . Indeed, if y 2 Y , then, since f (X) is an 2"-net in Y , 9 x 2 X s.t. d(y; f (x)). Now, since X0 is an "-net in X, 9xi 2 X0 , s.t. d(x; xi ) ". Therefore: d(y; yi ) = d(y; f (xi )) d(y; f (x)) + d(f (x); f (xi )) 2" + d(x; xi ) + dis f 2" + " + 2" 5". Remark 2.43. Prop. 2.42. () (Xn
! X) GH
, (8 " > 0; Xn is an "-approximation, 8 n large enough.)
More precisely we have the following Proposition: Proposition 2.44. Let X; fXn g1 1 compact metric spaces. Then: Xn !X () 8" > 0; 9 a nite "-net S X and 9 a nite "-net Sn GH ! Sn S and, moreover, jSn j = jSj, for large enough n.
Xn , s.t.
GH
2:42:
Proof ((=) If S; Sn exist, then =) Xn is an "-approximation of X =) Xn
!X GH
8 n.
(=)) Let S be an nite "=2-net in X. We construct in Xn corresponding nets Sn (to be more precise, we de ne: S n = fn (X), where fn is an "n -approximation, fn : X ! Xn ; "n ! 0.) Then Sn !S GH and, in addition, Sn is an "-net in S (for n large enough). We make the following extremely important Remark:
10
EMIL SAUCAN
Remark 2.45. Let M(n; k; D) be an n-dimensional manifold, of (sectional, Ricci) curvature k, and s.t. diam M D. Then (M; dGH ) is compact. However, it should be noted that this result doesn’t hold for curvature < k. (only for V ol(M) V0 ) and injectivity radius r0 . Note With the notations of the precedent Proposition, the distances in Sn con! X, therefore The Geometric Proprieties verge to the distances in S, as Xn GH of Sn will converge to those of S. Thus we can use the Gromov-Hausdor each and every time The Geometric Proprieties of Xn can be expressed in term of a nite number of points, and, by passing to the limit, automatically obtain proprieties of X. 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 R3 .) The induced (intrisic) metric
The Euclidian metric
c
S
3
R
Figure 4. On a more formal note, we have the following characterization of intrinsic metrics: Theorem 2.46. Let (X; d) be a complete metric space. (1) If 8 x; y 2 X; 9 21 xy, then d is strictly intrinsic. (2) If 8 x; y 2 X and 8" > 0; 9 the "-middle of xy, then d is intrinsic. Where we used the following de nitions and notations: De nition 2.47. (1) Given x; y points in (X; d), the middle (or midpoint) of the segment xy (more correctly: ’a midpoint between "x" and "y" ’) is de ned as: 1 xy = z; d(x; z) = d(z; y) : 2 (2) d is called strictly intrinsic i the length structure is associated with is complete. (3) Let d be an intrinsic metric. z is an "-middle (or an "-midpoit) for xy i : j 2d(x; z) d(x; y) j " and j 2d(y; z) d(x; y) j ". Remark 2.48. The converse of Thm. 2.46. holds in any metric space, more precisely we have: Proposition 2.49. If d is an intrinsic metric, then
1 2 xy
exists, 8x; y.
SURFACE TRIANGULATION { THE METRIC APPROACH
11
The following Theorem shows that length spaces are closed in the GH-topology : Theorem 2.50. Let fXn g be length spaces and let X be a complete metric space s.t. Xn !X. GH Then X is a length space. Proof We have already presented the idea of the proof: it is su cient to show that for every x; y there exist an "-midpoit (8" > 0). " . Then, from the a preceding result, it follows Indeed, let n be such that dGH < 10 that there exist a correspondence Xn R!X s.t. dis R < 5" . R R Let x; y 2 Xn , x$ x, y $ y. Since Xn is a length space, =) 9z 2 Xn s.t. z = 5" R midpoint of xn yn . Consider z 2 X; z $ z. Then: 1 " 2" 1 jxyj jxzj jxyj + 2dis R < + < ": jxzj 2 2 5 5 (Here we write jxyj instead of d(x; y), etc.) In a similar manner we show that: jyzj 21 jxyj < "; i.e. "-midpoit of xy. The next Theorem and its Corollary are of paramount importance: Theorem 2.51. Any compact length space is the GH-limit of a sequence of nite graphs. Proof Let "; ( ") small enough, and let S be a -net in X. Let G = (V; E) be the graph with V = S and E = f(x; y) j d(x; y) < "g. we shall prove that G is an "-approximation of X, for small enough (i.e. for < "2 4 diam(X)). (See Fig. 5.)
Sn
G
X
Figure 5. But, since S is an "-net both in X and in G, and since dG (x; y) dX (x; y), it is su cient to prove that: dG (x; y) dX (x; y) + " : Let be the shortest path between x and y, and let x1 ; :::; xn 2 s.t. n length( )=" (and jxi ; xi+1 )j "=2). Since 8 xi 9yi 2 S s.t. jxi ; yi j , it follows that jyi yi+1 j jxi xi+1 j + 2 < ": (See Fig. 6)
12
EMIL SAUCAN
Therefore, (for < "=4) 9 an edge e 2 G; e = yi yi+1 . From this we get the following upper bound for dG (x; y): n 0 jyi yi+1 j
dG (x; y) But n < 2length( )="
+2 n
2diam(X)="; therefore:
dG (x; y) (because
n 0 jxi xi+1 j
jxyj +
4diam(X) < jxyj + " : "
< "2 =4diam(X)).
y=x
xn-1 x i+1
.
xi
. . . .. x y
3
x
2
n
yn
...
S
yi+1
yn-1
y
i
G
y2
x=x 1 y1
x Figure 6. So, for any " > 0; 9 G = G" an "-approximation of X. Then, Gn ! " X. Corollary 2.52. Let X be a compact length space. Then X is the GromovHausdor limit of a sequence fG n gn 1 of nite graphs, isometrically embedded in X. Remark 2.53.
