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 1 - 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 Let K = {(u, w) I I w = }. For every (u, w) K, we must define the transition map ϕ wu : w Ω wu, which is a C k -diffeomorphic function that takes w onto Ω wu, where k is a positive integer or k =. Here, we will assume that ϕ wu is a composition of five distinct maps: two rotations, a double reflection, and two more general planar transformations. The latter transformations are the main obstacles for obtaining maps satisfying condition () of Definition 7.1. The picture in the next slide illustrates the roles of each of the five maps. Computational Manifolds and Applications (CMA) , IMPA, Rio de Janeiro, RJ, Brazil 2

3 f u (w) f u (y) r uw p f u (x) f u (u) E y u x w (h g u r uw )(p) f w (y) f w (u) Ω w f w (w) (rwu 1 gw 1 h g u r uw )(p) rwu 1 gw 1 h g u r uw f w (x) h g u r uw g 1 w h g u r uw g u r uw (g u r uw )(p) (1, 0) (1, 0) r uw (p) Computational Manifolds and Applications (CMA) , IMPA, Rio de Janeiro, RJ, Brazil (g 1 w h g u r uw )(p) Ω w

4 Let [u, w] be an edge of K. Then, r uw : is the rotation around the origin that identifies the edge [ f u (u) =u, f u (w)] of K u with the edge [u, u0 ], where u and u0 have coordinates and (1, 0), respectively. f u (w) f u (x) f u (y) f u (u) Computational Manifolds and Applications (CMA) , IMPA, Rio de Janeiro, RJ, Brazil 4

5 If f u (w) =u i, for some i {0,..., n u 1}, then the rotation angle, θ uw, of r uw is θ uw = i 2π n u. f u (w) f u (x) f u (y) f u (u) Computational Manifolds and Applications (CMA) , IMPA, Rio de Janeiro, RJ, Brazil 5

6 Note that f u (u) = f u (x) = cos (i 1) 2π n u, sin f u (w) = cos i 2π n u, sin i 2π n u f u (y) = cos (i + 1) 2π n u, sin (i 1) 2π n u (i + 1) 2π n u. f u (w) f u (x) f u (y) f u (u) Computational Manifolds and Applications (CMA) , IMPA, Rio de Janeiro, RJ, Brazil 6

7 So, r uw takes the quadrilateral Q uw =[f u (u), f u (x), f u (w), f u (y)] onto the quadrilateral r uw (Q uw )=, cos 2π, sin 2π 2π, (1, 0), cos, sin 2π n u n u n u n u. f u (w) f u (x) f u (u) f u (y) Computational Manifolds and Applications (CMA) , IMPA, Rio de Janeiro, RJ, Brazil 7

8 The next map, g u :, is ideally a C k -diffeomorphism of the plane. This map must take the interior of the quadrilateral, r uw (Q uw ), onto the so-called canonical quadrilateral, Q =, cos π, sin π π π, (1, 0), cos, sin. (1, 0) Q Computational Manifolds and Applications (CMA) , IMPA, Rio de Janeiro, RJ, Brazil 8

9 The reason for requiring that Q be the codomain of g u is somewhat arbitrary and unclear at this moment. But, it has to do with the cocycle condition. We will get back to this point in a few minutes. For the time being, let us assume that such a map g u exists. (1, 0) Q Computational Manifolds and Applications (CMA) , IMPA, Rio de Janeiro, RJ, Brazil 9

10 Finally, we have the map h : given by h(x, y) =(1 x, y). The map h performs a reflection with respect to the x axis and another with respect to the line x = 0.5. (1, 0) (1, 0) Note that h is a rotation of 180 o around the point (0.5, 0). So, we have that h = h 1. Computational Manifolds and Applications (CMA) , IMPA, Rio de Janeiro, RJ, Brazil 10

11 f u (w) f u (y) r uw p f u (x) f u (u) E y u x w (h g u r uw )(p) f w (y) f w (u) Ω w f w (w) (rwu 1 gw 1 h g u r uw )(p) rwu 1 gw 1 h g u r uw f w (x) h g u r uw g 1 w h g u r uw g u r uw (g u r uw )(p) (1, 0) (1, 0) r uw (p) Computational Manifolds and Applications (CMA) , IMPA, Rio de Janeiro, RJ, Brazil 11 (g 1 w h g u r uw )(p) Ω w

