ergun akleman, jianer chen and vinod srinivasan- a new paradigm for changing topology during...

8/3/2019 Ergun Akleman, Jianer Chen and Vinod Srinivasan- A New Paradigm for Changing Topology During Subdivision Mod

1/11

A New Paradigm for Changing Topology During Subdivision Modeling

ERGUN AKLEMAN

Visualization Laboratory

College of Architecture

JIANER CHEN

Department of Computer Science

College of Engineering

VINOD SRINIVASAN

Visualization Laboratory

College of Architecture

Abstract

In this paper, we present a new paradigm that allows dy-

namically changing the topology of 2-manifold polygonal

meshes. Our new paradigm always guarantees topological

consistency of polygonal meshes. Based on our paradigm,

by simply adding and deleting edges, handles can be cre-

ated and deleted, holes can be opened or closed, polygonalmeshes can be connected or disconnected.

These edge insertion and edge deletion operations are

highly consistent with subdivision algorithms. In particu-

lar, these operations can be easily included into a subdi-

vision modeling system such that the topological changes

and subdivision operations can be performed alternatively

during model construction.

We demonstrate practical examples of topology changes

based on this new paradigm and show that the new

paradigm is convenient, effective, efficient, and friendly to

subdivision surfaces.

1 Introduction

Parametric representations such as Bezier surfaces, B-

splines and NURBS have long been popular for designing

smooth shapes [16, 4]. The greatest power of parametric

representations is that they allow real time smooth shape

design [4]. Unfortunately, these widely used tensor product

parametric representations do not provide a large variety of

topologies since the control meshes of tensor product para-

metric surfaces must be organized as a regular rectangular

structure [29].

Address: 216 Langford Center, College Station, Texas 77843-3137.

email: [email protected]. Supported in part by the Texas A&M, Schol-

arly & Creative Activities Program.

Address: Department of Computer Science, College Station, TX

77843-3112. email: [email protected]. Supported in part by the National

Science Foundation under Grant CCR-9613805.

Address: 216 Langford Center, College Station, Texas 77843-3137.

email: [email protected]. Supported in part by the Texas A&M, Inter-

disciplinary Activities Program.

Subdivision methods [7, 13, 28, 22, 36, 35, 12] solve the

fundamental problem of tensor product parametric surfaces

without sacrificing the speed of shape computation. Unlike

tensor product surfaces, in subdivision surfaces, the control

meshes do not have to have a regular rectangular structure.

Subdivision algorithms can smooth out 2-manifold (or 2-

manifold with boundary) polygonal meshes [44].

An important property of subdivision surfaces is thatthey are a generalization of parametric surfaces. If the con-

trol mesh is organized as a regular rectangular structure, any

parametric surface can be represented by subdivision algo-

rithms. For instance, the Doo-Sabin surface [13] is a gener-

alization of quadric B-splines and the Catmull-Clark surface

is a generalization of cubic B-splines [7] and Non uniform

rational subdivision surfaces [36] are the generalization of

NURBS.

The main problem with subdivision schemes is that they

do not support topology change. This restriction means that

with subdivision algorithms the designers can only change

the shape of the objects. They cannot add or delete handles,

open and close holes, connect or disconnect two objects.

Since these topology changing operations are essential for

designing unusual and interesting shapes, it is important to

combine these operations with subdivision algorithms.

In this paper, we present a purely polygonal and non-

implicit approach changing the topology of polygonal

meshes. The topological changes we demonstrate include

opening and closing holes, creating and deleting handles,

connecting two disjoint meshes into one, and separating

one mesh into two disconnected ones. These topological

changes are simply done by inserting or deleting edges to a

polygonal mesh. During these operations, we also guaran-

tee that polygonal meshes remain valid 2-manifolds. There-fore, unlike implicitly based topological operations, our op-

erations are subdivision friendly, i.e., subdivision opera-

tions can always smooth out the meshes obtained by our

operations.

Our approach to handling topological changes makes a

natural partner with the subdivision approach within a 3D

modeling system. Our approach is not only suitable for

modeling initial 2-manifold polygonal meshes, but also use-


2/11

ful for changing the topology of a 2-manifold polygonal

mesh that are smoothed by subdivision operations. Since

subdivision operations are essential for the improvement of

the quality of meshes, performing the topological change

operations and the subdivision operations alternatively pro-

vides a powerful shape modeling approach that supports a

hierarchy of topology changes and quality improvements atdifferent levels of details.

Our approach is based on graph rotation systems. A

graph with a rotation system is embedded to a unique 2-

manifold shape [14]. It always guarantees the representa-

tion of valid 2-manifold polygonal meshes. Moreover, it is

easy to change the topology of meshes by simply inserting

or deleting edges in the corresponding graph rotation sys-

tem. An edge insert operation can either (1) combine two

faces by inserting a hole, (2) combine two faces by adding

a handle, (3) combine two faces by joining two separate 2-

manifold meshes, or (4) subdivide a single face into two.