(1) If Gn ! " X, Gn = (Vn ; En ). If 9N0 2 N s.t. (?) jEn j
N0 ; 8n 2 N ;
then X is a nite graph. (2) If condition (?) is replaced by: (??) jVn j
N0 ; 8n 2 N ;
then X will still be always a graph, but not necessarily nite(!)
SURFACE TRIANGULATION { THE METRIC APPROACH
13
3. The Embedding Curvature 3.1. Theoretical Setting. This is basically a comparison-curvature (as is the more "modern" CAT 10 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 classically11 the "minimal" geometric gures that allow the di erentiation 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 R3 .12 De nition 3.1. Let (M; d) be a metric space, and let Q = fp1 ; :::; p4 g M, together with the mutual distances: dij = dji = d(pi ; pj ); 1 i; j 4. The set Q together with the set of distances fdij g1 i;j 4 is called a metric quadruple. Remark 3.2. One can de ne metric quadruples in slightly more abstract manner, without the aid of the ambient space: a metric quadruple being a 4 point metric space; i.e. Q = fp1 ; :::; p4 g; fdij g , where the distances dij verify the axioms for a metric. Before we proceed to the next de nition, let us introduce the following Notation S denotes the complete, simply connected surface of constant curvature , if < 0. Here H2p , i.e. S R2 , if = 0; S S2p , if > 0; and S p 2 2 p p stands for the S S denotes the Sphere of radius R = 1= , and S H p Hyperbolic Plane of curvature , as represented by the Poincare Model of the p plane disk of radius R = 1= De nition 3.3. The embedding curvature (Q) of the metric quadruple Q is dened be the curvature of S into which Q can be isometrically embedded. (See Figures 7 and 8 for embeddings of the metric quadruple in S and H , respectively.) We can now de ne the embedding curvature at a point in a natural way by passing to the limit (but without neglecting the existence conditions), more precisely: De nition 3.4. Let (M; d) be a metric space, and let p 2 M be an accumulation point. Then p is said to have Wald curvature W (p) i (i) =9 N 2 N (p); N linear13 ; (ii) 8 " > 0; 9 > 0 s.t. Q = fp1 ; :::; p4 g M ; and s.t. d(p; pi ) < (i = 1; :::; 4) =) j (Q) (p)j < ". W Remark 3.5. (1) If one uses the second (abstract) de nition of the metric curvature of quadruples, then the very existence of (Q) is not assured, as it is shown by the following Counterexample 3.6. The metric quadruple of lengths d12 = d13 = d14 = 1; d23 = d24 = d34 = 2 10 i.e. Cartan-Alexandrov-Topogonov 11 as illustrated by the time-honored principles of Projective Geometry... 12 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.. 13 The neighborhood N of p is called linear i N is contained in a geodesic.
14
EMIL SAUCAN
X
p3
d 34 d12 23
d24
d14
p1 d12
d
p4
p2
f d34 d12
d23 R=
1
d14
d24
d12 k
2
Sk
Figure 7. admits no embedding curvature. (2) Even if a quadruple has an embedding curvature, it still may be not unique (even if Q is not liniar), indeed, one can study the following examples: Example 3.7. (a) The quadruple Q of distances dij = =2; 1 i<j is isometrically embeddable both in S0 = R2 and in S1 = S2 .
X
p3
d 34 d12 d
23
p1 d12
p4 d24
d14
p2
d23
f
d13 d12
d34
R=
d14
d24
2
Hk
Figure 8.
1 -k
4
SURFACE TRIANGULATION { THE METRIC APPROACH
15
(b) The quadruple Q of distances d13 = d14 = d23 = d24 = ; d 12 = d34 = 3 =2 admits exactly two embedding curvatures: 1 2 (1:5; 2) and 2 = 3. (See [BM].) However, for "good" metric spaces14 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): Proposition 3.8. Let 4(p1 ; q1 ; r1 ) S 1 and 4(p2 ; q2 ; r2 ) p1 q1 = p2 q2 ; p1 r1 = p2 r2 and \(q1 ; p1 ; r1 ) = \(q2 ; p2 ; r2 ). Then: 1 < 2 =) q1 r1 > q2 r2 .
S 2 , s.t.
Proposition 3.9. Let p1 ; q1 ; r1 2 S 1 ; p2 ; q2 ; r2 2 S 2 two isometric triples of points, s.t. the triple p1 ; q1 ; r1 is not linear. Then: \(q1 ; p1 ; r1 ) < \(q2 ; p2 ; r2 ); \(p1 ; q1 ; r1 ) < \(p2 ; q2 ; r2 ); \(q1 ; r1 ; p1 ) < \(q2 ; r2 ; p2 ). Proposition 3.10. Let Q1 = fp1 ; q1 ; r1 ; s1 g; Q2 = fp2 ; q2 ; r2 ; s2 g be non-linear and non-degenerate quadruples in S 1 ; S 2 , respectively. If 4(p1 ; q1 ; r1 ) = 4(p2 ; q2 ; r2 ) and 1 < 2 , then: (1) p1 s1 = p2 s2 ; q1 s1 = q2 s2 =) r1 s1 > r2 s2 ; (2) r1 s1 = r2 s2 ; q1 s1 = q2 s2 =) p1 s1 > p2 s2 ; (3) p1 s1 = p2 s2 ; r1 s1 = r2 s2 =) q1 s1 < q2 s2 . In order that we fully exploit the results above we need the following de nition: De nition 3.11. A metric quadruple Q = Q(p1 ; p2 ; p3 ; p4 ), of distances dij = dist(pi ; pj ); i = 1; :::; 4, is called semi-dependent (a sd-quad, for brevity), i 3 of its points are on a common geodesic, i.e. there exist 3 indices { e.g. 1,2,3 { s.t.: d12 + d23 = d13 . Now we can easily formulate the following immediate consequence of Prop. 3.10. : Corollary 3.12. A sd-quad admits at most one embedding curvature. Unfortunately { as we have already noticed { in the general case the uniqueness of the embedding curvature is not guaranteed. However we can be a bit more explicit using the following de nition: De nition 3.13. Let Q = fp; q; r; sg be a non-linear and non-degenerate quadruple. Q is called planar i \(q; p; r) + \(q; p; s) + \(s; p; r) = 2 . Then we have Proposition 3.14. Let Q = fp; q; r; sg be a a non-linear and non-degenerate quadruple in S . Then (1) If Q is planar, then it admits no isometric embedding in S 1 ; 1 > . 14 i.e. spaces that are locally "plane like"
16
EMIL SAUCAN
(2) If Q is not planar, then it admits no isometric embedding in S 2 ;
2
< .
