Introduction to Computational Manifolds and Applications

Transcription

1 IMPA - Institto de Matemática Pra e Aplicada, Rio de Janeiro, RJ, Brazil Introdction to Comptational Manifolds and Applications Part 1 - Constrctions Prof. Marcelo Ferreira Siqeira mfsiqeira@dimap.frn.br Departmento de Informática e Matemática Aplicada Universidade Federal do Rio Grande do Norte Natal, RN, Brazil

2 Simplicial Srfaces We will start investigating the constrction of 2-dimensional PPM s in E 3. In the previos lectre, we considered a polygon as a sketch of the shape of the crve we wanted to bild. Now, we need another object to play the same role the polygon did. We can think of a few choices, bt the easiest one is argably a polygonal mesh. So, let s start with a triangle mesh, which is a formally known as a simplicial srface. Comptational Manifolds and Applications (CMA) , IMPA, Rio de Janeiro, RJ, Brazil 2

3 Simplicial Srfaces Definition 9.1. Given a finite family, (a i ) i I, of points in E n, we say that (a i ) i I is affinely independent if the family of vectors, (a i a j ) j (I {i}), is linearly independent for some i I. E 2 E 2 E 2 a 1 a 0 a 0 a 2 a 1 A. I. a 0 E 2 E 2 a 3 a 1 a 0 a 2 a 1 a 2 a 0 NOT A. I. Comptational Manifolds and Applications (CMA) , IMPA, Rio de Janeiro, RJ, Brazil 3

4 Simplicial Srfaces Definition 9.2. Let a 0,...,a d be any d + 1 affinely independent points in E n, where d is a non-negative integer. The simplex spanned by the points a 0,...,a d is the convex hll of these points, and is denoted by [a 0,...,a d ]. The points a 0,...,a d are the vertices of. The dimension, dim( ), of the simplex is d, and is also called a d-simplex. In E n, the largest nmber of affinely independent points is n + 1. So, in E n, we have simplices of dimension 0, 1,..., n. A 0-simplex is a point, a 1- simplex is a line segment, a 2-simplex is a triangle, and a 3-simplex is a tetrahedron. Frthermore, the convex hll of any nonempty sbset of vertices of a simplex is a simplex. Comptational Manifolds and Applications (CMA) , IMPA, Rio de Janeiro, RJ, Brazil 4

5 Simplicial Srfaces Definition 9.3. Let =[a 0,...,a d ] be a d-simplex in E n. A face of is a simplex spanned by a nonempty sbset of {a 0,...,a d }; if this sbset is proper then the face is called a proper face. A face of whose dimension is k, i.e., a k-simplex, is called a k-face. a 2 a 0 a 1 a 2 a 2 a 2 a 0 a 1 a 2-face: [a 0, a 1, a 2 ] a 0 a 1 3 proper 1-faces: [a 0, a 1 ], [a 0, a 2 ], [a 1, a 2 ] a 0 a 1 3 proper 0-faces: [a 0 ], [a 1 ], [a 2 ] Comptational Manifolds and Applications (CMA) , IMPA, Rio de Janeiro, RJ, Brazil 5

6 Simplicial Srfaces Definition 9.4. A simplicial complex K in E n is a finite collection of simplices in E n sch that (1) if a simplex is in K, then all its faces are in K; (2) if, K are simplices sch that =, then is a face of both and. violates (1) violates (2) a simplicial complex Comptational Manifolds and Applications (CMA) , IMPA, Rio de Janeiro, RJ, Brazil 6

7 Simplicial Srfaces Definition 9.5. The dimension, dim(k), of a simplicial complex, K, is the largest dimension of a simplex in K, i.e., dim(k) =max{dim( ) K}. We refer to a d-dimensional simplicial complex as simply a d-complex. The set consisting of the nion of all points in the simplices of K is called the nderlying space of K, and it is denoted by K. The nderlying space, K, of K is also called the geometric realization of K. a 2-complex Comptational Manifolds and Applications (CMA) , IMPA, Rio de Janeiro, RJ, Brazil 7

