Introduction to Computational Manifolds and Applications

Transcription

1 IMPA - Instituto de Matemática Pura e Aplicada, Rio de Janeiro, RJ, Brazil Introduction to Computational Manifolds and Applications Part - Constructions Prof. Marcelo Ferreira Siqueira mfsiqueira@dimap.ufrn.br Departmento de Informática e Matemática Aplicada Universidade Federal do Rio Grande do Norte Natal, RN, Brazil

2 Recall that S = {(x, y) E x + y = } Computational Manifolds and Applications (CMA) - 0, IMPA, Rio de Janeiro, RJ, Brazil

3 S is a one-dimensional parametric pseudo-manifold in E. To see why, let us define it as such. We need to define a set of gluing data and a family of parametrizations. The gluing data will define the topology of S, while the parametrizations the geometry. We will start with the gluing data. Computational Manifolds and Applications (CMA) - 0, IMPA, Rio de Janeiro, RJ, Brazil 3

4 We need to define the p-domains, {Ω i } i I, the gluing domains, {Ω ij } (i,j) I I, and the transition functions, {ϕ ij } (i,j) K. Computational Manifolds and Applications (CMA) - 0, IMPA, Rio de Janeiro, RJ, Brazil 4

5 Let I = {,, 3, 4}. Parametric Pseudo-Manifolds Define the p-domains Ω, Ω, Ω 3, and Ω 4 as distinct copies of the open interval, ], [ E. Ω Ω Ω 3 Ω 4 Computational Manifolds and Applications (CMA) - 0, IMPA, Rio de Janeiro, RJ, Brazil

6 Define the gluing domains Ω, Ω 3, Ω 34, and Ω 4 as the open interval ] 0, [ contained in the p-domains Ω, Ω, Ω 3, and Ω 4, respectively. 0 Ω 0 Ω 3 0 Ω 34 0 Ω 4 Ω Ω Ω 3 Ω 4 Computational Manifolds and Applications (CMA) - 0, IMPA, Rio de Janeiro, RJ, Brazil 6

7 Similarly, define the gluing domains Ω, Ω 3, Ω 43, and Ω 4 as the open interval ],0[ contained in the p-domains Ω, Ω 3, Ω 4, and Ω, respectively. Ω 4 Ω Ω 3 Ω 43 0 Ω 0 Ω 3 0 Ω 34 0 Ω 4 Ω Ω Ω 3 Ω 4 Finally, let Ω ii = Ω i, for i =,, 3, 4. Computational Manifolds and Applications (CMA) - 0, IMPA, Rio de Janeiro, RJ, Brazil 7

8 We let Ω 3 = Ω 3 = Ω 4 = Ω 4 =. What does this "gluing" look like? Ω 4 Ω Ω 3 Ω 43 0 Ω 0 Ω 3 0 Ω 34 0 Ω 4 Ω Ω Ω 3 Ω 4 Ω Ω 3 Ω 43 Ω 4 Ω Ω 3 Ω 34 Ω 4 Computational Manifolds and Applications (CMA) - 0, IMPA, Rio de Janeiro, RJ, Brazil 8

9 Our intuition is... Ω Ω 4 Ω Ω 3 Computational Manifolds and Applications (CMA) - 0, IMPA, Rio de Janeiro, RJ, Brazil 9

10 What transition functions can "realize" our intuition? Ω 4 Ω Ω 3 Ω 43 0 Ω 0 Ω 3 0 Ω 34 0 Ω 4 Ω Ω Ω 3 Ω 4 Ω Ω 3 Ω43 Ω 4 Ω Ω 3 Ω 34 Ω 4 Computational Manifolds and Applications (CMA) - 0, IMPA, Rio de Janeiro, RJ, Brazil 0

