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 More on Transition Maps We ve seen a class of complex functions that can play the role of the g maps in our transition functions. It is worth mentioning that we still have to check assumption (5) for them. Recall that we had to change the geometry of the p-domains, so that we could define a C k -diffeomorphism between Q and the gluing domains, where k is a positive integer or k =. However, as we shall see in a coming lecture, this change in geometry imposes some difficulties for defining bump functions, shape functions, and parametrizations on the p-domains. Now, we present an alternative choice for the g maps. This alternative also requires a change in the geometry of the p-domains. But, this change is more natural and less troublesome. Computational Manifolds and Applications (CMA) , IMPA, Rio de Janeiro, RJ, Brazil 2

3 More on Transition Functions The key idea is to consider the p-domain as an open disk in the underlying space of K u. More specifically, Ω u is the interior of the circle, C u, inscribed in K u : π 2 Ω u = (x, y) E 2 x 2 + y 2 < cos. n u E 3 E 2 u K u f u (u) Ω u Computational Manifolds and Applications (CMA) , IMPA, Rio de Janeiro, RJ, Brazil 3

4 More on Transition Functions E 2 f u (w) f u (y) p f u (x) f u (u) Ω u (0, 1) E 3 u x y w (h g u r uw )(p) f w (y) f w (u) E 2 (rwu 1 gw 1 h g u r uw )(p) Ω w f w (w) f w (x) (0, 1) r uw rwu 1 gw 1 h g u r uw h g u r uw (0, 0) g u r uw (0, 1) (0, 0) (g u r uw )(p) (1, 0) (0, 0) (1, 0) g 1 w h g u r uw (0, 0) (0, 1) Ω u r uw (p) Computational Manifolds and Applications (CMA) , IMPA, Rio de Janeiro, RJ, Brazil 4 (g 1 w h g u r uw )(p) Ω w

5 More on Transition Functions Like we did before, let g u : E 2 {(0, 0)} E 2 {(0, 0)} be given by the composition g u (p) =(Π 1 Γ u Π)(p), for every p R 2 {(0, 0)}. However, Γ u : R + ] π, π [ R + ] π, π [ is given by Γ u (r, θ) = cos(π/6) cos(π/n u ) r, n u 6 θ where (r, θ) =Π(p) are the polar coordinates of p., Computational Manifolds and Applications (CMA) , IMPA, Rio de Janeiro, RJ, Brazil 5

6 More on Transition Functions Function Γ u maps Ω u {(0, 0)} onto C {(0, 0)}, where C = (x, y) E 2 x 2 + y 2 π 2 cos 6. E 2 cos πnu p cos π 6 Γ u (p) (0, 0) (0, 0) Computational Manifolds and Applications (CMA) , IMPA, Rio de Janeiro, RJ, Brazil 6

7 More on Transition Functions For any (u, w) I I, the gluing domain Ω uw is defined as the image, (ruw 1 gu 1 )( E ), of the interior, E, of the canonical lens, E, under the composite function ruw 1 gu 1, where and E = C D, C = {(x, y) x 2 + y 2 (cos(π/6)) 2 } and D = {(x, y) (x 1) 2 + y 2 (cos(π/6)) 2 }. Computational Manifolds and Applications (CMA) , IMPA, Rio de Janeiro, RJ, Brazil 7

8 More on Transition Functions So, for any (u, w) I I, the gluing domain Ω uw is defined as (r 1 uw g 1 u )( E) Ω uw = Ω u if u = w, (ruw 1 gu 1 )( E) if [u, w] is an edge of K, otherwise. Computational Manifolds and Applications (CMA) , IMPA, Rio de Janeiro, RJ, Brazil 8

9 For any (u, w) K, the transition map, More on Transition Functions ϕ wu : Ω uw Ω wu, is such that, for every p Ω uw, we let p if u = w, ϕ wu (p) = (rwu 1 gw 1 h g u r uw )(p) otherwise. Computational Manifolds and Applications (CMA) , IMPA, Rio de Janeiro, RJ, Brazil 9

