Ergun Akleman and Jianer Chen- Guaranteeing 2-Manifold Property for Meshes

8/3/2019 Ergun Akleman and Jianer Chen- Guaranteeing 2-Manifold Property for Meshes

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Guaranteeing 2-Manifold Property for Meshes

ERGUN AKLEMANVisualization Sciences

College of Architecture

Texas A&M University

College Station, Texas 77843-3137

email: [email protected]

phone: +(409) 845-6599

fax: +(409) 845-4491

JIANER CHENDepartment of Computer Science

College of Engineering

Texas A&M University

College Station, TX 77843-3112

email: [email protected]

phone: +(409) 845-4259

fax: +(409) 847-8578

Abstract

Meshes are the most commonly used objects in computer

graphics. They generalize polyhedra by using non-planar

faces. Modeling 2-dimensional manifold meshes with a

simple user interface is an important problem in computer

aided geometric design. In this work we propose a concep-

tual framework for mesh modeling systems that guarantees

topologically correct 2-dimensional manifolds.

Our solution is based on graph rotation systems devel-

oped in topological graph theory. As an internal represen-

tation of meshes, we use Doubly Linked Face List (DLFL).

We have also developed a visual representation of the topol-

ogy that provides a powerful tool for developing a user in-

terface to manipulate the topology of the mesh.

1 Significance

Most current shape modeling systems are based on

tensor product parametric surfaces such as Tensor Prod-

uct NURBS modeler. However tensor product surfaces

can only support quadrilateral meshes as control meshes

for smooth surfaces, and do not provide general topology

[23, 22]. The restriction to quadrilateral meshes makes the

modeling process extremely difficult for the user. There has

been a significant effort to solve this problem by still stay-ing in parameteric surface realm such as [23, 22]. Instead

of parametric surfaces, subdivision surfaces have currently

be resurfaced as an alternative for providing general topolo-

gies. Subdivision surfaces were first proposed by Doo and

Sabin [4] and by Catmull and Clark [5] as a generalization

of B-spline surfaces in 1978. Their potential as an alter-

native has recently been realized and huge amount of work

about subdivision surfaces has been published[21, 15].

Subdivision surfaces assume that the users provide an ir-

regular control mesh. These initial control meshes can becreated by either direct modeling or obtained by scanning

a sculpted real object. A smoother version of this initial

mesh without changing orginal topology is obtained by sub-

division operations. The effective usage of subdivision sur-

faces requires guaranteeing 2-manifold property for initial

meshes. Since the quality and topology of the smooth sur-

face resulting from subdivision rules greatly depend on the

initial control mesh, theoretical assurance of the quality of

initial meshes is extremely important. In other words, the

process of obtaining initial mesh must be robust and guar-

antee 2-manifold meshes. Unfortunately, set operations,

which are most commonly used operations in mesh mod-

eling, can result in non-manifold surfaces. Moreover, theexisting powerful data structures are specifically developed

to be able to represent the non-manifold surfaces resulting

from set operations and they clearly do not guarantee 2-

manifold surfaces. Because of this fundemental problem,

in the process of obtaining the initial mesh, unwanted ar-

tifacts can be generated. These artifacts include wrongly-

oriented polygons, intersecting or overlapping polygons,

missing polygons, cracks, and T-junctions. There have been

recent research efforts to correct these artifacts [25, 1].

In this paper, we develop a theoretical approach for

artifact-free modeling. Our approach guarantees the

topological consistency of initial 2-dimensional manifoldmeshes. Our approach is based on a simple data structure,

the Doubly Linked Face List (DLFL). This data structure

is based on graph embeddings and guarantees topological

consistency of initial 2-dimensional manifold meshes. In

order to avoid non-manifold surfaces, we avoid set oper-

ations. Instead, we propose only to use simple topolog-

ical operations based on graphs. DLFL allows us com-

pute related topological operations efficiently. Furthermore,



DLFL and related topological operators provide us with a

paradigm for a new user-interface for changing topological

properties of the meshes.