11 Let K = {(, ), (, ), (3, 3), (4, 4), (, ), (, 3), (3, 4), (4, ), (, 4), (, ), (3, ), (4, 3)}. For each (i, j) K and for all x Ω ij, the transition map ϕ ji : Ω ij Ω ji is given by ϕ ji (x) = x if i = j, x if j = i + or j = and i = 4, x + if j = i or j = 4 and i =. Note that our transition maps are affine functions. We claim that the triple G = (Ω i ) i I, (Ω ij ) (i,j) I I, (ϕ ji ) (i,j) K is a set of gluing data. Computational Manifolds and Applications (CMA) - 0, IMPA, Rio de Janeiro, RJ, Brazil

12 Checking... () For every i I, the set Ω i is a nonempty open subset of E n called parametrization domain, for short, p-domain, and any two distinct p-domains are pairwise disjoint, i.e., Ω i Ω j =, for all i = j. Our p-domains are (connected) open intervals of E, which do not overlap (since they were assumed to be in distinct copies of E). So, condition () of Definition 7. is satisfied. Computational Manifolds and Applications (CMA) - 0, IMPA, Rio de Janeiro, RJ, Brazil

13 Checking... () For every pair (i, j) I I, the set Ω ij is an open subset of Ω i. Furthermore, Ω ii = Ω i and Ω ji = if and only if Ω ij =. Each nonempty subset Ω ij (with i = j) is called a gluing domain. Our gluing domains are open intervals of E. Furthermore, Ω ii = Ω i and Ω ij = if and only if Ω ji =, for i, j =,, 3, 4. So, condition () of Definition 7. is also satisfied. Computational Manifolds and Applications (CMA) - 0, IMPA, Rio de Janeiro, RJ, Brazil 3

14 Checking... Parametric Pseudo-Manifolds (3) If we let K = {(i, j) I I Ω ij = }, then ϕ ji : Ω ij Ω ji is a C k bijection for every (i, j) K called a transition (or gluing) map. Computational Manifolds and Applications (CMA) - 0, IMPA, Rio de Janeiro, RJ, Brazil 4

15 Checking... Recall that, for each (i, j) K and for all x Ω ij, we have ϕ ji (x) = x if i = j, x if j = i + or j = and i = 4, x + if j = i or j = 4 and i =. These maps are either the identity function or a "translation". In either case, they are C bijective functions. But, to satisfy condition (3), we still have to check three more cases. Computational Manifolds and Applications (CMA) - 0, IMPA, Rio de Janeiro, RJ, Brazil

16 Checking... Parametric Pseudo-Manifolds (a) ϕ ii = id Ωi, for all i I, Ω i ϕ ii = id Ωi By definition, ϕ ji (x) =x whenever i = j. So, condition 3(a) is satisfied. Computational Manifolds and Applications (CMA) - 0, IMPA, Rio de Janeiro, RJ, Brazil 6

17 Checking... Parametric Pseudo-Manifolds (b) ϕ ij = ϕ ji, for all (i, j) K, and Ω i ϕ ij Ω j p ϕ ji Computational Manifolds and Applications (CMA) - 0, IMPA, Rio de Janeiro, RJ, Brazil 7

18 Checking... Parametric Pseudo-Manifolds If i = j then condition 3(b) is trivially satisfied by our definition of ϕ ji. If j = i + or j = and i = 4 then ϕ ji (x) =x (/). So, ϕ ji (x) =x +(/). If j = i or j = 4 and i = then ϕ ji (x) =x +(/). So, ϕ ji (x) =x (/). Thus, ϕ ji (x) = x if i = j, x if j = i or j = 4 and i =, x + if j = i + or j = and i = 4. Computational Manifolds and Applications (CMA) - 0, IMPA, Rio de Janeiro, RJ, Brazil 8

19 Checking... Parametric Pseudo-Manifolds But, since ϕ ji (x) = x if i = j, x if j = i + or j = and i = 4, x + if j = i or j = 4 and i =, we must have that So, ϕ ij (x) = x if j = i, x if j = i or i = and j = 4, x + if j = i + or i = 4 and j =. ϕ ij (x) =ϕ ji (x). Computational Manifolds and Applications (CMA) - 0, IMPA, Rio de Janeiro, RJ, Brazil 9