Conversely, an edge delete operation can either (1) separate

one face into two by closing a hole, (2) separate one faceinto two by deleting a handle, (3) separate one face into

two by disconnecting one 2-manifold object into two, or (4)

combine two faces into one. In either operations the first

three cases change the topology of a mesh. The last one

does not change the topology, it only changes the number

of polygons.

In order to represent graph rotation systems and effi-

ciently implement edge insertion and deletion operations,

we have developed the Doubly Linked Face List (DLFL)

data structure. Using DLFL, we demonstrate examples of

topology changes based on this new paradigm and show that

the paradigm is convenient, effective, and efficient for topo-

logical changes for subdivision surfaces.We first introduce the concepts behind subdivision meth-

ods and topological requirements imposed by subdivision

methods. We then introduce the previous works in graph ro-

tation system. Then, we demonstrate why and how edge in-

sert and edge delete operations work by illustrating each of

these cases with figures. The discussions in these sections

are formal and abstract. The readers who are interested in

only implementation issues can skip them and go directly to

the section that covers implementation issues and examples.

2 Topological Consistency and Subdivision

Methods

With the introduction of subdivision surfaces and wider

usage of implicit surfaces, topology has become an impor-

tant element of computer graphics research, development

and production. There has been various studies in topologi-

cal modeling during the last decade [17, 38, 37, 19].

Subdivision surfaces assume that the users provide an

irregular control mesh. These initial control meshes can ei-

ther be created by direct modeling or obtained by scanning

a sculpted real object. A smoother version of this initial

mesh without changing the original topology is obtained by

subdivision operations. All subdivision schemes can be ex-

pressed by a set of linear difference equations. More for-

mally, each new point is computed as a linear combination

of a set of points in a local topological region. The schemecan be written as a linear system

where and are the vectors of respectively the old

points and the new points in the local topological region

and is the transformation matrix [44]. Note here, the lo-

cal topological region should correspond to a simple disk

(topologically). This implies that underlying structure must

be a valid 2-manifold.

Since the quality and topology of the smooth surface re-

sulting from subdivision rules depend greatly on the initial

control mesh, theoretical assurance of the quality of initialcontrol meshes is extremely important. In other words, the

process of obtaining the initial control mesh must be robust

and guarantee valid 2-manifolds. Unfortunately, set oper-

ations, which are the most commonly used operations in

mesh modeling, can result in non-manifold surfaces. More-

over, the existing data structures in mesh modeling are

specifically developed in such a way that they can represent

non-manifold surfaces resulting from the set operations. In

particular, they do not guarantee valid 2-manifold surfaces.

Because of this fundamental problem, in the process of ob-

taining the initial control mesh, unwanted artifacts can be

generated. These artifacts include wrongly-oriented poly-

gons, intersecting or overlapping polygons, missing poly-gons, cracks, and T-junctions. There have been recent re-

search efforts to correct these artifacts [3, 33].

Besides guaranteeing topological consistency, data

structures for mesh modeling should also support topologi-

cal operations efficiently.

The classical view of mesh representation is based on

adjacency relationships between points, edges and faces.

For instance, the vertex-edge adjacency relationship spec-

ifies two adjacent vertices for each edge. There exist nine

such adjacency relationships, but it is sufficient to maintain

only three of the ordered adjacency relationships to obtain

the others [41].

In most practical computer graphics applications,meshes are often represented with one adjacency relation-

ship. The data structure is generally organized as an un-

ordered list of polygons where each polygon is specified by

an ordered sequence of vertices, and each vertex is speci-

fied by its , ! , and " coordinates [3]. 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 degeneraciessuch as cracks, holes and over-


3/11

laps [3, 33].

These degeneracies can be partly eliminated by adding

an additional adjacency relationship: edge lists to vertex-

polygon lists [3]. 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 guarantee valid manifold surfaces ei-

ther. It is still possible to specify a non-manifold surface in

terms of the vertex-polygon-edge list.

One of the oldest formalized data structures that supports

manifold surfaces is the winged-edge representation [5].

Baumgart also suggested using a winged-edge structure and

Euler operators in order to obtain coordinate free operations

[6]. Winged-edge data structures support 2-manifold sur-

faces, and starting from a valid 2-manifold mesh, winged-

edge can only create valid 2-manifolds with Euler opera-

tors. However, like vertex-polygon-edge lists winged-edge

structures can also accept non-manifold surfaces [5, 40].

To represent Voronoi diagrams and Delauney triangulation,

Guibas and Stolfi introduced the quad-edge data structure

and two topological operators, make-edge and splice [21].