8 Simplicial Srfaces A simplicial complex is a combinatorial object (i.e., a finite collection of simplices). The nderlying space of a simplicial complex is a topological object, a sbset of some E n. a 2-complex Comptational Manifolds and Applications (CMA) , IMPA, Rio de Janeiro, RJ, Brazil 8

9 Simplicial Srfaces Definition 9.6. Let K be a simplicial complex in E n. Then, for any simplex in K, we define two other complexes, the star, st(, K), and the link, lk(, K), of in K, as follows: st(, K) ={ K in K sch that is a face of and is a face of } and lk(, K) ={ K is in st(, K) and and have no face in common}. K st(, K) lk(, K) Comptational Manifolds and Applications (CMA) , IMPA, Rio de Janeiro, RJ, Brazil 9

10 Simplicial Srfaces Definition 9.7. A 2-complex K in E n is called a simplicial srface withot bondary if every 1-simplex of K is the face of precisely two simplices of K, and the nderlying space of the link of each 0-simplex of K is homeomorphic to the nit circle, S 1 = {x E 2 x = 1}. The set consisting of the 0-, 1-, and 2-faces of a 3-simplex is a simplicial srface withot bondary. Comptational Manifolds and Applications (CMA) , IMPA, Rio de Janeiro, RJ, Brazil 10

11 Simplicial Srfaces The simplicial complex consisting of the proper faces of two 3-simplices (i.e., two tetrahedra) sharing a common vertex is not a simplicial srface withot bondary as the link of the common vertex of the two 3-simplices is not homeomorphic to the nit circle, S 1. Comptational Manifolds and Applications (CMA) , IMPA, Rio de Janeiro, RJ, Brazil 11

12 Simplicial Srfaces From now on, we will refer to a simplicial srface withot bondary as simply a simplicial srface. The nderlying space of a simplicial srface is called its nderlying srface. The nderlying srface of a simplicial srface is a topological 2-manifold in E n. K K Comptational Manifolds and Applications (CMA) , IMPA, Rio de Janeiro, RJ, Brazil 12

13 Simplicial Srfaces Definition 9.8. Let K be a simplicial complex in E n. For each integer i, with 0 i dim(k), we define K (i) as the simplicial complex consisting of all j-simplices of K, for every j sch that 0 j i. Moreover, if L is a simplicial complex in E m, then a map f : K (0) L (0) is called a simplicial map if whenever [a 0,...,a d ] is a simplex in K, then [ f (a 0 ),..., f (a d )] is a simplex in L. A simplicial map is a simplicial isomorphism if it is a bijective map, and if its inverse is also a simplicial map. Finally, if there exists a simplicial isomorphism from K to L, then we say that K and L are simplicially isomorphic. Comptational Manifolds and Applications (CMA) , IMPA, Rio de Janeiro, RJ, Brazil 13

14 Simplicial Srfaces a 2 E 3 E 2 b 1 b 2 a 3 a 0 a 1 b 0 b 5 a 4 b 3 K a 5 b 4 L K and L are simplicially isomorphic. Comptational Manifolds and Applications (CMA) , IMPA, Rio de Janeiro, RJ, Brazil 14

15 Simplicial Srfaces a 2 E 3 E 2 b 1 b 2 a 3 a 0 a 1 b 0 b 5 a 4 b 3 K a 5 b 4 L Let f : K (0) L (0) be given by f (a 0 )=b 5, f (a 1 )=b 3, f (a 2 )=b 2, f (a 3 )=b 1, f (a 4 )=b 0, f (a 5 )=b 4. Comptational Manifolds and Applications (CMA) , IMPA, Rio de Janeiro, RJ, Brazil 15

16 Simplicial Srfaces a 2 E 3 E 2 b 1 b 2 a 3 a 0 a 1 b 0 b 5 a 4 b 3 K a 5 b 4 L It is easily verified that f is a simplicial isomorphism. Comptational Manifolds and Applications (CMA) , IMPA, Rio de Janeiro, RJ, Brazil 16