12 By definition, r uw and h are C -diffeomorphisms of the plane. So, if the map g u is a C k -diffeomorphism of the plane satisfying g u (r uw ( Q uw )) = Q, we could tentatively define ϕ wu : w Ω wu such that for every x w. ϕ wu (x) = id Ωuw if u = w, (rwu 1 gw 1 h g u r uw )(x) if u = w, As we shall prove, the map ϕ wu satisfies conditions (a) and (b) of Definition 7.1. But, without knowing the map g u, we cannot say anything about condition (c) yet. Computational Manifolds and Applications (CMA) , IMPA, Rio de Janeiro, RJ, Brazil 12

13 Checking... () 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. By assumption, the g u maps are C k -diffeomorphisms of the plane, for some integer k or k =. Since rotations and reflections are C -diffeomorphisms of the plane, the composition of these maps yield a C k -diffeomorphic transition map. So, condition holds. Computational Manifolds and Applications (CMA) , IMPA, Rio de Janeiro, RJ, Brazil 1

14 Checking... (a) ϕ ii = id Ωi, for all i I, Ω i ϕ ii = id Ωi By definition, ϕ uw (x) =x, for every x w, whenever u = w. So, condition (a) holds. Computational Manifolds and Applications (CMA) , IMPA, Rio de Janeiro, RJ, Brazil 14

15 Checking... Parametric Pseudo-Manifolds (b) ϕ ij = ϕ 1 ji, for all (i, j) K, and Ω i ϕ ij Ω j p ϕ 1 ji Computational Manifolds and Applications (CMA) , IMPA, Rio de Janeiro, RJ, Brazil 15

16 Checking... If u = w then condition (b) is trivially true. So, let us assume that u = w. Then, for every point p Ω wu, we have (by definition) that ϕ 1 wu(p) =(rwu 1 gw 1 = ruw 1 gu 1 =(ruw 1 gu 1 = ϕ uw (p). h g u r uw ) 1 )(p) h 1 (g 1 w ) 1 (r 1 wu) 1 (p) h g w r wu )(p) This implies that condition (b) holds when u = w too. Computational Manifolds and Applications (CMA) , IMPA, Rio de Janeiro, RJ, Brazil 16

17 What about condition (c)? (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. We cannot really verify condition (c) without knowing the map g u. However, it is possible to understand what this condition requires from the map without knowing g u. The picture in the next slide should help us clarify the above claim. Computational Manifolds and Applications (CMA) , IMPA, Rio de Janeiro, RJ, Brazil 17

18 f u (w) f u (v) f u (z) f u (u) f u (y) z w u x y v f v (x) f v (w) Ω v f v (u) f v (v) f v (y) f w (x) f w (v) f w (w) f w (u) Ω w f w (z) Computational Manifolds and Applications (CMA) , IMPA, Rio de Janeiro, RJ, Brazil 18

19 Note that the intersection of the stars of u, v, and w consists of exactly one triangle and its edges and vertices. This means at most p-domains overlap at the same point. z u y w v x So, the cocycle condition must hold for points that belong to the triangles w v, Ω vu Ω vw, and Ω wu Ω wv, which are pairwise identified by the transition functions. Computational Manifolds and Applications (CMA) , IMPA, Rio de Janeiro, RJ, Brazil 19

20 But... Ω vu Ω vw =[f v (v), f v (u), f v (w)] f u (w) f u (v) f u (z) f u (u) f u (y) z w u x y v f v (x) f v (w) Ω v f v (u) f v (v) f v (y) w v =[f u (u), f u (w), f u (v)] Ω wu Ω wv =[f w (w), f w (v), f w (u)] f w (v) f w (x) f w (u) Ω w f w (w) f w (z) Computational Manifolds and Applications (CMA) , IMPA, Rio de Janeiro, RJ, Brazil 20

21 For the cocycle condition... (c) For all i, j, k, if then Ω ji Ω jk =, 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. let us set i = u, w = j, and v = k. Computational Manifolds and Applications (CMA) , IMPA, Rio de Janeiro, RJ, Brazil 21