Corollary 3.15. Let Q = fp; q; r; sg be a a non-linear and non-degenerate quadruple. Then Q has at most two di erent embedding curvatures. In fact we can state a much stronger assertion, of which Example 3.7.(a) is just a very particular case: Proposition 3.16. 8 p 2 S , and 8 > 0; 9 U 2 N (p) s.t. 9 a nonlinear, non-degenerate quadruple Q U of embedding curvature 0. Proof. Let 1 ; 2 2 U , two great-circle arcs s.t. 1 \ 2 = p. p Let q1 ; q2 2 1 s.t. pq1 = pq2 6= 0 and let q 2 2 s.t. pq < =2 . 15 Consider 4(q10 q20 ; q 0 ) R2 ; 4(q10 q20 ; q 0 ) = 4(q1 q2 ; q), let p0 = 12 q10 q20 , and let h = q 0 p0 .
x0
q1
U
γ
q2
p
1
γ
2
q
Sκ 2
q'' h x'0
q'1
p'
q'2
h
R
2
q'
Figure 9. Then since 0 < , Proposition 3.10.(3) applied to the quadruples fq; q 1 ; q2 ; pg and fq 0 ; q10 ; q20 ; p0 g implies that h < pq. Now let x 2 2 , x between p and q, and let x0 2 R2 s.t. 4(q10 q20 ; x0 ) = 4(q1 q2 ; x) ! s.t. x and q 0 are on di erent sides of the line q1 q2 . Then, x = p ) xq > x0 q 0 , and x = q ) xq = 0 < x0 q 0 = 2h, where, in this case x0 = q 00 . (See Figure 9.) 15 i.e a quarter of the length of a great circle in S
SURFACE TRIANGULATION { THE METRIC APPROACH
17
Then it follows from a continuity argument that 9 x0 2 2 ; x0 between p and q, s.t. x0 q = x00 q 0 , thus implying that fq1 ; q2 ; q; xg = fq10 ; q20 ; q 0 ; x0 g. Remark 3.17. fq1 ; q2 ; q; xg is planar, while fq10 ; q20 ; q 0 ; x0 g is not planar. 3.2. The Wald Curvature vs. Gauss Curvature. The discussion above would be nothing more than a nice intellectual exercise where it not for the fact that the metric (Wald) and the classical (Gauss) curvatures coincide whenever the second notion makes sense, that is for smooth (i.e. of class C2 ) surfaces in R3 . More precisely the following theorem holds: Theorem 3.18. (Wald) Let S R3 ; S 2 C m ; m 2 be a smooth surface. Then W (p) exists, for all p 2 S, and W (p) = G (p); 8 p 2 M . Moreover, Wald also proved that a partial reciprocal theorem holds, more precisely he proved the following: Theorem 3.19. Let M be a compact and convex metric space. If W (p) exists, for all p 2 M , then M is a smooth surface and W (p) =
G (p); 8 p
2 M.
Remark 3.20. I one tries to restrict oneself, in the building of De nition 3.4. only to sd-quads, then Theorem 3.19. holds only if the following presumption is added: Condition 3.21. M is locally homeomorphic to R2 . However the proof of this facts is involved and, as such, beyond the scope of this presentation. Therefore we shall restrict ourselves to a succinct description of the principal steps towards the proofs. The basic idea is to show that if a metric M space admits a Wald curvature at any point, than M is locally homeomorphic to R2 , thus any metric proprieties of R2 can be translated to M , (in particular the rst fundamental form). The rst of these partial results is: Theorem 3.22. Let M be a convex metric space. Then M admits at most one Wald curvature W (p); 8 p 2 M . Proof By Corollary 3.12. it su ces to prove that any disk neighborhood B(p ; ) 2 N (p) contains a non degenerate sd-quad. Without loss of generality one can assume that B(p ; ) contains three points p 1 ; p2 ; p3 s.t. d(p; pi ) < =2; i = 1; 2; 3.16 Then, by the convexity of M it follows that 9 q 2 M s.t. p 6= p2 ; p3 and p2 q + p3 q = p2 p3 . But p2 p3 pp2 + pp3 < =) (pq < =2) _ (pp2 < =2). In the rst inequality holds, then pq pp2 + p2 q < , i.e q 2 B(p ; ); and if the second one holds, then pd pp3 + p3 q < , i.e. q 2 B(p ; ). But p 6= q, therefore p; p2 ; p3 ; q are not linear. Our next step will be to analyze those neighborhoods that display "a normal behavior", both metrically and curvature-wise: that is precisely those disk neighborhoods in which the Wald curvature is de ned and ranges over a small, bounded set of values prescribed by the very radius of the disk: De nition 3.23. A disk neighborhood B(p ; ); > 0 is called regular i 8 nondegenerate quadruple Q B(p ; ) , W (Q) exists and j W (Q)j < 2 =16 2 . 16 See [B]
18
EMIL SAUCAN
Remark 3.24. If regular.
