topological ppt

7/31/2019 Topological ppt

1/18

Topological Algorithm

David Gu1

1Mathematics Science CenterTsinghua University

Tsinghua University

David Gu Conformal Geometry
http://find/http://goback/


2/18

Algebraic Topology

http://find/


3/18

orientation

Suppose Ck, = [v0,v1, ,ck] Let p be a permutation,then if {p(0),p(1), ,p(k)} differs from {0,1, ,k} by even

number of swaps, then [vp(0),vp(1), ,vp(k)] has the sameorientation with ; if they differ by an odd number of swaps,

then [vp(0),vp(1), ,vp(k)] = 1.

http://find/


4/18

Computing Homology basis

Suppose M is a triangle mesh, with vertices, edges and faces

V,E,F respectively. Given two simplices i Ck, j Ck1,denote the adjacency number,

[i,j] =

+1 j i1 1j i0 ji = /0

http://find/


5/18


V = {vi} form the basis of C0, E = {ej} form the basis of C1,F = {fk} form the basis of C2. 1 : C1 C0 and 2 : C2 C1are linear maps, and represented as matrices.

2 = ([fk,ej]),1 = ([ej,vi]).

construct the following linear operator : C1 C1,

= 2 T2 +

T1 1,

the eigen vectors corresponding to zero eigen values are thebasis of H1(,).

http://find/


6/18


Definition (Cut Graph)

A cut graph G of a mesh is a graph formed by non-orientededges of , such that /G is a topological disk.

Figure: Cut graph


C G
http://find/


7/18

Cut Graph

Algorithm:Cut graph

Input: A triangular mesh .Output: A cut graph G

1 Compute the dual mesh , each edge e has a uniquedual edge e .

2 Compute a spanning tree T of .

3 The put graph is the union of all edges whose dual edges

are in T.

G= {e e T}.


W d
http://find/


8/18

Wedge

Definition (Wedge)

On a face f, a corner at vertex v is denoted as a pair (f,v), and

represented using halfedge data structure. Given a vertex v,the corners adjacent to it are ordered counter-clockwisely. A

maximal sequence of adjacent corners without sharp edges

form a wedge.

Figure: Cut graphDavid Gu Conformal Geometry

F d t l D i


9/18

Fundamental Domain

Algorithm: Fundamental Domain

Input: A mesh and a cut graph G.Output: A fundamental domain .

1 Compute the cut graph G of , label all the edges in G assharp edges.

2 Compute the wedges of formed by sharp edges.

3 Construct an empty mesh .

4 For each wedge w insert a vertex into .

5

For each face f = [v0,v1,v2] in , insert a facef = [w0,w1,w2] into , such that the corner (f,vk) belongsto the wedge wk, (f,vk) wk.


F d t l D i
http://find/


10/18

Fundamental Domain

input mesh cut graph fundamental domain

Figure: Fundamental domain


H m l b i
http://find/


11/18

Homology basis

Theorem (Homology basis)

Suppose is a closed mesh, G is a cut graph of, then thebasis of loops of G (assigned with an orientation) is also ahomology basis of.

Proof.

The computation of the cut graph in fact find a CW-celldecomposition

= GD2,

where D2 is a 2-cell. Suppose {1,2, ,n} are the loop basis

of G, D2 is a loop in G, which is represented as a word in1(G), then

1() = 1,2, ,nD2.


Homology basis
http://find/


12/18

Homology basis

Algorithm: Loop basis for a graph G

Input: A graph G.

Output: A basis of loops of G.

1 Compute a spanning tree T of G.2 G/T = {e1,e2, ,en}.

3 eiT has a unique loop, denoted as i.

4 {1,2, ,n} form a basis of 1(G).


Homology Basis


13/18

Homology Basis

Figure: Homology basis


Universal Covering Space
http://find/


14/18


Algorithm: Universal covering space

Input: A mesh .

Output: A finite portion of the universal cover .

1 Compute a cut graph G of . We call a vertex on G withvalence greater than 2 a knot. The knots divide G to

segments, assign an orientation to each segment, label the

segments as {s1,s2, ,sn}.2 Compute a fundamental domain , induced by G, whose

boundary is composed of sks.

3 Initialize , , represented using sks.

4 Glue one copy of to the current along only onesegment sk , sk , sk .

5 Update , if sk are adjacent in , glue them. Repeatthis step, until there is no adjacent sk in the boundary .

6

Repeat gluing copies of until

is large enough.




15/18


input mesh one fundamental domain universal cover

Figure: universal covering space


Cohomology Basis


16/18

Cohomology Basis

Algorithm: Cohomology Basis

Input: A homology basis {1,2, ,n} Output: A cohomologybasis {1,2, ,n}

1 select k, slice the mesh open along k, to get an open

mesh k, k = +k k .

2 Set a 0-form fk : k , such that v +k , fk(v) = 1;

v k , fk(v) = 0.

3 0fk is defined on k, because it is consistent on the

corresponding edges on

+

k and

k , so it is well defined on as well.


Double Covering
http://find/


17/18

Double Covering

Algorithm: Double Cover

Input: An open mesh with boundaries. Output: The double

covering of the mesh, which is a closed symmetric mesh.

1 construct a copy of the input mesh

2

reverse the orientation of the copy, each [v0,v1,v2] isconverted to [v1,v0,v2].

3 Identify each boundary vertex in with the correspondingone in ,

4 each boundary halfedge e has a uniquecorresponding halfedge e . glue the correspondingboundary halfedges to the same boundary edge.


double cover


18/18

double cover

Figure: Double Covering


topological ppt

Documents