20 Checking... (c) For all i, j, k, if Ω ji Ω jk =, then ϕ ij (Ω ji Ω jk )=Ω ij Ω ik and ϕ ki (x) =ϕ kj ϕ ji (x), for all x Ω ij Ω ik. Note that if Ω ji Ω jk =, then we get i = k. In other words, at most two gluing domains overlap at the same point of any p-domain. As a result condition 3(c) holds trivially. Computational Manifolds and Applications (CMA) - 0, IMPA, Rio de Janeiro, RJ, Brazil 0

21 Checking... Parametric Pseudo-Manifolds What about condition 4 of Definition 7. (the Hausdorff condition)? (4) For every pair (i, j) K, with i = j, for every x (Ω ij ) Ω i and y (Ω ji ) Ω j, there are open balls, V x and V y, centered at x and y, so that no point of V y Ω ji is the image of any point of V x Ω ij by ϕ ji. Computational Manifolds and Applications (CMA) - 0, IMPA, Rio de Janeiro, RJ, Brazil

22 Checking... Let j = i + or j = and i = 4. So, if x (Ω ij ) Ω i and y (Ω ji ) Ω j, then x = y = 0. Ω ij Ω ji 0 0 Ω i Ω j Thus, if we let V x = V y =], [, where < (/4), then we have that ϕ ji (V x ) V y =. If j = i or j = 4 and i =, we can proceed in a similar manner. So, condition (4) holds. Computational Manifolds and Applications (CMA) - 0, IMPA, Rio de Janeiro, RJ, Brazil

23 We defined a gluing data that captures the topology of S. Now, we need to define the geometry of S. For that, we must define parametrizations that take the Ω i s to E Computational Manifolds and Applications (CMA) - 0, IMPA, Rio de Janeiro, RJ, Brazil 3

24 Recall that a parametric C k pseudo-manifold of dimension n in E d (PPM) is a pair, M =(G, (θ i ) i I ), such that G = (Ω i ) i I, (Ω ij ) (i,j) I I, (ϕ ji ) (i,j) K is a set of gluing data, for some finite set I, and each θ i : Ω i E d is C k and satisfies (C) For all (i, j) K, we have θ i = θ j ϕ ji. The key is to define parametrizations that respect condition (C). In the case of S, this is a particularly easy job! Computational Manifolds and Applications (CMA) - 0, IMPA, Rio de Janeiro, RJ, Brazil 4

25 Indeed, let θ (x) = cos((x + 0.) π), sin((x + 0.) π), θ (x) = cos((x +.0) π), sin((x +.0) π), θ 3 (x) = cos((x +.) π), sin((x +.) π), θ 4 (x) = cos((x +.0) π), sin((x +.0) π), for all x ], [. Computational Manifolds and Applications (CMA) - 0, IMPA, Rio de Janeiro, RJ, Brazil

26 Let x be any point in Ω =]0, / [. Then, θ ϕ (x) = cos(ϕ (x)+.0) π, sin(ϕ (x)+.0) π = cos((x 0.)+.0) π, sin((x 0.)+.0) π = cos(x + 0.) π, sin(x + 0.) π = θ (x). We can proceed in a similar way to show that θ i = θ j ϕ ji, for i, j =,, 3, 4. So, S is in fact a one-dimensional parametric pseudo-manifold in E. Computational Manifolds and Applications (CMA) - 0, IMPA, Rio de Janeiro, RJ, Brazil 6

27 It turns out that θ, θ, θ 3 and θ 4 are all injective and they also satisfy conditions (C ) For all (i, j) K, (C ) For all (i, j) K, θ i (Ω i ) θ j (Ω j )=θ i (Ω ij )=θ j (Ω ji ). θ i (Ω i ) θ j (Ω j )=. So, our pseudo-manifold is actually a manifold, which comes with no surprise! Computational Manifolds and Applications (CMA) - 0, IMPA, Rio de Janeiro, RJ, Brazil 7