W (p)
exists, then for any su ciently small , B(p ; ) will be
It turns out that regular neighborhoods, in compact, convex spaces have the following "nice" (i.e. Real Plane like) proprieties: Proposition 3.25. Let M be a compact, convex metric space and let B(p ; ) M be a regular neighborhood. Then if a non-degenerate quadruple Q B(p ; ) contains two linear triples of points, then Q is linear. Proposition 3.26. Let M be a compact, convex metric space. Then: 8 p 2 M and 8 B(p ; ) regular, 9 q; r 2 B(p ; ) s.t. p; q; r are not linear. Proposition 3.27. Any regular neighborhood B(p ; ) of a compact, convex metric space is strictly convex, i.e. q; r 2 B(p ; ) =) int(qr) B(p ; ). While the proof of this last Proposition is lengthy, that of the following important Corollary is not: Corollary 3.28. Let B(p ; ) be a regular neighborhood. Then, 8 q; r 2 B(p ; ); 9! qr and int(qr) B(p ; ). Proof. By the convexity of B(p ; ) it follows the existence of at least one geodesic qr; 8 q; r 2 B(p ; ). If s 2 int(q)r, then by the proposition above we have that s 2 B(p ; ). It follows that B(p ; ) contains all the geodesics with end points q; r. Hence, by Proposition 3.25. , the geodesic segment qr is unique. We can now begin to prove that a compact, convex metric space locally mimics R2 . We start by showing that the sinus function is de ned on M , thus allowing for angle measure (hence for the de nition of Polar Coordinates on regular neighbourhoods17). First, let M be as before, and let p 2 M s.t. W p exists. Let q; r 2 B(p ; ); q 6= p 6= r, where B(p ; ) is a regular neighborhood of p. Then, 8 x 2 [0; minfpq; prg), de ne q(x) 2 pq; r(x) 2 pr by: d(p; q(x)) = x = d(p; r(x)), and let d(x) = d(; q(x); r(x)) (see Figure 10 bellow). Proposition 3.29. The following limit exists: d(x) : lim x!0 x We omit the proof since it is rather involved (but canonical for any axiomatic approach to Euclidian Geometry { see, for instance, [B], [RR].) Now we can de ne the measure of angles at p : De nition 3.30. The measure of the angle \(q; p; r)) is given by: 1 d(x) : lim 2 x!0 x Remark 3.31. The De nition above enables us to de ne Polar Coordinates on regular neighborhoods18 in the following manner: Let p1 ; p2 2 B(p ; ) s.t. p; p1 ; p2 are not collinear. (Such points exist by Proposition 3.26.). To every point q 2 B(p ; ) we associate the following pair of real 4
m(\(q; p; r)) = 2arcsin
17 In the same way geodesic polar coordinates are used on classical surfaces. 18 ... and once Coordinates (be they Polar or Cartesian) are introduced, the (local) homomor-
phism with R2 is immediate.
SURFACE TRIANGULATION { THE METRIC APPROACH
B(p;ρ)
19
q
q(x)
d(x)
x x
p
r(x)
r
M
Figure 10. numbers (de ning the Polar Coordinates of q relative to the frame determined by p; p1 ; p2 ): (r(q); (q)), where 4
r(q) = d(p; q) and 4
(q)) =
2
m(\(q; p; p1 )) if jm(\(p2 ; p; p1 )) m(\(q; p; p1 )) if jm(\(p2 ; p; p1 ))
m(\(q; p; p1 ))j = m(\(q; p; p1 )) ; m(\(q; p; p1 ))j 6= m(\(q; p; p1 )) .
We can now safely state the foretold homomorphism result: Proposition 3.32. Any convex, compact metric space is locally homeomorphic to the real plane. 4. Computing Embedding Curvature In this section we develop formulas for the computation of Embedding Curvature of Quadruples. First we follow the classical approach of Wald-Blumenthal that employs the so-called Cayley-Menger determinants (see bellow). Unfortunately, the formulas obtained, albeit precise are transcendental. Therefore we present, in the next subsection, the approximate formulas developed by C.V. Robinson. 4.1. Embedding Curvature { The Determinant Approach. Given a general metric quadruple Q = Q(p1 ; p2 ; p3 ; p4 ), of distances dij = dist(pi ; pj ); i = 1; :::; 4, we denote by D(Q) = D(p1 ; p2 ; p3 ; p4 ) the following determinant:
(4.1)
D(p1 ; p2 ; p3 ; p4 ) =
0 1 1 0 1 d212 1 d213 1 d214
1 d212 0 d223 d224
1 d213 d223 0 d234
1 d214 d224 d234 0
20
EMIL SAUCAN
Then the embedding curvature (Q) of Q is given { depending upon the embedding space (i.e. upon the sign of the curvature) { by the following formulae:
(4.2)
(Q) =
8 > > < > > :
; ;
0 if D(Q) = 0p; < 0 if det(coshp 0; ijd) =p d d > 0 if det(cos ) and ij ij and all the principal minors of order 3 are
0.
The determinant D(Q) = D(p1 ; p2 ; p3 ; p4 ) is called the Cayley-Menger determinant (of the points p1 ; :::p4 )19 and, in order to prove (4.2) we need rst to investigate some of its properties. We start with the following Lemma 4.1. Let: p1 ; :::; p4 be points in R3 . Then: ! ! 1 (4.3) Gram(p1 p! 2 ; p1 p3 ; p1 p4 ) = D(p1 ; p2 ; p3 ; p4 ) ; 8 where ! ! def ! ! Gram(p1 p! 2 ; p1 p3 ; p1 p4 ) = det(p1 pi p1 pj )i;j=2;3;4 3 (Here " " denotes the standard scalar (dot) product in R .)