10 More on Transition Functions It is now time for checking our assumptions regarding g u : (1) The g u map is a C k -diffeomorphism of R 2 {(0, 0)}, for every u I (2) The g u map takes r uw (Ω uw ) onto E for every (u, w) K. (3) The g u map satisfies (g u r 2π n u g 1 u )(q) =r π 3 (q), where q g u (Ω u ). (4) If f u (w) precedes f u (v) in a counterclockwise enumeration of the vertices of lk(u, K), then (g u r uw )(p) =(r π 3 g u r uv )(p), for every point p in the gluing domain Ω uw. Computational Manifolds and Applications (CMA) , IMPA, Rio de Janeiro, RJ, Brazil 10

11 More on Transition Functions We have not checked the following assumption: (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 )=Ω uv Ω uw. We will also explore that in a homework. Computational Manifolds and Applications (CMA) , IMPA, Rio de Janeiro, RJ, Brazil 11

12 More on Transition Functions ( t. Computational Manifolds and Applications (CMA) , IMPA, Rio de Janeiro, RJ, Brazil 12

13 As far as we know, Cindy Grimm and John Hughes presented the first construction of parametric pseudo-manifolds from gluing data (see the Ph. D. thesis of Grimm, 1996). Here, we will give an overview of this construction. We refer the audience to the aforementioned Ph. D. thesis and to Grimm and Hughes SIGGRAPH 1995 paper for details. Pointer to these references can be found on the course web page. The construction of the gluing data is very intricate. So, reading the above references may be crucial for an in-depth understanding of their work and for implementation purposes. Computational Manifolds and Applications (CMA) , IMPA, Rio de Janeiro, RJ, Brazil 13

14 The input for the construction is any polygonal mesh. But, since we have not yet defined such meshes, we will restrict our attention to triangle meshes (i.e., simplicial surfaces). As usual, let us denote the given simplicial surface by K. The simplicial surface K is "refined" by one step of the Catmull-Clark subdivision rule, and then the dual of the resulting (cell) complex is considered for defining the gluing data. The object resulting from the Catmull-Clark subdivision and its dual can be thought of as "graphs" with straight edges immersed in E 3. Here, we will not define them in a formal way. Instead, we will illustrate how they are obtained using the subdivision rule. Computational Manifolds and Applications (CMA) , IMPA, Rio de Janeiro, RJ, Brazil 14

15 K Computational Manifolds and Applications (CMA) , IMPA, Rio de Janeiro, RJ, Brazil 15

16 face point Computational Manifolds and Applications (CMA) , IMPA, Rio de Janeiro, RJ, Brazil 16

17 edge point Computational Manifolds and Applications (CMA) , IMPA, Rio de Janeiro, RJ, Brazil 17

18 Computational Manifolds and Applications (CMA) , IMPA, Rio de Janeiro, RJ, Brazil 18

19 Computational Manifolds and Applications (CMA) , IMPA, Rio de Janeiro, RJ, Brazil 19

20 Dual Mesh Computational Manifolds and Applications (CMA) , IMPA, Rio de Janeiro, RJ, Brazil 20

21 Let us denote the dual mesh by K. Computational Manifolds and Applications (CMA) , IMPA, Rio de Janeiro, RJ, Brazil 21

22 Note that all vertices of K have degree four. This is the reason for defining K. Computational Manifolds and Applications (CMA) , IMPA, Rio de Janeiro, RJ, Brazil 22

23 The gluing data defined by Grimm s constructions consists of one p-domain per each component of K ; that is, we assign a p-domain with each dual mesh vertex, edge, and face. Here, we view a "face" as a disk-like region bounded by a simple cycle of edges of K, which is the dual of a vertex of the graph obtained from the Catmull-Clark subdivision. Computational Manifolds and Applications (CMA) , IMPA, Rio de Janeiro, RJ, Brazil 23

24 Computational Manifolds and Applications (CMA) , IMPA, Rio de Janeiro, RJ, Brazil 24

25 So, a face can be identified with a regular n-sided polygon in E 2. The p-domains associated with vertices, edges, and faces have distinct geometry. Furthermore, the geometry of the p-domains associated with edges (resp. faces) can also differ. Let V be the set of vertices of K. Then, for each v V, we define the p-domain Ω v as Ω v =] 0.5, 0.5 [ 2. ( 0.5, 0.5) y (0.5, 0.5) ( 0.5, 0.5) (0, 0) x (0.5, 0.5) Computational Manifolds and Applications (CMA) , IMPA, Rio de Janeiro, RJ, Brazil 25