28 Reconstructing a curve homeomorphic to S S is certainly one of the easiest curves we could reconstruct with the gluing process. This is because the topology and (mainly) the geometry of S are quite simple. But, what if the curve has the same topology as S, but a more "challenging" shape? In particular, what if we do not know any equation that captures the exact shape of the curve? To illustrate this situation and how we can deal with it, let us consider a "sketch" of a shape. Computational Manifolds and Applications (CMA) - 0, IMPA, Rio de Janeiro, RJ, Brazil 8

29 Reconstructing a curve homeomorphic to S (, 3) (, 3) ( 3, ) (, ) (, ) (3, ) ( 3, ) (, ) (, ) (3, ) (, 3) (, 3) Computational Manifolds and Applications (CMA) - 0, IMPA, Rio de Janeiro, RJ, Brazil 9

30 Reconstructing a curve homeomorphic to S Let us assume that we do not know any equation that approximates the global shape, but that we know how to approximate it locally, using, for instance, arcs of Bézier curves. In particular, assume that we can approximate the shape with four Bézier curves of degree. Computational Manifolds and Applications (CMA) - 0, IMPA, Rio de Janeiro, RJ, Brazil 30

31 Reconstructing a curve homeomorphic to S Recall that a Bézier curve, C : [0, ] E 3, of degree is expressed by the equation C(t) = i=0 where p i are the so-called control points. i= t i ( t) i p i i, Computational Manifolds and Applications (CMA) - 0, IMPA, Rio de Janeiro, RJ, Brazil 3

32 Reconstructing a curve homeomorphic to S In our example, the curves are c j (t) = i=0 t i ( t) i p (j) i i, for j =,, 3, 4, where the control points are the following: p () 0 = (3, ) p () = (, ) p () = (, 3) p () 3 = (, 3) p () 4 = (, ) p () 0 = (, 3) p () = (, ) p () = ( 3, ) p () 3 = ( 3, ) p () 4 = (, ) p (3) 0 = ( 3, ) p (3) = (, ) p (3) = (, 3) p (3) 3 = (, 3) p (3) 4 = (, 3) p (4) 0 = (, 3) p (4) = (, ) p (4) = (3, ) p (4) 3 = (3, ) p (4) 4 = (, ) p () = ( 3, ) p () = (, 3) p (3) = (3, ) p (4) = (, 3) Computational Manifolds and Applications (CMA) - 0, IMPA, Rio de Janeiro, RJ, Brazil 3

33 Reconstructing a curve homeomorphic to S c ([0, ]) (, 3) (, 3) ( 3, ) (, ) (, ) (3, ) c ([0, ]) c 4 ([0, ]) ( 3, ) (, ) (, ) (3, ) (, 3) (, 3) c 3 ([0, ]) Computational Manifolds and Applications (CMA) - 0, IMPA, Rio de Janeiro, RJ, Brazil 33

34 Reconstructing a curve homeomorphic to S As we can see, the trace of the four Bézier curves do not match "exactly". But, before we worry about that, let us define a set of gluing data for the curve we want to build. Since the topology is the same as before and since we also have four "pieces" of curve, we can basically re-use the same gluing data. We will make a slight modification only. Our p-domains will be distinct copies of the open interval ] 0, [ E. 0 Ω 0 Ω 0 Ω 3 0 Ω 4 Our gluing domains will change too. Computational Manifolds and Applications (CMA) - 0, IMPA, Rio de Janeiro, RJ, Brazil 34

35 Reconstructing a curve homeomorphic to S We will let Ω, Ω 3, Ω 34, and Ω 4 be the subsets ] 3,[ of Ω, Ω, Ω 3, and Ω 4, respectively. 3 Ω 3 Ω 3 3 Ω 34 3 Ω 4 0 Ω 0 Ω 0 Ω 3 0 Ω 4 Similarly, we let Ω, Ω 3, Ω 43, and Ω 4 be the subsets ] 0, [ of Ω, Ω 3, Ω 4, and Ω, respectively. Ω 4 3 Ω Ω 3 Ω 3 Ω 3 3 Ω 34 Ω 43 3 Ω 4 0 Ω 0 Ω 0 Ω 3 0 Ω 4 Computational Manifolds and Applications (CMA) - 0, IMPA, Rio de Janeiro, RJ, Brazil 3