Proof. Use expansion and manipulation of determinants. Since is a known fact that: ! ! Gram(p1 p! 2 ; p1 p3 ; p1 p4 ) = V ol(p1 ; p2 ; p3 ; p4 )
2
;
where V ol(p1 ; p2 ; p3 ; p4 ) denotes the (un-oriented) volume of the parallelepiped de! ! termined by the vertices p1 ; :::; p4 (and with edges p1 p! 2 ; p1 p3 ; p1 p4 ); formula (2.3) shows that: (4.4)
D(p1 ; p2 ; p3 ; p4 ) = 8 V ol(p1 ; p2 ; p3 ; p4 )
2
:
Therefore the following assertion is immediate: Proposition 4.2. The points p1 ; :::; p4 are the vertices of a simplex in R3 i D(p1 ; p2 ; p3 ; p4 ) 6= 0_. However, we can prove the much strong result bellow: Theorem 4.3. Let dij > 0 ; 1 4 ; i 6= j. Then there exists a simplex T = T (p1 ; :::; p4 ) D(pi ; pj ) < 0 ; (8) fi; jg f1; :::; 4g and D(pi ; pj ; pk ) > 0 ; (8) fi; j; kg f1; :::; 4g; where, for instance, 0 1 D(p1 ; p2 ) = 1 0 1 d212 and 0 1 1 0 D(p1 ; p2 ; p3 ) = 1 d212 1 d213
R3 s.t. dist(xi ; xj ) = dij ; i 6= j; i
1 d212 0 1 d212 0 d223
1 d213 d223 0
;
19 This de nition readily generalizes to any dimension, as do the results bellow.
SURFACE TRIANGULATION { THE METRIC APPROACH
21
etc... In fact, the necessary and su cient condition above can be relaxed, indeed one can also show that the following holds20: Proposition 4.4.21 Let dij > 0 ; 1 4 ; i 6= j. Then there exists a simplex T = T (p1 ; :::; p4 ) R3 s.t. dist(xi ; xj ) = dij ; i 6= j; i D(p1 ; p2 ; p3 ; p4 ) 6= 0 and sign D(p1 ; p2 ; p3 ; p4 ) = +1 . Proof.(Sketch) Su cient to show (by using standard operations on determinants) that: 2 D(p1 ; p2 ; p3 )D(p1 ; :::; p^i ; :::; p4 ) = Mi4 + D(p1 ; :::; p^i ; :::; p4 )D(p1 ; p2 ; p3 ; p4 ) ;
where Mi4 is the cofactor (in D) of d2i4 , and were we used the notation: fp1 ; :::; p^i ; :::; p4 g = fp1 ; p2 ; p3 ; p4 g n fpi g. Proving the formula for the spherical and hyperbolical cases would prove to be to technical for this limited exposition; su ce to say that they essentially reproduce the proof given in the Euclidian case, and tacking into account the fact that performing computations in the spherical (resp. hyperbolic) metric one has to replace the distances dij by cos dij (resp. cosh dij )22. 4.2. Embedding Curvature { Approximate Formulas. The formulas we just developed in are not only transcendental, but also the computed curvature may fail to be unique (see preceding section). However, uniqueness is guaranteed for sd-quads. Moreover, the relatively simple geometric setting of sd-quads allows for the development of simple (i.e. rational) formulas for the approximation of the Embedding Curvature. Proposition 4.5. Given the metric quadruple Q = Q(p1 ; p2 ; p3 ; p4 ), of distances dij = dist(pi ; pj ); i = 1; :::; 4, the embedding curvature (Q) is well approximated by: 6(cos \0 2 + cos \0 20 ) (4.5) K(Q) = d24 (d12 sin2 (\0 2) + d23 sin2 (\0 20 )) where: \0 2 = \(p1 p2 p4 ) ; \0 20 = \(p3 p2 p4 ) represent the angles of the Euclidian triangles of sides d12 ; d14 ; d24 and d23 ; d24 ; d34 , respectively. The error R can be estimated by using the following inequality: (4.6)
jRj = jR(Q)j = j (Q)
K(Q)j < 4 2 (Q)diam2 (Q)= (Q)
where we put: (Q) = d 24 (d12 sin \0 2+d23 sin \0 20 )=S 2 , and where S = M axfp; p0 g; 2p = d12 + d14 + d24 ; 2p0 = d32 + d34 + d24 . Proof The basic idea of the proof is to recreate, in a general metric setting, the Gauss Map { in this case one measures the curvature by the amount of "bending" one has to apply to a general planar quadruple so that it may be "straightened" (i.e. isometrically embedded as a sd-quad) in some S . Consider two plane23 triangles 4p1 p2 p4 and 4p2 p3 p4 , and denote by 4p1;k p2;k p4;k 20 For the direct proof of Theorem 4.3., see [Be] or, alternatively [B] 21 We formulate this result { for convenience and practicality { for the case n = 3, only.