22 So, we must show that (c) for all u, v, and w such that [u, v, w] is a triangle of K, if Ω wu Ω wv = then ϕ uw (Ω wu Ω wv )=v w and ϕ vu (p) =(ϕ vw ϕ wu )(p), for every p v w. Assume that the statement "if Ω wu Ω wv = then ϕ uw (Ω wu Ω wv )=v w " holds, and consider the condition "ϕ vu (p) =(ϕ vw ϕ wu )(p), for every p v w ". Computational Manifolds and Applications (CMA) , IMPA, Rio de Janeiro, RJ, Brazil 22

23 An illustration: f u (w) f u (z) ϕ vu ϕ vu (p) = (ϕ vw ϕ wu )(p) f v (w) f v (u) f u (v) p f u (u) f u (y) ϕ wu f w (x) ϕ vw f v (x) Ω v f v (v) f v (y) f w (v) ϕ wu (p) f w (u) Ω w f w (w) f w (z) Computational Manifolds and Applications (CMA) , IMPA, Rio de Janeiro, RJ, Brazil 2

24 What is the "anatomy" of this composition? f u (w) p f u (v) f u (z) f u (u) f u (y) (g u r uw )(p) f v (x) f v (w) Ω v f v (u) f v (v) f v (y) ϕ wu f w (x) (rwu 1 gw 1 h g u r uw )(p) f w (v) f w (u) Ω w f w (w) f w (z) Computational Manifolds and Applications (CMA) , IMPA, Rio de Janeiro, RJ, Brazil 24

25 h g u r uw applied to w can be visualized as fol- So, the composition rwu 1 gw 1 lows: out! f u (w) f u (u) g u r uw f u (v) h f w (v) f w (w) rwu 1 gw 1 out! Ω w f w (u) Computational Manifolds and Applications (CMA) , IMPA, Rio de Janeiro, RJ, Brazil 25

26 What is the "anatomy" of this composition? f u (w) f u (z) (g w r wv rwu 1 gw 1 h g u r uw )(p) f v (w) f v (u) f u (v) p f u (u) f u (y) ϕ wu f w (v) f w (x) f w (w) ϕ vw f v (x) Ω v f v (v) f v (y) (rvw 1 gv 1 h g w r wv rwu 1 gw 1 h g u r uw )(p) (rwu 1 gw 1 h g u r uw )(p) f w (u) Ω w f w (z) Computational Manifolds and Applications (CMA) , IMPA, Rio de Janeiro, RJ, Brazil 26

27 What is the "anatomy" of this composition? f u (w) f u (z) (g u r uv )(p) ϕ vu f v (w) f v (u) f u (v) p f u (u) f u (y) f w (x) f v (x) (rvu 1 gv 1 h g u r uv )(p) Ω v f v (v) f v (y) f w (v) f w (w) f w (u) Ω w f w (z) Computational Manifolds and Applications (CMA) , IMPA, Rio de Janeiro, RJ, Brazil 27

28 So, we must show that (rvu 1 gv 1 h g u r uv )(p) =(rvw 1 gv 1 h g w r wv r 1 wu g 1 w h g u r uw )(p). This means that if you have a candidate for the g map, you must verify if the above expression holds for it. Obviously, this expression may be too complicated to be useful. Nevertheless, it can help us derive a simple sufficient condition for testing candidate maps. Computational Manifolds and Applications (CMA) , IMPA, Rio de Janeiro, RJ, Brazil 28

29 We want to show that (rvu 1 gv 1 h g u r uv )(p) =(rvw 1 gv 1 h g w r wv r 1 wu g 1 w h g u r uw )(p). First, note that r wv r 1 wu is equal to a rotation of 2π n w around. f w (x) f w (v) f w (w) So, f w (u) Ω w f w (z) rvw 1 gv 1 h g w r wv rwu 1 gw 1 h g u r uw = rvw 1 gv 1 h g w r 2π n w g 1 w h g u r uw. Computational Manifolds and Applications (CMA) , IMPA, Rio de Janeiro, RJ, Brazil 29