36 Reconstructing a curve homeomorphic to S Finally, we let Ω 3 = Ω 3 = Ω 4 = Ω 4 =, and Ω ii = Ω i, for i =,, 3, 4. The intuition for defining the gluing domains comes from the fact that the Bézier curves overlaps in a (linear) region that corresponds to roughly of their parametric domain. (, 3) (, 3) ( 3, ) (, ) (, ) (3, ) ( 3, ) (, ) (, ) (3, ) (, 3) (, 3) Computational Manifolds and Applications (CMA) - 0, IMPA, Rio de Janeiro, RJ, Brazil 36

37 Reconstructing a curve homeomorphic to S What does this "gluing" look like? Ω 4 3 Ω Ω 3 Ω 3 Ω 3 3 Ω 34 Ω 43 3 Ω 4 0 Ω 0 Ω 0 Ω 3 0 Ω 4 Ω Ω 3 Ω 43 Ω 4 Ω Ω 3 Ω 34 Ω 4 Computational Manifolds and Applications (CMA) - 0, IMPA, Rio de Janeiro, RJ, Brazil 37

38 Reconstructing a curve homeomorphic to S Ω Ω Ω 4 Ω 3 Computational Manifolds and Applications (CMA) - 0, IMPA, Rio de Janeiro, RJ, Brazil 38

39 Reconstructing a curve homeomorphic to S We are now left with the transition functions. Well, they are also affine maps (the identity and some translations). Let K = {(, ), (, ), (3, 3), (4, 4), (, ), (, 3), (3, 4), (4, ), (, 4), (, ), (3, ), (4, 3)}. For each (i, j) K and for all x Ω ij, the transition map ϕ ji : Ω ij Ω ji is given by ϕ ji (x) = x if i = j, x 3 if j = i + or j = and i = 4, x + 3 if j = i or j = 4 and i =. Computational Manifolds and Applications (CMA) - 0, IMPA, Rio de Janeiro, RJ, Brazil 39

40 Reconstructing a curve homeomorphic to S We can show that we do have a set of gluing data, but the proof is similar to what we did before. So, it will be left as an easy problem for one of the following homeworks. Let us now deal with a new challenge: our Bézier curves do not yield consistent parametrizations. So, they cannot be used as parametrizations. What should we do then? We will resort to an approach that is often used in the gluing of -dimensional PPMs. The idea is to create parametrizations by averaging the Bézier curves wherever their domains overlap (according to the gluing). For that, we will use the notion of partition of unity. Computational Manifolds and Applications (CMA) - 0, IMPA, Rio de Janeiro, RJ, Brazil 40

41 Reconstructing a curve homeomorphic to S Definition 8.. Given a subset U of E n,apartition of unity {α k } k K on U is a set of nonnegative compactly supported functions α k : E n R that add up to at every point of U. More precisely, for each k K and for each point p U, we have that α k (p) 0, α k (p) =, and {supp(α k )} k K k K is a locally finite cover of U, where the support supp(α k ) of α k is the closure of the point set {p U α k (p) = 0}. U Computational Manifolds and Applications (CMA) - 0, IMPA, Rio de Janeiro, RJ, Brazil 4

42 Reconstructing a curve homeomorphic to S We will (indirectly) define a partition of unity on each p-domain, Ω, Ω, Ω 3, and Ω 4. First, we need a bump function: For every t R, we define ξ : R R as ξ(t) = if t H 0 if t H /( + e s ) otherwise where H, H are constant, with 0 < H < H <, s = and H = H H t H H H. Computational Manifolds and Applications (CMA) - 0, IMPA, Rio de Janeiro, RJ, Brazil 4