However it is readily generalized to any dimension. 22 See [B] for the full details 23i.e. embedded in R2 S0
22
EMIL SAUCAN
and 4p2;k p3;k p4;k their respective isometric embeddings into Sk . Then pi;k pj;k will denote the geodesic (of Sk ) through pi;k and pj;k . Also, let \k 2 and \k 20 denote, respectively, the following angles of 4p1;k p2;k p4;k and 4p2;k p3;k p4;k : \k 2 = \p1;k p2;k p4;k and \k 20 = \p2;k p3;k p4;k . (See Fig. 11) d
4
d 34
d
14
d 24 p3
p
1
d
2
2'
12
p
d 23
2
Figure 11. But \k 2 and \k 20 are strictly increasing as functions of k. Therefore the equation \ k 2 + \ k 20 =
(4.7)
has at most one solution k , i.e. k represents the unique value for which the points p1 ; p2 ; p4 are on a geodesic in Sk (for instance on p1 p4 ). But that means that k is precisely the Embedding Curvature, i.e. k = (Q) , where Q = Q(p1 ; p2 ; p3 ; p4 ). Equation (4:7) is equivalent to \k 20 \k 2 + cos2 = 1 2 2 The basic idea being the comparison between metric triangles with equal sides, embedded in S0 and Sk , respectively, it is natural to consider instead of the previous equation, the following: cos2
(4.8)
(k; 2)
2 \0 2 cos + 2
(k; 20 )
2 cos
\0 20 = 1 2
where we denote: (k; 2) :=
\k 2 2 cos2 \20 2
cos2
;
0
(k; 2 ) :=
\k 20 2 cos2 \20 2
cos2
:
Since we want to approximate (Q) by K(Q) we shall resort { naturally { to expansion into MacLaurin series. We are able to do this because of the existence of the following classical formulas: p p sin(p k) sin(d k) 2 \k 2 p ; k > 0; p = cos 2 sin(d12 k) sin(d 24 k)
SURFACE TRIANGULATION { THE METRIC APPROACH
p \k 2 sinh(p k) p cos = 2 sinh(d12 k) 2
23
p sinh(d k) p ; k < 0; sinh(d 24 k)
and, of course cos2
pd \0 2 = ; 2 d12 d24
were: d = p d14 = (d12 + d24 d14 )=2 :24 By using the development into series of f1 (x) = (easily) gets the desired expansion for (k; 2):
p sin p x x
1 (k; 2) = 1 + kd12 d24 cos(\0 2) 6
(4.9)
and f2 (x) =
p sinh p x; x
one
1 +r;
where: jrj < 83 k 2 p4 , for jkp2 j < 1=16 . By applying (4.9.) to (4.8), we receive: 1 [1 + k d12 d24 cos(\0 2) 6
(4.10)
1 + r] cos2
\0 2 + 2
1 \0 20 [1 + k d23 d24 cos(\0 20 ) 1 + r0 ] cos2 = 1; 6 2 for: jrj + jr0 j < 34 (k )2 (M axfp; p0 g)4 = 34 (k )2 S 4 . By solving linear equation (in variable k ) (4.10) and using some elementary trigonometric transformation one has: k =
6(cos \0 2 + cos \0 20 ) +R d24 (d12 sin2 (\0 2) + d23 sin2 (\0 20 ))
where: jRj
0 and, moreover, (Q) ! 0 ) Q ! linearity.] (b) Since (Q) 6= 0 it follows that: K(Q) 2 R for any quadrangle Q. In addition: sign(k(Q)) = sign(K(Q)). (c) If Q is any sd-quad, then 2 (Q)diam2 (Q)= (Q) < 1. Moreover, 25 if (Q) = 0, then 2 (Q)diam2 (Q)= (Q) = 0 i.e. jRj is small if Q is not close to linearity. In this case jR(Q)j diam2 (Q) (for any given Q). 24and the analogous formulas for cos 2 \k0 2 . 2 25i.e. for not very small values of (Q)
24
EMIL SAUCAN
Since the Gaussian curvature kG (p) at a point p is given by: kG (p) = lim (Q n ) ; n!0
where Qn ! Q = p1 pp3 p4 ; diam(Qn ) ! 0,from Remark 4.6.(c) we immediately infer that the following holds26: Theorem 4.7. Let S be a di erentiable surface. Then, for any point p 2 S: kG (p) = lim K(Qn ) ; n!0
for any sequence fQn g of sd-quads that satisfy the following condition: Qn ! Q =
p1 pp3 p4 ; diam(Qn ) ! 0 :
Remark 4.8. In the following special cases even "nicer" formulas are obtained: (1) If d12 = d32 , then cos \0 2 + cos \0 20 ; d13 24 d sin2 \0 2 + sin2 \0 20 (here we have of course: d13 = 2d12 = 2d32 ); or, expressed as a function of distances alone: 2d212 + 2d224 d214 d213 (4.12) K(Q) = 12 2 2 2 8d12 d24 (d12 + d224 d214 )2 (d212 + d224 d234 )2
(4.11)
K(Q) =
12
(2) If d12 = d32 = d24 and if the following condition also holds: (3) \0 20 = =2; i.e. if d 234 = d212 + d224 or, considering (2), also: d234 = 2d212 then 2d212 d214 6 cos \0 2 = 4 (4.13) K(Q) = : 2 4d12 + 4d214 d212 d414 d12 (1 + sin \0 2) 5. Appeendix 1 { The Menger and Haantjes Curvatures Better known than the Wald Curvature, the Menger Curvature is a metric definition of curvature of curves, as is the Haantjes Curvature. As such they can be employed as sectional curvatures to approximate curvature of triangulated surfaces. We begin by introducing the the Menger Curvature: this is a metric expression for the circum-radius of a triangle27, based upon elementary high-school formulas: De nition 5.1. Let (M; d) be a metric space, and let p; q; r 2 M be three distinct points. Then: p (pq + qr + rp)(pq + qr rp)(pq qr + rp)( pq + qr + rp) ; KM (p; q; r) = pq qr rp is called the Menger Curvature of the points p; q; r. We can now de ne the Menger Curvature at a given point by passing to the limit: De nition 5.2. Let (M,d) be a metric space and let p 2 M be an accumulation point. Then M has at p Menger Curvature M (p) i 8 " > 0; 9 > 0 s.t. d(p; pi ) < ; i = 1; 2; 3 =) jK(Q) M (p)j < ". 26and gives theoretical justi cation to the algorithm
27 thus giving in the limit a metric de nition of the Osculatory Circle
SURFACE TRIANGULATION { THE METRIC APPROACH
25
Remark 5.3. The apparent equivalent notion of Alt Curvature, in which one uses only two points converging to the third, is in fact more general, where we de ne the Arp curvature by: De nition 5.4. Let (M,d) be a metric space and let P 2 M be an accumulation point. Then M has at p Alt Curvature A (p) i the following limit exists A (p)
4
= lim K(p; q; r) : q;r!p
However, both M (p) and A (p) su er from the same imperfection: since they are both modelled closely after the Euclidian Plane, they convey this Euclidian type of curvature upon the space they are de ned on. However, the next de nition doesn’t mimic closely R2 so it better tted for generalizations: De nition 5.5. Let (M,d) be a metric space and let c : I = [0; 1] ! M be a homeomorphism, and let p; q; r 2 c(I); q; r 6= p. Denote by qr b the arc of c(I) between q and r, and by qr segment from q to r. (See Figure 12 bellow.)