2 Introduction

Modeling meshes requires development of data struc-

tures that support topological operations efficiently and

guarantee topological consistency of meshes. The classi-

cal view of mesh representation is based on adjacency re-

lationships between points, edges and faces. For instance,

the vertex-edge adjacency relationship specifies two adja-

cent vertices for each edge. There exist nine such adjacency

relationships, but, it is enough to maintain only three of the

ordered adjacency relationships to obtain the others [31].

In most current practical Computer Graphics applica-

tions, meshes are often represented with one adjacency re-

lationship. The data structure is generally organized as an

unordered list of polygons where each polygon is specified

by an ordered sequence of vertices, and each vertex is spec-

ified by its x, y, and z coordinates [1]. Let us call this data

structure a vertex-polygon list . Vertex-polygon lists do not

always guarantee topological consistency. In addition, they

can even create degeneracies such as cracks, holes and over-

laps.

These degeneracies can partly be eliminated by adding

an additional adjacency relationship: edge lists to vertex-

polygon lists [1]. In a vertex-polygon-edge list structure, a

list of vertices, a list of directed edges, and a list of poly-

gons are described. Vertices are specified by their three

coordinates, directed edges are specified by two vertices,and polygons are specified by an ordered sequence of edges.

Each polygon is oriented in a consistent direction, typically

counter-clockwise when viewed from outside of the model.

Because of the last condition, vertex-polygon-edge lists are

more powerful than vertex-polygon lists. However, the rep-

resentation does not gurantee valid manifold surfaces either,

i.e., it is possible to specify a non-manifold surface by giv-

ing wrong specifications.

One of the oldest formalized data structure that supports

manifold surfaces is the winged-edge representation [3].

Winged-edge data structures support 2-dimensional mani-

fold surfaces, but like vertex-polygon-edge lists they can

also accept non-manifold surfaces [3, 30].Baumgart suggested using a winged-edge structure and

Euler operators in order to obtain coordinate free operations

[2]. Guibas and Stolfi introduced the quad-edge data struc-

ture and topological operators such as splice operators [13].

Simplicial complexes are also used to specify the connectiv-

ity of the vertices, edges and faces.

When using set operations, resulting solids can have

non-manifold boundaries [28, 27, 17, 18]. Therefore, it is

interesting to note that although the data structures, such

as winged-edge, can handle some non-manifold surfaces,

they still can complicate the algorithms for solid mod-

eling [24, 18]. Therefore, data structures that can sup-

port a wider range of non-manifold surfaces have been in-

vestigated. Examples of such works are Weiler’s radial-

edge structure [32], Karasick’s star-edge structure [19] andVanecek’s edge-based data structure [29].

In our research, we propose to return to the basic concept

of coordinate free operations over 2-dimensional manifold

surfaces and ignore set-operations. Similar to our approach,

the usage of Morse operators instead of set operations has

recently been investigated [20, 12]. We use topological

graph operations which are similar to Euler’s operations and

based on graph embeddings. The biggest advantage of our

operations is that they are extremely simple. Only two op-

erations, INSERT(edge) and DELETE(edge), are enough to

change the topology. If an inserted or deleted edge adds or

removes a handle, we can efficiently find the new topology

by using graph embeddings. This efficient computation isdue to our Doubly Linked Face List (DLFL) data structure

[6]. We propose to use this data structure to support a repre-

sentation in which the basic topological operations related

to computer graphics, such as surface subdivision, adding

or removing a handle, can all be done very efficiently.

DLFL allows not just efficient computation of topologi-

cal changes. Unlike the winged-edge data structure, DLFL

also always guarantees topological consistency, i.e. it al-

ways gives a valid 2-dimensional manifold. The DLFL rep-

resentation is based on the concept of graph rotation sys-

tems in classical topological graph theory [14]. In addition,

DLFL uses the minimum amount of computer memory [6].

