topological representations for meshestutte’s representation of surfaces an abstract set of...

Topological Representations forMeshes

David Cardoze

Gary Miller

Todd Phillips

Topological Representations for Meshes – p.1/26

What is meshing?

Meshing is the decomposition of a geometric domain intobasic building blocks.

Topological Representations for Meshes – p.2/26

What is meshing?

Meshing is the decomposition of a geometric domain intobasic building blocks.

Topological Representations for Meshes – p.2/26

Meshing is ubiquitous

Numerical solutions to partial differential equations

Geographic Information Systems

Data interpolation

Topological Representations for Meshes – p.3/26

Meshing is ubiquitous

Numerical solutions to partial differential equations

Geographic Information Systems

Data interpolation

Topological Representations for Meshes – p.3/26

Meshing is ubiquitous

Numerical solutions to partial differential equations

Geographic Information Systems

Data interpolation

Topological Representations for Meshes – p.3/26

Cells

A

�

-cell is a region homeomorphic to the open�

-ball

� � � ��� � � � � � � � �

-cell

-cells

-cells

Topological Representations for Meshes – p.4/26

Cells

A

�

-cell is a region homeomorphic to the open�

-ball

� � � ��� � � � � � � � �

�

-cell

-cells

-cells

Topological Representations for Meshes – p.4/26

Cells

A

�

-cell is a region homeomorphic to the open�

-ball

� � � ��� � � � � � � � �

�

-cell

-cells

-cells

Topological Representations for Meshes – p.4/26

Cells

A

�

-cell is a region homeomorphic to the open�

-ball

� � � ��� � � � � � � � �

�

-cell

-cells

�

-cells

Topological Representations for Meshes – p.4/26

Cell Complex

A cell complex is the decomposition of a region into cells.

Topological Representations for Meshes – p.5/26

Cell Complex

A cell complex is the decomposition of a region into cells.

Topological Representations for Meshes – p.5/26

Cell Complex

A cell complex is the decomposition of a region into cells.

Topological Representations for Meshes – p.5/26

Barycentric Subdivisions of Cells

Creates a triangulation of the cell.

Topological Representations for Meshes – p.6/26

Barycentric Subdivisions of Cells

Creates a triangulation of the cell.

Topological Representations for Meshes – p.6/26

Barycentric Subdivisions of Cells

Creates a triangulation of the cell.

� �

� �

�

Topological Representations for Meshes – p.6/26

Why Barycentric Subdivisions?

Provide an efficient and elegant way to accesstopological and order information by means of switchoperators

Topological Representations for Meshes – p.7/26

Why Barycentric Subdivisions?

Provide an efficient and elegant way to accesstopological and order information by means of switchoperators

Topological Representations for Meshes – p.7/26

Why Barycentric Subdivisions?

Provide an efficient and elegant way to accesstopological and order information by means of switchoperators

Topological Representations for Meshes – p.7/26

Numbered Simplicial Sets

Barycentric subdivisions of cell complexes are specialcases of numbered simplicial sets.

A numbered -simplex is a simplex whose vertices areuniquely labeled with numbers between and .

Topological Representations for Meshes – p.8/26

Numbered Simplicial Sets

Barycentric subdivisions of cell complexes are specialcases of numbered simplicial sets.

A numbered

�

-simplex is a simplex whose vertices areuniquely labeled with numbers between

�and

�

.

� � � �

� �

�

Topological Representations for Meshes – p.8/26

Numbered Simplicial Sets

Barycentric subdivisions of cell complexes are specialcases of numbered simplicial sets.

A numbered

�

-simplex is a simplex whose vertices areuniquely labeled with numbers between

�and

�

.

� � � �

� �

�

Topological Representations for Meshes – p.8/26

Numbered Simplicial Sets

A numbered

�

-simplicial set is a collection numberedsimplices glued along

� ��� �

-faces with compatiblelabels.

Topological Representations for Meshes – p.9/26

Numbered Simplicial Sets

A numbered

�

-simplicial set is a collection numberedsimplices glued along

� ��� �

-faces with compatiblelabels.

Topological Representations for Meshes – p.9/26

Faces of a Simplicial Set

� �

� �

�

� �

� �

�

Topological Representations for Meshes – p.10/26

Not a Cell Complex

Topological Representations for Meshes – p.11/26

Recognizing Cell Complexes

Easy in two and three dimensions.

Rather complicated in four dimensions (Poincare’sconjecture).

Not known in five dimensions.

It is recursively unsolvable for dimensions six andhigher.

Topological Representations for Meshes – p.12/26

Recognizing Cell Complexes

Easy in two and three dimensions.

Rather complicated in four dimensions (Poincare’sconjecture).

Not known in five dimensions.

It is recursively unsolvable for dimensions six andhigher.

Topological Representations for Meshes – p.12/26

Recognizing Cell Complexes

Easy in two and three dimensions.

Rather complicated in four dimensions (Poincare’sconjecture).

Not known in five dimensions.

It is recursively unsolvable for dimensions six andhigher.

Topological Representations for Meshes – p.12/26

Recognizing Cell Complexes

Easy in two and three dimensions.

Rather complicated in four dimensions (Poincare’sconjecture).

Not known in five dimensions.

It is recursively unsolvable for dimensions six andhigher.

Topological Representations for Meshes – p.12/26

So what do we do?