C
r qr qr
p
q
Figure 12. Then c has Haantjes Curvature 2 H (p)
H (p)
at the point p i :
= 24 lim
l(qr) b
q;r!p
d(q; r)
l(qr) b
3
;
where "l(qr)" b denotes the length28 of qr. b Remark 5.6. H exists only for recti able curves, but if of c, then c is recti able.
M
exists at any point p
Remark 5.7. Evidently we have the following relationship between curvatures: 9
M
=) 9
9
A
=) = 9
A
:
while However, we can prove the following theorem: 28 given by the intrinsic metric induced by d
M
:
26
EMIL SAUCAN
Theorem 5.8. Let c : I ! M be a recti able curve, and let p 2 M . If A (or M ) exists, then H (p) exists and A
=
H (p) :
Remark 5.9. This last result and the Remark preceding it allow as to employ any of the curvatures above in estimating curvatures of smooth curves on triangulated surfaces. 6. Appendix 2 { The Rinow Curvature The curvatures introduced before may seem a bit archaic in comparison to the more fashionable approach of comparison triangles, with their ar reaching applications. We present here one of these comparison criteria and show its equivalence with the Wald curvature. We start with the following de nition: De nition 6.1. Let (M; d) be a metric space, together with the intrinsic metric induced by d. Let R = int(R) M be a region of M . We say that R is a region of curvature ( 2 R) i (1) 8p; q 2 R ; 9 a geodesic segment pq R; (2) 8 T (p; q; r) R is isometrically embeddable in S ; (3) If T (p; q; r) R and x 2 pq; y 2 pr, and if the points p ; q ; r ; x ; y 2 S satisfy the following conditions: (a) T (p; q; r) = T (p ; q ; r ); (b) T (p; q; x) = T (p ; q ; x ); (c) T (p; r; y) = T (p ; r ; y ); then xy xy . By replacing the condition: "xy x y " with: "xy x y ", we obtain the de nition of a region of curvature . (See Fig. 13.)
xκ
qκ
yκ
pκ
2
Sκ
rκ
q x
R
p
M
y
r
Figure 13.
(κ > 0)
SURFACE TRIANGULATION { THE METRIC APPROACH
27
We now pass to the localization of the De nition above: De nition 6.2. Let (M; d) be a metric space, together with the intrinsic metric induced by d, and let p 2 M be an accumulation point. Then M has at p Rinow Curvature R (p) i (i) =9 N 2 N (p); N linear; (ii) 8 " > 0; 9 :0, s.t.B(p; ) is (a) a region of Rinow curvature R (p) + " and (b) a region of Rinow curvature (p) ". R While its greater generality endows the Rinow curvature with more exibility in applications and makes it easier in generalization, it is even more di cult to compute than Wald Curvature. However this quandary was has an almost ideal solution, due to Kirk (see [K]), solution which we brie y expose here: De nition 6.3. Let M be a compact, convex metric space, and let p 2 M . If W (p) exists, then R (p) exists, and R (p) = W (p). Unfortunately, since R (p) makes no presumption of dimensionality, the existence of R (p) does not imply the existence of W (p). Counterexample 6.4. Let M R3 . Then R (p) 0 but W (p) does not exist at any point, since every neighborhood contains linear quadruples. The solution (due to Kirk) of this problem is to consider the Modi ed Wald curvature W K , de ned as follows: De nition 6.5. Let (M; d) be a metric space, together with the intrinsic metric induced by d, and let p 2 M . Then M has at p Modi ed Wald curvature Curvature W K (p) i (i) =9 N 2 N (p); N linear; (ii) 8 " > 0; 9 :0, s.t. if Q B(p; ) is a non-degenerate sd-quad, then W (Q) exists and j W K (p) W (Q)j < ". Remark 6.6. 9
W (p)
=) 9
W K (p)
but 9
KW (p)
=) =
W (p).
This modi ed curvature indeed represents the wished for solution, as proved by the following to Theorems: Theorem 6.7. Let (M; d) be a metric space. Then: 9 R (p) =) 9 W K (p) and R (p) = W K (p). Theorem 6.8. Let (M; d) be a metric space together with the associated intrinsic metric, and let p 2 M . Then, if (i) W K (p); and if (ii) 9 B(p; ) 2 N (p), s.t. qr B(p; ); 8q; r 2 B(p; ); then R (p) exists and R (p) = W K (p). 7. Appendix 3 { The Radius Formula The Cayley-Menger determinant allows one to express not only the volume and area29 of simplices in Rn but (as expected) it may be used to compute the radius 29The 2-dimensional analogue of Formula (4.4) for the area of the triangle T (p ; p ; p ) being: 1 2 3
Area (p1 ; p2 ; p3 )
2
=
D(p1 ; p2 ; p3 ) :
28
EMIL SAUCAN
of the circumscribed sphere around an Euclidian simplex. To be more precise, we have the following result30: Theorem 7.1. (1) The radius R = R(p1 ; p2 ; p3 ; p4 ) of the sphere circumscribed around the tetrahedron T (p1 ; p2 ; p3 ; p4 ) 2 R3 is given by: R2 =
1 (p 1 ; p2 ; p3 ; p4 ) 2 D(p1 ; p2 ; p3 ; p4 )
where:
0 d212 d213 d214
(p 1 ; p2 ; p3 ; p4 ) =
d212 0 d223 d224
d213 d223 0 d234
d214 d224 d234 0
(2) The points p1 ; p2 ; p3 ; p4 ; p5 2 R3 are coplanar or co-spherical i (p 1 ; p2 ; p3 ; p4 ; p5 ) = 0 : Proof (1) If p0 2 R3 is s.t. d0i = R; i = 1; :::; 5, then by direct computation we obtain: (p 0 ; :::; p5 ) =
2R2 (p 1 ; :::; p5 )
(p 1 ; p2 ; p3 ; p4 ; p5 ) ;
from which the desired formula follows immediately if we chose p0 as the center of the sphere circumscribed around the points p1 ; p2 ; p3 ; p4 ; p5 . (2) (=)) Let fx1 ; x2 ; x3 g be any orthonormal coordinate frame for R3 , and let pji ; i = 1; :::; 5; j = 1; 2; 3; represent the coordinates of the points p1 ; p2 ; p3 ; p4 ; p5 relative to this coordinate system. Then p1 ; p2 ; p3 ; p4 ; p5 belong to the same sphere or plane i 9 (a; b; c j ) 6= (0; 0; 0); j = 1; 2; 3; s.t. ajjpi jj2 + b +
3 X
cj pji ; i = 1; :::; 5:
j=1
Then
1
=
2
= 0, where:
1 (p1 ; p2 ; p3 ; p4 ; p5 )
and 2 (p1 ; p2 ; p3 ; p4 ; p5 )
Therefore proven.