26 Let E be the set of edges of K. Then, for each e E, we define the p-domain Ω e as a diamond-shaped region that consists of the interior of an hexagon with vertices p 0,...,p 5 : p 4 p 3 y (0, 0) p 5 p 0 =(0.5 h, h cot(π/n u )) p 2 p 0 x p 1 p 1 =(0.5 h, h cot(π/n l )) p 2 = 0, cot(π/n l) 2 p 3 =( h, h cot(π/n l )) p 4 =( h, h cot(π/n u )) p 5 = 0, cot(π/n u) 2 So, the coordinates p 0,...,p 5 depend on the parameters h, n u, and n l. Computational Manifolds and Applications (CMA) , IMPA, Rio de Janeiro, RJ, Brazil 26

27 By construction, each edge e of K is incident with exactly two faces, say f u and f l, of K. e f u f l Computational Manifolds and Applications (CMA) , IMPA, Rio de Janeiro, RJ, Brazil 27

28 We let n u and n l be the number of vertices of f u and f l, respectively. Now, consider two regular polygons in E 2, with n u and n l sides, respectively. If the their sides have unit length, then the distance from the center of the polygon to the middle point of any edge of the polygon is equal to 1 2 cot(π/n u) and 1 2 cot(π/n l ). This is why the second coordinate of p 2 and p 5 are given as 1 2 cot(π/n u) and 1 2 cot(π/n l). l l = 1 2 cot(π/6) Computational Manifolds and Applications (CMA) , IMPA, Rio de Janeiro, RJ, Brazil 28

29 The parameter h is related to the transition functions and will be defined later. Let F be the set of faces of K. Then, for each f F, we define the p-domain Ω f as the interior of a regular n-sided polygon centered at (0, 0) and whose sides are 1 2h units long. The parameter n is the number of vertices of f. ( h, 0.5 h) y (0.5 h, 0.5 h) n = 4 = (0, 0) x ( h, h) (0.5 h, h) Computational Manifolds and Applications (CMA) , IMPA, Rio de Janeiro, RJ, Brazil 29

30 Just like before, gluing domains are determined by the adjacency relations of vertices, edges, and faces of K. In particular, p-domains associated with the same class of elements of K (i.e., vertices, edges, and faces) are not identified by the gluing process. So, if v 1 and v 2 are two distinct vertices in V, then Ω v1 v 2 = Ω v2 v 1 =. Likewise, if e 1 and e 2 are two distinct edges in E and if f 1 and f 2 are two distinct faces in F, then we get Ω e1 e 2 = Ω e2 e 1 = Ω f1 f 2 = Ω f2 f 1 =. Furthermore, if v 1 = v 2 then Ω v1 v 2 = Ω v2 v 1 = Ω v1. The same is true for edges and faces. Computational Manifolds and Applications (CMA) , IMPA, Rio de Janeiro, RJ, Brazil 30

31 There are only three possibilities for nonempty gluing domains: (1) The p-domain, Ω v, associated with vertex v V is glued to Ω e, the p-domain associated with edge e E. (2) The p-domain, Ω v, associated with vertex v V is glued to Ω f, the p-domain associated with face f F. (3) The p-domain, Ω e, associated with edge e E is glued to Ω f, the p-domain associated with face f F. Computational Manifolds and Applications (CMA) , IMPA, Rio de Janeiro, RJ, Brazil 31

32 In particular, we have (1) Ω ve = if and only if vertex v is a vertex of edge e. (2) Ω vf = if and only if vertex v is a vertex of face f. (3) Ω ef = if and only if edge e is an edge of face f. (4) Ω ev = if and only if edge e is incident with vertex v. (5) Ω fv = if and only if face f is incident with vertex v. (6) Ω fe = if and only if face f is incident with edge e. Computational Manifolds and Applications (CMA) , IMPA, Rio de Janeiro, RJ, Brazil 32