In order to provide an intuitive interface to users for

manipulating the topology of 2-dimensional manifolds, we

add parameter coordinates into vertex specifications. These

parameter coordinates come from the polygonal represen-

tations of orientable 2-dimensional manifolds which are

4n-sided polygons that are obtained from transforming the

torus with n handles and 2n cuts [11] .

The combination of the graph rotation system represen-

tation and the polygonal representation of meshes can pro-

vide a new approach for defining modeling meshes.

The topology of meshes can be updated automatically

during modeling process simply by adding or remov-

ing new vertices and edges to the graph.

A new topology can be obtained by merging two

polygonal representations of two 2-dimensional mani-

folds.

The polygonal representation can be used for texture

coordinates. By changing the parametric coordinates

of the vertices, users can change the texture mapping.



Moreover, the users can directly paint the parameter

space in order to create a texture.

Our new proposed data structure will be extremely use-

ful in the computation of the subdivision surface when

the mesh is used as a subdivision surface control mesh.

3 New Representations and Their Data

Structures

We suggest a new conceptual framework to develop

mesh modeling system based on Doubly Linked Face List

data structure (DLFL), and a new user-interface paradigm.

DLFL is based on a graph rotational system which guaran-

tees the topological consistency of the meshes.

3.1 Graph Rotation System

The graph rotation system is a powerful tool for guaran-

teeing topological consistency. In this subsection, we intro-

duce historical background and some mathematical funda-

mentals of it.

The concept of rotation systems of a graph originated

from the study of graph embeddings and it is implicitly due

to Heffter [16] who used it in Poincare dual form. Start-

ing with a graph embedding in an orientable surface, there

corresponds an obvious rotation system, namely, the one

in which the rotation at each vertex is consistent with the

cyclic order of the neighboring vertices in the embedding.Edmonds [10] was the first to call attention explicitly to

studying rotation systems of a graph.

LetG

be a graph. A rotation at a vertexv

of G

is a

cyclic permutation of the edge-ends incident onv

. A rota-

tion system of G is a list of rotations, one for each vertex of

G

. Given a rotation system of a graphG

, to each oriented

edge u ; v in G , one assigns the oriented edge v ; w such

that vertexw

is the immediate successor of vertexu

in the

rotation at vertex v . The result is a permutation on the set of

oriented edges, that is, on the set in which each undirected

edge appears twice, once with each possible direction. In

each edge-orbit under this permutation, the consecutive ori-

ented edges line up head to tail, from which it follows thatthey form a directed cycle in the graph. If there are

r

ori-

ented edges in an orbit, then an r -sided polygoncan be fitted

into it. Fitting a polygon to every such edge-orbit results in

polygons on both sides of each edge, and collectively the

polygons form a 2-dimensional manifold.

For example, consider the graph G given in Figure 1,

whereG

is drawn in such a way that the rotation at each

vertex can be traced by traversing the incident edges in the

clockwise order. More specifically, the rotation system of

G is:

a : a

0

c d b b : b

0

a e c c : c

0

b f a

d : d

0

e a f e : e

0

f b d f : f

0

d c e

a

0

: a b

0

d

0

c

0

b

0

: b c

0

e

0

a

0

c

0

: c a

0

f

0

b

0

d

0

: d f

0

a

0

e

0

e

0

: e d

0

b

0

f

0

f

0

: f e

0

c

0

d

0

This rotation system has twelve orbits, which are given as

follows:

O

1

= a d f c O

2

= d a b e O

3

= e b c f

O

4

= a

0

d

0

e

0

b

0

O

5

= d

0

a

0

c

0

f

0

O

6

= f

0

c

0

b

0

e

0

O

7

= d d

0

f

0

f O

8

= d

0

d e e

0

O

9

= f f

0

e

0

e

O

1 0

= a a

0

b

0

b O

1 1

= a

0

a c c

0

O

1 2

= b b

0

c

0

c

If we associate each orbit with a 4-sided polygon, we ob-

tain an embedding of the graphG

on the torus (i.e., the 2-

dimensional manifold of genus 1), as shown in Figure 2.

! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !

! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! ! ! ! !

! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! !d’

a’

a

d

e’

b’

b

e

f

c

c’

f’

Figure 1. The graphG

.

e’

b’

b

f

c

c’

f’

e

d’

d

a

a’

Figure 2. The embedding of the graphG

on

the torus.

Edmonds [10] has shown that every rotation system

of a graph gives a unique orientable 2-dimensional mani-

fold. Moreover, the corresponding orientable 2-dimensional

manifold is constructible, as we described above. For any

2-dimensional manifold S , there is a graph G and a rotation



system

of G

such that

corresponds to an embedding of

G on S [7].

Therefore, the existence of the bijective correspondence

between graph embeddings on orientable 2-dimensional

manifolds and graph rotation systems enables us to rep-

resent topological objects by combinatorial ones. In par-

ticular, every 2-dimensional manifold can be representedby a rotation system of a graph, and every rotation sys-

tem of a graph corresponds to a valid 2-dimensional man-

ifold. In consequence, the presentation of graph rotation

systems always guarantees topological consistency. In re-

cent research, we have developed a very efficient algorithm

that, given a rotation system of a graph, constructs the cor-

responding 2-dimensional manifold [9].

We should also point out that each edge of a graph ap-

pears exactly twice in any of its rotation systems. There-

fore, the amount of memory used by graph rotation systems

is very small.

3.2 Topological Operations and Internal Representa-

tion of Meshes: The Doubly Linked Face List

Graph rotation systems are an effective tool for repre-

senting 2-dimensional manifolds. We have identified the

following topological operations on rotation systems as es-

sential for 2-dimensional manifold mesh modeling:

FACE-TRACE f

outputs a boundary walk of the face

f

. This is related to reconstructing a polygon in a 2-

dimensional manifold and triangulation of a polygon

in a 2-dimensional manifold.

VERTEX-TRACE v outputs the edges incident onthe vertex

v

in the (circular) ordering of the rotation

at v . This is useful when a 2-dimensional manifold

needs reshaping.

INSERT c

1

; c

2

; e

inserts the new edgee

between the

face cornersc

1

andc

2

. This is used in triangulation

of a polygon in a 2-dimensional manifold, changing

the topological character of a 2-dimensional manifold

(i.e., adding a handle to a manifold), and gluing two

seperated manifolds. Figure 3 shows two examples of

edge insertion. The first insertion does not change the

topology, but the second insertion changes the topol-

ogy.

DELETE e

deletes the edgee

from the current em-

bedding. This is the converse operation of edge inser-

tion and is also used in changing the topological char-

acter of a manifold.

We have initiated the study of efficient algorithms on

graph rotation systems in order to implement these opera-

tions efficiently[6]. We have first observed that if a rotation

No Needfor a Changein Topology

Changein

Topology

Figure 3. Topological effect of edge insertion

in a rotation system.

system of a graph is represented in the edge-list form, which

for each vertexv

contains the list of its incident edges, ar-

ranged in the order according to the rotation atv

, the rep-

resentation does not efficiently support the operations listed

above.

Another data structure, which is closely related to

the winged-edge structure, the doubly-connected-edge-list

(DCEL) has been widely used in computational geometry

[26]. DCEL can also be directly used for representing a ro-

tation system. However, we observed that the DCEL struc-

ture does not support the operations INSERT and DELETE

efficiently [6].To efficiently support all the operations listed above, we

have introduced a new data structure called the doubly-

linked-face-list (DLFL) [6]. Essentially, in DLFL structure,

each face of an embedding is represented by a balanced

search tree so that searching, insertion, and deletion of an

item from the tree can be done very efficiently. We have

shown that the DLFL structure and the DCEL structure can

be converted from one to the other very efficiently (in linear



time). In consequence, the DLFL structure for graph rota-

tion systems can be built in linear time and requires linear

storage.