17 Gling Data Given a simplicial srface, K, in E 3, we are interested in bilding a parametric psedo-srface, M, in E 3 sch that the image, M, of M is homeomorphic to the nderlying srface, K, of K, and sch that M also approximates the geometry of K. K M Comptational Manifolds and Applications (CMA) , IMPA, Rio de Janeiro, RJ, Brazil 17

18 Gling Data As we did before, let s first focs on the definition of a set of gling data. Unfortnately, this task is not as easy as it was in the one-dimensional case. The key is to notice that the simplicial srface, K, which is a combinatorial object, explicitly defines a topological strctre on K (via the adjacency relations of all simplices). So, we shold define p-domains, gling domains, and transition fnctions based on K. Comptational Manifolds and Applications (CMA) , IMPA, Rio de Janeiro, RJ, Brazil 18

19 Gling Data As we will see dring the next lectres, there are many choices for p-domains. Bt, in general, p-domains are associated with simplices of K. For instance, the vertices of K. We can define a one-to-one correspondence between p-domains and vertices of K. E 3 E 2 v w v w Comptational Manifolds and Applications (CMA) , IMPA, Rio de Janeiro, RJ, Brazil 19

20 Gling Data The previos correspondence implies that the nmber of p-domains is eqal to the nmber of vertices of K. A distinct choice of correspondence may yield a different nmber. The choice of a geometry for the p-domains is a key decision too. E 3 E 2 v w v w Comptational Manifolds and Applications (CMA) , IMPA, Rio de Janeiro, RJ, Brazil 20

21 Gling Data Intitively, each p-domain is an open "disk" that is consistently gled to other p- domains in order to define the topology of the image, M, of the parametric psedosrface. Since a vertex of K is connected only to the vertices of K that belong to the link, lk(, K), of in K, it is natral to think of the p-domain,, which is associated with vertex, as the interior of a polygon in E 2 with the same nmber of vertices as lk(, K). E 3 E 2 Comptational Manifolds and Applications (CMA) , IMPA, Rio de Janeiro, RJ, Brazil 21

22 Gling Data To simplify calclations, we can assme that is a reglar polygon inscribed in a nit circle centered at the origin of a local coordinate system of E 2. We can also assme that one vertex of is located at the point (0, 1). Now, is niqely defined. E 3 E 2 (0, 0) (0, 1) Comptational Manifolds and Applications (CMA) , IMPA, Rio de Janeiro, RJ, Brazil 22

23 Gling Data Formally, let I = { is a vertex in K}, n be the nmber of vertices of the link, lk(, K), of in K, and P be the reglar, n -polygon whose vertices are located at the points cos i 2 n, sin i 2 n, for all i = 0, 1,..., n 1. Then, we can define = P, where P is the interior of P. E 3 E 2 (0, 0) (0, 1) Comptational Manifolds and Applications (CMA) , IMPA, Rio de Janeiro, RJ, Brazil 23

24 Gling Data Checking... (1) For every i I, the set i is a nonempty open sbset of E n called parametrization domain, for short, p-domain, and any two distinct p-domains are pairwise disjoint, i.e., i j =, for all i = j. Or p-domains are (connected) open sbsets of E 2. If we assme that they live in distinct copies of E 2, then they will not overlap, and hence condition (1) of Definition 7.1 holds. Comptational Manifolds and Applications (CMA) , IMPA, Rio de Janeiro, RJ, Brazil 24

25 Gling Data What abot gling domains? choice: The following pictre shold help s find a good E 3 w (0, 0) (0, 1) (0, 0) (0, 1) E 2 w Comptational Manifolds and Applications (CMA) , IMPA, Rio de Janeiro, RJ, Brazil 25