Allow for building blocks that are more general thancells.

Retain as many nice properties from cell complexes aspossible.

Topological Representations for Meshes – p.13/26

So what do we do?

Allow for building blocks that are more general thancells.

Retain as many nice properties from cell complexes aspossible.

Topological Representations for Meshes – p.13/26

Explicit Models

Faces of different dimensions are represented explicitlytogether with their adjacency relations.

Typically represented as a DAG.

Topological Representations for Meshes – p.14/26

Explicit Models

Faces of different dimensions are represented explicitlytogether with their adjacency relations.

Typically represented as a DAG.

Topological Representations for Meshes – p.14/26

Implicit Models

There is only one basic class of objects.

Faces are represented implicitly in terms of operationson these objects.

Topological Representations for Meshes – p.15/26

Implicit Models

There is only one basic class of objects.

Faces are represented implicitly in terms of operationson these objects.

Topological Representations for Meshes – p.15/26

Implicit Models

There is only one basic class of objects.

Faces are represented implicitly in terms of operationson these objects.

Topological Representations for Meshes – p.15/26

Tutte’s Representation of Surfaces

An abstract set

�

of crosses and three permutations

�

,

�

and

�

of

�

satisfying:

and .

, , and are all distinct for all crosses .

.

The orbits of and under are distinct for allcrosses .

Topological Representations for Meshes – p.16/26

Tutte’s Representation of Surfaces

An abstract set

�

of crosses and three permutations

�

,

�

and

�

of

�

satisfying:� � � � � � �

and

� � � � �

.

, , and are all distinct for all crosses .

.

The orbits of and under are distinct for allcrosses .

Topological Representations for Meshes – p.16/26

Tutte’s Representation of Surfaces

An abstract set

�

of crosses and three permutations

�

,

�

and

�

of

�

satisfying:� � � � � � �

and

� � � � �

.�

,

� �

,

� �

and

� � �

are all distinct for all crosses

�

.

.

The orbits of and under are distinct for allcrosses .

Topological Representations for Meshes – p.16/26

Tutte’s Representation of Surfaces

An abstract set

�

of crosses and three permutations

�

,

�

and

�

of

�

satisfying:� � � � � � �

and

� � � � �

.�

,

� �

,

� �

and

� � �

are all distinct for all crosses

�

.� � � � � � �

.

The orbits of and under are distinct for allcrosses .

Topological Representations for Meshes – p.16/26

Tutte’s Representation of Surfaces

An abstract set

�

of crosses and three permutations

�

,

�

and

�

of

�

satisfying:� � � � � � �

and

� � � � �

.�

,

� �

,

� �

and

� � �

are all distinct for all crosses

�

.� � � � � � �

.

The orbits of

�

and

� �

under�

are distinct for allcrosses

�

.

Topological Representations for Meshes – p.16/26

Example

�� �

� � ! �� " # �

�� �

! � �! �

! �� " $ �

! �� �

! # � �! # �

! $ �! $ � �

Topological Representations for Meshes – p.17/26

Torus

�� �

� �� � �

%� %

� %� � %

Topological Representations for Meshes – p.18/26

-dimensional Generalized Maps

Extend Tutte’s representation to higher dimensionalobjects.

Allow for boundaries and non-manifold objects.

Faces need not be cells.

Topological Representations for Meshes – p.19/26

-dimensional Generalized Maps

Extend Tutte’s representation to higher dimensionalobjects.

Allow for boundaries and non-manifold objects.

Faces need not be cells.

Topological Representations for Meshes – p.19/26

-dimensional Generalized Maps

Extend Tutte’s representation to higher dimensionalobjects.

Allow for boundaries and non-manifold objects.

Faces need not be cells.

Topological Representations for Meshes – p.19/26

-dimensional Generalized Maps

A tuple

& � �' ( )* (+ + + ( ) � � where

'

is an set of dartsand the ), ’s are involutions.

whenever .

Topological Representations for Meshes – p.20/26

-dimensional Generalized Maps

A tuple

& � �' ( )* (+ + + ( ) � � where

'

is an set of dartsand the ), ’s are involutions.

), )- � )- ), whenever

� . / � /0 � . 1 . �.

Topological Representations for Meshes – p.20/26

Maps and Numbered Simplicial Sets

Every map corresponds to a numbered simplicial set.

)2 )*) �

Topological Representations for Meshes – p.21/26

Maps and Numbered Simplicial Sets

Not every numbered simplicial set corresponds to amap.

Topological Representations for Meshes – p.22/26

Removing Undesirable Cases

good configurations

� � � ���

�bad configurations

�

��

�

Topological Representations for Meshes – p.23/26

Additional Constraints

), is fixed point free for

� . / � �

Topological Representations for Meshes – p.24/26

Removing Undesirable Cases

Topological Representations for Meshes – p.25/26

Additional Constraints

), is fixed point free for

� . / � �

For any dart and , if andthen iff and

.

Topological Representations for Meshes – p.26/26

Additional Constraints

), is fixed point free for

� . / � �

For any dart 3 and

� . / � �

, if ) � � ) * (+ + + ( ),54 2 6 and7 � � )8,9 2 (+ + + ( ) � 6 then ) 7 � 3 � � 3 iff ) � 3 � � 3 and7 � 3 � � 3.

Topological Representations for Meshes – p.26/26