1
t 2
= 0. But
1
=
=
1 1 1 1 1 t 2
jjp1 jj2 jjp2 jj2 jjp3 jj2 jjp4 jj2 jjp5 jj2 jjp1 jj2 jjp2 jj2 jjp2 jj2 jjp2 jj2 jjp2 jj2
1 1 1 1 1
p11 p12 p13 p14 p15 2p11 2p12 2p13 2p14 2p15
p21 p22 p23 p24 p25 2p21 2p22 2p23 2p24 2p25
p31 p32 p33 p34 p35 2p31 2p32 2p33 2p34 2p35
= (p 1 ; p2 ; p3 ; p4 ; p5 ), so this implication is
30that can be readily generalized to higher dimensions
SURFACE TRIANGULATION { THE METRIC APPROACH
((=) (p 1 ; p2 ; p3 ; p4 ; p5 ) = 0 =) (0; 0; 0); j = 1; 2; 3; s.t. ajjpi jj2 + b +
1
3 X
29
= 0 and there exist numbers (a; b; cj ) 6=
cj pji = 0 ; i = 1; :::; 5;
j=1
i.e. p1 ; p2 ; p3 ; p4 ; p5 belong to the plane or the sphere given by the equation ajjXjj2 + b + c
3 X
cj X = 0 ; X = (x1 ; x2 ; x3 ) :
j=1
References [Ba1] Bancho , T.A. { Critical points and curvature for embedded polyhedra, J. Di erential Geometry, 1 (1967), 257-268. [Ba2] Bancho , T.A. { Critical Points and Curvature for Embedded Polyhedral Surfaces, Amer. Math. Monthly, 77 (1970), 475-485. [Be] Berger, M. { Geometry I, Universitext, Spinger-Verlag, 1987. [B] Blumenthal, L. M. Distance Geometry { Theory and Applications, Claredon, 1953. [BM] Blumenthal, L. M. and Menger, K. { Studies in Geometry, Freeman and Co., 1970. [BH] Bridson, M. R. and Hae iger, A. { Metric spaces of non-positive curvature , Grundlehren der mathematischen Wissenschaften, Springer-Verlag, Berlin, 1999. [BBI] Burago, D. , Burago, Y. and Ivanov, S. { A Course in Metric Geometry, GSM, AMS, RI, 2000. [BCM] Borrelli, V. Cazals, F. and Morvan, J.-M. { On the angular defect of triangulations and the poitwise approximation of Curvatures, Computer Aided Geometric Designs, 20, pp. 319341, 2003. [CMS] Cheeger, J. , Muller, W. , and Schrader, R. { On the Curvature of Piecewise Flat Spaces, Comm. Math. Phys. , 92, 1984, 405-454. [C-SM] Cohen-Steiner, D. and Morvan, J.-M. { Restricted Delaunay triangulations and normal cycle, preprint, 2003. [F] Fu, J. H. G. { Convergence of Curvatures in Secant Approximation, J. Di erential Geometry, 37, 1993, 177-190. [G+] Mikhail Gromov { Metric structures for Riemannian and non-Riemannian spaces, Progress in Mathematics 152, Birkhauser, Boston, 1999. [K] Kirk, W. A. { On Curvature of a Metric Space at a Point, Paci c J. Math. 14: 195-198, 1964. [LWZL] Liu, G.H. , Wong, Y.S. , Zhang, Y.F. and Loh, H.T. { Adaptive fairing of digitized point data with discrete curvature, Comuter Aided Design, vol. 34(4), 309-320, 2002. [MD] Maltret, J.-L. and Daniel, M. { Discrete curvatures and applications: a survey, preprint, 2003. [P] Pajot, H. { Analytic Capacity, Recti cabilility, Menger Curvature and the Cauchy Integral, LNM 1799, Springer, Berlin, 2002. [RR] Ramsay, A. and Richtmayer, R.D. { Introduction to Hyperbolic Geometry, Universitext, Spinger-Verlag, 1991. [Rat] Ratcli e, J.C. : Foundations of Hyperbolic Manifolds, GTM 194, Springer Verlag, N.Y., 1994. [R] Robinson, C.V. { A Simple Way of Computing the Gauss Curvature of a Surface, Reports of a Mathematical Colloquium, Second Series, Issue 5-6, 16-24, 1944. [SMSER] Surazhsky, T. , Magid, E. , Soldea, O. , Elber, G. and Rivlin, E. { A Comparison of Gaussian and Mean Curvatures Estimation Methods on TRiangular Meshes, preprint, 2003. [T] Troyanov, M. { Tangent Spaces to Metric Spaces: Overview and Motivations, preprint, 2003.
30
EMIL SAUCAN
Department of Mathematics,Technion & Department of Software Engineering, Ort Braude College, Karmiel E-mail address: [email protected]