26 Gling Data To precisely define gling domains, we associate a 2-dimensional simplicial complex, K, with each p-domain. The complex K satisfies the following two conditions: (1) K, is the closre,, of and (2) K is isomorphic to the star, st(, K), of in K. An obvios choice for K is the canonical trianglation of : E 3 E 2 K Comptational Manifolds and Applications (CMA) , IMPA, Rio de Janeiro, RJ, Brazil 27

27 Gling Data Fix any conterclockwise enmeration, 0, 1,..., m, of the vertices in lk(, K). E 3 E K Comptational Manifolds and Applications (CMA) , IMPA, Rio de Janeiro, RJ, Brazil 28

28 Gling Data Let 0 be the vertex of K located at the point (0, 1). Let 0, 1,..., m be the conterclockwise enmeration of the vertices of lk(, K ) starting with 0. E E K (0, 1) 3 4 Comptational Manifolds and Applications (CMA) , IMPA, Rio de Janeiro, RJ, Brazil 29

29 Gling Data Let f : st(, K) (0) K (0) be the simplicial map given by f () = and f ( i )= i, for i = 0,..., m. It is easily verified that f is a simplicial isomorphism. E E K (0, 1) 3 4 Comptational Manifolds and Applications (CMA) , IMPA, Rio de Janeiro, RJ, Brazil 30

30 Gling Data Let and w be any two vertices of K sch that [, w] is an edge in K. Let x and y be the other two vertices of K that also belong to both st(, K) and st(w, K). E 3 y x w Assme that x precedes w in a conterclockwise traversal of the vertices of lk(, K) starting at y. Comptational Manifolds and Applications (CMA) , IMPA, Rio de Janeiro, RJ, Brazil 31

31 Gling Data We can now define the gling domains, w and w, as w =Q w and w =Q w, where Q w =[f (), f (x), f (w), f (y)] and Q w =[f w (w), f w (y), f w (), f w (x)] are the qadrilaterals given by the vertices f (), f (x), f (w), f (y) of K and the vertices f w (w), f w (y), f w (), f w (x) of K w, and Q w and Q w are the interiors of Q w and Q w. E 2 E 3 f (x) f y (w) f () x w f w (w) f w (y) (0, 1) (0, 1) f (y) f w () f w (x) Comptational Manifolds and Applications (CMA) , IMPA, Rio de Janeiro, RJ, Brazil 32 w E 2

32 Gling Data Formally, for every (, w) I I, we let if = w, w = if = w and [, w] is not an edge of K, Q w if = w and [, w] is an edge of K. E 2 f (w) f (x) f () (0, 1) E 3 x y w f w (y) w E 2 f w (w) (0, 1) f (y) f w () f w (x) Comptational Manifolds and Applications (CMA) , IMPA, Rio de Janeiro, RJ, Brazil 33

33 Gling Data Checking... (2) For every pair (i, j) I I, the set ij is an open sbset of i. Frthermore, ii = i and ji = if and only if ij =. Each nonempty sbset ij (with i = j) is called a gling domain. By definition, the sets,, and Q w are open in E 2. Frthermore, the sets Q w and Q w are well-defined and nonempty, for every, w I sch that [, w] is an edge of K. So, for every, w I, we have that w = iff [, w] is an edge of K. Ths, for every, w I, w = iff w =, and hence condition (2) of Definition 7.1 also holds. Comptational Manifolds and Applications (CMA) , IMPA, Rio de Janeiro, RJ, Brazil 34

34 Gling Data Or definitions of p-domain and gling domain natrally lead s to a gling process indced by the gling of the stars of the vertices of K along their common edges and triangles. The gling strategy we adopted here does not depend on the geometry of the p- domains and gling domains, bt on the adjacency relations of vertices and edges of K. However, the geometry of the p-domains and gling domains have a strong inflence in the level of difficlty of the transition maps and parametrizations we choose to se. Despite of or commitment to a particlar geometry, we will present next an axiomatic way of defining the transition maps. Or axiomatic definition shold be as mch independent of the geometry of the p-domains and gling domains as possible. Comptational Manifolds and Applications (CMA) , IMPA, Rio de Janeiro, RJ, Brazil 35