For the operations listed above, we are able to show that

the DLFL structure supports all these operations efficiently

[6]. In particular, based on the DLFL structure, the oper-

ation FACE-TRACE f

can be done in time linear in thesize of the face f , and the operation VERTEX-TRACE v

can be done in time linear in the degree of the vertexv

,

and the DLFL structure supports the operations INSERT

and DELETE in logarithmic time. In fact, DLFL seems

to be the first data structure that supports all of above op-

erations on topological surfaces in optimal time. Due to

the space limit, we omit the formal definition of DLFL and

pseudocodes for the operations FACE-TRACE, VERTEX-

TRACE, INSERT, and DELETE. Interested readers can find

them in the published literature [6].

3.3 User Interface for Topological Operations

INSERT and DELETE operations in internal represen-

tation change the topology of the mesh. INSERT operation

requires a special attention. An inserted edge must satisfy

some constraints to get a unique topological description. To

explain why we need some constraints, we need to look

at rotation systems more closely. Rotation system is de-

fined by the order of connections for every edge. In other

words, from the 3-dimensional positions and orientations

of the edges, it should be possible to get a unique order of

connection to update the rotational system. If the user draw

any edge freely in 3-dimensions, it is not possible to give a

unique order. Therefore, in such a situation, ambiguities canoccur and edge insertion may not correspond to one unique

topology as shown in Figure 4. In other words, we need to

ensure that the user defines a unique order for the connec-

tions between vertices.

The example we have shown in Figure 3 illustrates a

constraint that can be used: the new inserted edge must be

drawn on the surface of 2-manifold in which the mesh be-

longs. In order to enforce such a constraint we must force

the user to draw each new edge over the 2-manifold. As it

can easily be seen, this constraint is too restrictive for inter-

active modeling. Fortunately, this constraint is sufficient but

not necessary to get a unique topological description. There

exists another simpler constraint that is both sufficient andnecessary.

To illustrate this simple constraint, let us look at the

property of rotation systems. Since, for any 2-manifold, a

small neighborhood around every point is homeomorphic

to an open disk, it is possible to define a disk around ev-

ery vertex of the mesh as shown in Figure 5. If every edge

originated from a given vertex is constrained to the disk of

this vertex, it is possible to give an order. For example as

shown in Figure 5 in a planar disk that includes a given ver-

tex, it is possible to determine the order of connections. In

other words, if the inserted edges should be constrained to

the disks that are in 2-manifold surface and include the end

vertices of the edges, it is possible to determine a unique ro-

tation system and a unique topology. This is not extremely

restrictive constraint. Each vertex can be defined by a pointand a tangent plane which are shown as a space circle. Each

edge leaving a vertex can be constrained to the plane. The

user can change the order of the connection and orienta-

tion of the tangent plane. Note that these edges need to be

represented at least a third degree parametric curve such as

Bezier or Hermitian curve.

The minimum constraint explained above suggests that

the topological operations can be effectively done even in 2-

dimensions. To provide an effective interface for handling

2-dimensional manifolds, we also propose a new visual rep-

resentation that will be used as an interface in the modeling

topology of 2-dimensional manifolds. This 2-dimensional

visual representation is useful since it can clearly separatetopological and geometrical operations.

Topological Ambiguity

?

Figure 4. An Edge insertion without any con-straint may not define one unique topology.

3.4 Visual Representation of Topology and 2D User-

Interface for Topological Operations

It is well-known in topology that the orientable 2-

dimensional manifold of genusn

can be represented by a

4 n sided polygon, called its polygonal representation [11].



Figure 5. A simple constraint around verticesis enough to define unique topology.

The 4n sided polygon (4n-gon) that corresponds to the 2-

dimensional manifold with genus n can be obtained by 2n

cuts on the manifold. For each handle, we need to make 2

cuts. For example, the torus in Figure 2 can be represented

in its polygonal representation in the rectangle shown in

Figure 6, with the opposite sides of the rectanglebeing iden-

tified. Using similar techniques as we developed for DLFL,

we can also apply topological operations efficiently to the

polygonal representations of 2-dimensional manifolds. Fig-

ure 7 shows the process of obtaining a polygonal represen-

tation going from a 4n-gon with one handle to a 4(n+1)-gon.