33 For any given vertex v V, there are exactly nine nonempty gluing domains in Ω v : f 3 e 3 f 2 Ω vf3 Ω ve3 Ω vf2 e 0 v e 2 Ω ve0 Ω v Ω ve2 f 0 f 1 e 1 Ω vf0 Ω ve1 Ω vf1 Computational Manifolds and Applications (CMA) , IMPA, Rio de Janeiro, RJ, Brazil 33

34 The gluing domain Ω vei corresponds to half a diamond-shaped region: f 3 e 3 f 2 Ω vf3 Ω ve3 Ω vf2 e 0 v e 2 Ω ve0 Ω v Ω ve2 f 0 f 1 e 1 Ω vf0 Ω ve1 Ω vf1 Computational Manifolds and Applications (CMA) , IMPA, Rio de Janeiro, RJ, Brazil 34

35 The gluing domain Ω vfi corresponds to a non-degenerated quadrilateral: f 3 e 3 f 2 Ω vf3 Ω ve3 Ω vf2 e 0 v e 2 Ω ve0 Ω v Ω ve2 f 0 f 1 e 1 Ω vf0 Ω ve1 Ω vf1 Computational Manifolds and Applications (CMA) , IMPA, Rio de Janeiro, RJ, Brazil 35

36 For any given edge e E, there are exactly five nonempty gluing domains in Ω e : f u Ω e Ω efu v l v r Ω evl Ω evr f l Ω efl Computational Manifolds and Applications (CMA) , IMPA, Rio de Janeiro, RJ, Brazil 36

37 The gluing domain Ω evl corresponds to the set of points (x, y) Ω e such that x < 0. f u v l v r Ω evl f l Computational Manifolds and Applications (CMA) , IMPA, Rio de Janeiro, RJ, Brazil 37

38 The gluing domain Ω evr corresponds to the set of points (x, y) Ω e such that x > 0. f u v l v r Ω evr f l Computational Manifolds and Applications (CMA) , IMPA, Rio de Janeiro, RJ, Brazil 38

39 The gluing domain Ω efl corresponds to the set of points (x, y) Ω e such that y < h cot(π/n l ). f u v l v r f l Ω efl Computational Manifolds and Applications (CMA) , IMPA, Rio de Janeiro, RJ, Brazil 39

40 The gluing domain Ω efr corresponds to the set of points (x, y) Ω e such that y > h cot(π/n u ). f u Ω efu v l v r f l Computational Manifolds and Applications (CMA) , IMPA, Rio de Janeiro, RJ, Brazil 40

41 For any given n-sided face f F, there are exactly 2n + 1 nonempty gluing domains in Ω f : v 2 Ω fv2 e 2 f e 1 Ω fe2 Ω f Ω fe1 Ω fv0 Ω fe0 Ω fv1 v 0 v 1 e 0 Computational Manifolds and Applications (CMA) , IMPA, Rio de Janeiro, RJ, Brazil 41

42 The gluing domains Ω fvi correspond to open quadrilaterals defined by connecting the center of Ω f (i.e., the origin (0, 0)) to the midpoint of the edges of the closure of Ω f. Ω fv2 Ω fv0 Ω fv1 Computational Manifolds and Applications (CMA) , IMPA, Rio de Janeiro, RJ, Brazil 42

43 The gluing domains Ω fei correspond to open triangles defined by connecting the center of Ω f (i.e., the origin (0, 0)) to the vertices of the closure of Ω f. Ω fe2 Ω fe1 Ω fe0 Computational Manifolds and Applications (CMA) , IMPA, Rio de Janeiro, RJ, Brazil 43

44 There are six types of transition functions: (1) vertex-vertex, (2) edge-edge, (3) faceface, (4) vertex-edge, (5) vertex-face, and (6) edge-face. The first 3 functions are the identity. Function (6) is an affine map (takes a triangle onto a triangle). Function (5) is a projective map (takes a quadrilateral onto a quadrilateral). Function (4) is defined as a weighted sum of two composite functions, each of which is the composition of an edge-face and vertex-face function. This function is bit complicated. Computational Manifolds and Applications (CMA) , IMPA, Rio de Janeiro, RJ, Brazil 44