30 Now, consider the composition (g w r 2π n w v ). g 1 w )(q), where q is a point in g w (w r 2π nw gw 1 (q) gw r 2π n w gw 1 (q) g 1 w (q) q Ω w This picture suggests that (g w r 2π n w g 1 w )(q) =r π (q) might be a reasonable choice. Computational Manifolds and Applications (CMA) , IMPA, Rio de Janeiro, RJ, Brazil 0

31 So, rvw 1 gv 1 h g w r 2π n w g 1 w h g u r uw = r 1 vw g 1 v h r π h g u r uw. The same assumption we made before leads us to one more property: (g u r uw )(p) =(r π g u r uv )(p), for every p w. f u (z) f u (w) p f u (u) r uw g u f u (v) f u (y) Computational Manifolds and Applications (CMA) , IMPA, Rio de Janeiro, RJ, Brazil 1

32 So, rvw 1 gv 1 h g w r 2π n w g 1 w h g u r uw = r 1 vw g 1 v h r π h g u r uw. The same assumption we made before leads us to one more property: (g u r uw )(p) =(r π g u r uv )(p), for every p w. f u (z) f u (w) p f u (u) r uv g u f u (v) f u (y) Computational Manifolds and Applications (CMA) , IMPA, Rio de Janeiro, RJ, Brazil 2

33 So, rvw 1 gv 1 h g w r 2π n w g 1 w h g u r uw = r 1 vw g 1 v h r π h g u r uw. The same assumption we made before leads us to one more property: (g u r uw )(p) =(r π g u r uv )(p), for every p w. r π f u (z) f u (w) p f u (u) r uv g u f u (v) f u (y) Computational Manifolds and Applications (CMA) , IMPA, Rio de Janeiro, RJ, Brazil

34 So, rvw 1 gv 1 h wv g w r 2π n w g 1 w h uw g u r uw = r 1 vw g 1 v h wv r π h uw r π g u r uv. For the exact same reason, g v r vw (p) =r π g u r vu (p), for every p Ω vw. So, vw gv 1 )(p) =(rvu 1 gv 1 r π )(p). (r 1 So, rvw 1 gv 1 h g w r 2π n w g 1 w h g u r uw = r 1 vu g 1 v r π h r π h r π g u r uv. But, h = r π h r π h r π. Computational Manifolds and Applications (CMA) , IMPA, Rio de Janeiro, RJ, Brazil 4

35 So, rvw 1 gv 1 h g w r 2π n w g 1 w h g u r uw = rvu 1 gv 1 = rvu 1 gv 1 = ϕ vu. r π h r π h r π g u r uv h g u r uv The above equality holds for every point p in v w, and hence the cocycle condition, ϕ vu (p) =(ϕ vw ϕ wu )(p), is satisfied (under the many assumptions we made along the way; which are they?) Computational Manifolds and Applications (CMA) , IMPA, Rio de Janeiro, RJ, Brazil 5

36 We made the following assumptions: (1) The g u map are is a C k -diffeomorphism of the plane, for every u I. (2) The g u map takes Q uw onto Q, for every (u, w) K. () The g u map satisfies (g u r 2π n u g 1 u )(q) =r π (q), where q g u ( ). (4) If f u (w) precedes f u (v) in a counterclockwise enumeration of the vertices of K u, then (g u r uw )(p) = (r π g u r uv )(p), for every point p in the gluing domain w. (5) For all u, v, w such that [u, v, w] is a triangle of K, if Ω wu Ω wv = then ϕ uw (Ω wu Ω wv )=v w. Computational Manifolds and Applications (CMA) , IMPA, Rio de Janeiro, RJ, Brazil 6

37 It turns out that (5) can be derived from (1)-(4). We will leave that as an exercise. So, the point of this lecture is: if we want to know if a candidate for the g map will satisfy condition of Definition 7.1, we can test the map against conditions (1) through (4). Conditions () and (4) are sufficient. Are they also necessary? We are now fully equipped to discuss the "candidate" functions in the next lecture. More specifically, we will go over projective and conformal transformations. Computational Manifolds and Applications (CMA) , IMPA, Rio de Janeiro, RJ, Brazil 7