43 Reconstructing a curve homeomorphic to S We use H = 0. and H = Computational Manifolds and Applications (CMA) - 0, IMPA, Rio de Janeiro, RJ, Brazil 43

44 Reconstructing a curve homeomorphic to S Using ξ, we define the bump function w : R [0, ] such that ξ( x) if x 0., w(x) = ξ(x ) otherwise Computational Manifolds and Applications (CMA) - 0, IMPA, Rio de Janeiro, RJ, Brazil 44

45 Reconstructing a curve homeomorphic to S Now, we are ready to define the parametrizations. The key idea is to assign a bump function, w i : R [0, ], with each p-domain, Ω i, such that w i (x) =w(x), for every x Ω i. 0 Ω 0 Ω 0 Ω 3 0 Ω 4 Computational Manifolds and Applications (CMA) - 0, IMPA, Rio de Janeiro, RJ, Brazil 4

46 Reconstructing a curve homeomorphic to S Finally, we assign a parametrization, θ i : Ω i E, with each p-domain, Ω i, such that c i (t) if t and t 3, θ i (t) = w i (t) c i (t)+w j (ϕ ji (t)) c j (ϕ ji (t)) w i (t)+w j (ϕ ji (t)) if t > 3 and t < and j = i + or i = 4 and j =, w i (t) c i (t)+w j (ϕ ji (t)) c j (ϕ ji (t)) w i (t)+w j (ϕ ji (t)) if t > 0 and t < and j = i or i = and j = 4, for i =,, 3, 4. We also claim that θ i (x) =θ j ϕ ji (x), for all x Ω ij and for i =,, 3, 4. Computational Manifolds and Applications (CMA) - 0, IMPA, Rio de Janeiro, RJ, Brazil 46

47 Reconstructing a curve homeomorphic to S Let j = i + or i = 4 and j = and t ] 3/, [. Then, we have s = ϕ ji (t) ] 0, /[ and θ j ϕ ji (t) =θ j (s) = w j(s) c j (s)+w i (ϕ ij (s)) c i (ϕ ij (s)) w j (s)+w i (ϕ ij (s)) = w j(s) c j (s)+w i (ϕ ji w j (s)+w i (ϕ ji (s)) (s)) c i (ϕ ji (s)) = w j(ϕ ji (t)) c j (ϕ ji (t)) + w i (t) c i (t) w j (ϕ ji (t)) + w i (t) = w i(t) c i (t)+w j (ϕ ji (t)) c j (ϕ ji (t)) w i (t)+w j (ϕ ji (t)) = θ i (t). Computational Manifolds and Applications (CMA) - 0, IMPA, Rio de Janeiro, RJ, Brazil 47

48 Reconstructing a curve homeomorphic to S If j = i or i = and j = 4 and t ] 0, /[, then we can proceed in a similar manner. So, our parametrizations are consistent. c ([0, ]) θ (Ω ) M = 4 i= θ i (Ω i ) c ([0, ]) c 4 ([0, ]) θ (Ω ) θ 4 (Ω 4 ) c 3 ([0, ]) θ 3 (Ω 3 ) Computational Manifolds and Applications (CMA) - 0, IMPA, Rio de Janeiro, RJ, Brazil 48

49 Reconstructing a curve homeomorphic to S Some important remarks: The partition of unity functions are "hidden" in the convex sum that defines the θ i s. Indeed, if we denote the function associated with Ω i by α i, then we have that α i (x) =, if x and x 3, w i (x)/(w i (x)+w j (ϕ ji (x)), if x > 0 and x <, j = i or j = 4 and i =,, w i (x)/(w i (x)+w j (ϕ ji (x)), if x > 3 and x <, j = i + or j = and i = 4,. The bump functions (the w i s), the transition maps (the ϕ ij s), and the Bézier curves (the c i s) are all C -functions. As a result the parametrizations (the θ i s) are C. In turn, these facts imply that the one-dimensional PPM we just built is C. Computational Manifolds and Applications (CMA) - 0, IMPA, Rio de Janeiro, RJ, Brazil 49