45 The edge-face transition map, ϕ fe : Ω fe Ω ef There is a unique affine transformation that takes the Ω ef onto Ω fe after a one-to-one correspondence between the vertices of the triangles corresponding to their closures is established. Computational Manifolds and Applications (CMA) , IMPA, Rio de Janeiro, RJ, Brazil 45

46 The vertex-face transition map, ϕ fv : Ω vf Ω fv Computational Manifolds and Applications (CMA) , IMPA, Rio de Janeiro, RJ, Brazil 46

47 To compute the projective transformation that takes Ω vf onto Ω fv, we consider two sets of points. The first contains the points p 0, p 1, p 2, p 3 that define the quadrant of Ω v containing Ω vf. The second contains the points q 0, q 1, q 2, and q 3, which define q quadrilateral in a regular, n-sided polygon centered at the origin and whose sides have length 1 p 3 p 2 q 1 q 0 Ω vf Ω fv p 0 p 1 q 2 q 3 Computational Manifolds and Applications (CMA) , IMPA, Rio de Janeiro, RJ, Brazil 47

48 The domain Ω vf is actually defined as ϕ vf (Ω fv ). The vertex-edge transition map, ϕ ev : ϕ fu e ϕ vfu ϕ ve ϕ fl e ϕ vfl Computational Manifolds and Applications (CMA) , IMPA, Rio de Janeiro, RJ, Brazil 48

49 How can we define ϕ ve so that the cocycle condition holds? Grimm did not define ϕ ve in a direct manner. Instead, she forced ϕ ve to be a weighted sum of two composite functions: ϕ vfl ϕ fl e and ϕ vfu ϕ fu e. Since the domain of these functions are disjoint (in Ω ev ), she blended the functions along the region Ω ev (Ω efl Ω efu ). Ω ev (Ω efl Ω efu ) Computational Manifolds and Applications (CMA) , IMPA, Rio de Janeiro, RJ, Brazil 49

50 The idea is to let ϕ ve (p) =(ϕ vfl ϕ fl e)(p) if p (Ω ev Ω efl ), ϕ ve (p) =(ϕ vfu ϕ fu e)(p) if p (Ω ev Ω efu ), and ϕ ve (p) equal to some "average" value if p (Ω ev (Ω efl Ω efu )). In particular, ϕ ve (p) =(1 β(t)) (ϕ vfl ϕ fl e)(p)+β(t) (ϕ vfu ϕ fu e)(p), where β : R [0, 1] is a function satisfying the following properties: β(t) =0 for t < h cot(π/n l ). β(t) =1 for t > h cot(π/n u ). Computational Manifolds and Applications (CMA) , IMPA, Rio de Janeiro, RJ, Brazil 50

51 β(t) is monotonically increasing. β is C k for a given k (the desired continuity of the manifold). The derivative of β is bounded by the function β (t) = h cot(π/6)+t h cot(π/6) h cot(π/6) t h cot(π/6) if t 0 if t > 0. Grimm shows that for k 0 the function ϕ ve is invertible, one-to-one, and onto. She also tells us how to build the function β. Computational Manifolds and Applications (CMA) , IMPA, Rio de Janeiro, RJ, Brazil 51

52 The choice of a value for the parameter h is related to the geometry of the resulting manifold. Grimm computes this value by solving the equation δk ϕ vf (0.5 h, h cot(π/6)) = 2, δ k 2, where δ k = 1 2 (2k + 3), where k is the degree of continuity of β. Computational Manifolds and Applications (CMA) , IMPA, Rio de Janeiro, RJ, Brazil 52

53 There are a few issues with this construction. First, the number of p-domains is large compared to the number of p-domains in the approach we saw before. Second, the definition of the map ϕ ve is not elegant. Third, the gluing regions are small (compared to the ones in other constructions), which may lead to visual artifacts in the resulting surfaces. Computational Manifolds and Applications (CMA) , IMPA, Rio de Janeiro, RJ, Brazil 53