The polygonal representation of a manifold can be ob-

tained efficiently using the method of graph embeddings.

In order to obtain the polygonal representation of an ori-

entable 2-dimensional manifold of genus k , we first embed

a bouquet of 2 k

edges into the manifold so that the embed-

ding has only a single face [8], then cut the manifold along

the edges of the bouquet to obtain a4 k

polygonal represen-

tation of the manifold.The polygonal representation of a 2-dimensional mani-

fold has the advantage that the figures drawn on a local area

in the manifold look exactly the same as on its polygonal

representation. Therefore, local manipulations on a mani-

fold can be performed conveniently and intuitively on the

corresponding polygonal representation. For local manipu-

lations interesting figures need to be moved to the center of

the polygon for easier manipulations. Moving in a polyg-

A A

B

B

c’ a’ b’

f’ d’ e’

f d e

c a b

e’

b’

b

f

c

c’

e

d’

d

a

f’ A

B

Figure 6. Polygonal representation of previ-

ous graph G .

New 4(n+1)-gon

A change in topology

A 4n-gon

Two cuts

Figure 7. Obtaining 4(n+1) sided polygon.



onal representation can be performed effectively by using

the fact that all the corners of the polygonal representation

correspond to the same point in the 3-dimensional space.

In other words, in order to move in a polygonal representa-

tion, it is possible to cut the 2-dimensional manifold sur-

face differently. Since it is possible to move the cutting

edges slowly, the user can perceive the change as a movein a polygonal representation as shown in Figure 8.

Figure 8. Moving a figure to the center bychanging cutting edges.

An additional advantage of the polygonal representation

is that the positions of the vertices in the4 n

sided polygon

can be used as the texture coordinates for the mesh surfaceand the users can control texture mapping by changing the

vertex positions in the polygonal representation.

The polygonal representation provides us a paradigm for

an interface for topology control. The interface can be simi-

lar to 2-dimensional graph drawing or drawing systems. By

using such an interface, the user can make manipulations

over the topology and change texture coordinates. The user

is also able to define and merge polygonal representations

of several meshes.

4 Future Work

Our theoretical framework does not take into account the

3-dimensional positions of the vertices. Some choices of

these positions may result in self-intersection of the meshes.

Such a self-intersection is not desirable unless the user par-

ticularly wants to achieve it.

The problem with self-intersections can easily be ex-

plained with 1-dimensional manifolds. As we know a poly-

gon in 2-dimensions is a 1-dimensional manifold. There-

fore, we expect that a polygon separates the 2-dimensional

space into three regions: inside the polygon and outside

the polygon and the polygon itself. However, by mov-

ing a vertex in 2-dimensions, the user can create a self-

intersecting polygon which has neither inside nor outside.

In 2-dimensional drawing systems, this is not considered

a problem since the user can easily correct it. If theuser does not correct a self-intersection, the result can

still be considered to be an acceptable shape and in some

systems, such self-intersecting polygons are filled by us-

ing geometry-based inside-outside tests. This approach

to self-intersection used in 2-dimensional drawing systems

is not useful for modeling meshes. Since some of the

faces of the mesh will be hidden, the user can create self-

intersection without even realizing it. Moreover, unlike the

2-dimensional case, it is not easy for the user to detect such

self-intersection. Therefore, if the user does not want to

create self-intersection, the system must prevent it from oc-

curing by restricting the positions of the vertices. We are

currently working on several possible solutions to this prob-lem.

5 Discussion and Conclusion

We have proposed a conceptual framework for the de-

velopment of systems for modeling topologically consistent

meshes. We combine the graph rotation system represen-

tation and the polygonal representation into one integrated

system, in which the graph rotation system is used as an

internal representation for manifolds, while the polygonal

representation is used in the user interface. We also provide

topological operations for edge insertion and deletion, andvertex and polygon tracing.

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Ergun Akleman and Jianer Chen- Guaranteeing 2-Manifold Property for Meshes

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