Like winged-edge structure, quad-edge structure creates

valid 2-manifolds with make-edge and splice operations if

it starts from a valid 2-manifold. Unfortunately, quad-edge

data structures can still support non-manifold surfaces.

When using set operations, resulting solids can havenon-manifold boundaries [24, 25]. It is worthwhile to note

that although the data structures, such as winged-edge, can

handle some non-manifold surfaces, they actually compli-

cate the algorithms for solid modeling [25, 32]. Therefore,

it is interesting to note that data structures that can sup-

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

investigated. Examples of such work are Weilers radial-

edge structure [42], Karasicks star-edge structure [31], and

Vaneceks edge-baseddata structure [39].

In the current paper, we propose to return back to the ba-

sic concept of coordinate free operations over 2-manifold

surfaces by ignoring set-operations. Similar to our ap-

proach, instead of set operations the usage of Morse opera-tors that describe the changes of cross-sectional contours at

critical sections (peaks, passes and pits) has recently been

investigated [38, 19]. We use topological graph operations

which are similar to Eulers operations and based on graph

embeddings. The biggest advantage of our operations is that

they are extremely simple and always guarantee topological

consistency. Only two operations, Insert edge and Delete

edge, are enough to change the topology. If an inserted or

deleted edge change the topology, we can efficiently find

the new topology by using graph embeddings. This efficient

computation is due to our Doubly Linked Face List (DLFL)

data structure [10]. We propose to use this data structure to

support a representation in which the basic topological op-

erations related to computer graphics, such as surface sub-

division, adding or removing a handle, can all be done veryefficiently.

DLFL not only supports efficient computations on 2-

manifolds, but also always guarantees topological consis-

tency, i.e. it always gives a valid 2-manifold.

DLFL structure is based on the classical theory ofgraph

rotation systems. An extensive research has been done in

the study of graph rotation systems. Among the extensive

research in the area by mathematicians, the most remark-

able result is that graph rotation systems provide necessary

and sufficient conditions in representing valid 2-manifolds.

Our DLFL structure is an efficient implementation of graph

rotation systems.

3 Graph Rotation Systems

In this section, we introduce historical background and

some mathematical fundamentals for graph rotation sys-

tems (see [20] for more detailed discussion).

The concept of rotation systems of a graph originated

from the study of graph embeddings and it is implicitly due

to Heffter [23] who used it in Poincare dual form. A graph

embedding in an orientable surface corresponds to an obvi-

ous rotation system, namely, the one in which the rotation at

each vertex is consistent with the cyclic order of the neigh-

boring vertices in the embedding. Edmonds [14] was thefirst to call attention explicitly to studying rotation systems

of a graph.

Let # be a graph. A rotation at a vertex $ of # is a

cyclic permutation of the edge-ends incident on $ . A rota-

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

# . Given a rotation system of a graph # , to each oriented

edge %& ( $ 1

in # one assigns the oriented edge %$ ( 3 1

such

that vertex 3 is the immediate successor of vertex & in the

rotation at vertex $ . 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 5 ori-

ented edges in an orbit, then an 5 -sided polygon can 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-manifold.

Edmonds [14] has shown that every rotation system of a

graph gives a unique orientable 2-manifold. Moreover, the

corresponding orientable 2-manifold is constructible, as we


4/11

described above. For any 2-manifold 6 , there is a graph #

and a rotation system 7 of # such that 7 corresponds to an

embedding of # on 6 [8].

Therefore, the existence of the bijective correspondence

between graph embeddings on orientable 2-manifolds and

graph rotation systems enables us to represent topologi-

cal objects by combinatorial ones. In particular, every 2-manifold can be represented by a rotation system of a graph,

and every rotation system of a graph corresponds to a valid

2-manifold. In consequence, the presentation of graph ro-

tation systems always guarantees topological consistency.

Recently, Chen, Gross and Riper have developed a very ef-

ficient algorithm that, given a rotation system of a graph,

constructs the corresponding 2-manifold [9].

v1

v2

v3

v4

v1 v3

v2

v4

Figure 1. A graph drawn in a rotation sys-tem and the corresponding embedding on thesphere.

Based on graph rotation systems, a graph that is drawn

in 2D can uniquely represent a 2-manifold shape. For ex-

ample, consider two graphs given in Figures 1 and 2. Thesegraphs are drawn in such a way that the rotation at each

vertex can be traced by traversing the incident edges in

counter-clockwise order. These are two different rotation

systems for the same graph. When we consider the rotation

order they correspond to two different 2-manifold shapes:

the sphere in Figure 1 and the toroid in Figure 2.

v1

v2

v3

v4

v1 v3

v2

v4

Figure2. The same graph as in Figure1 drawnin a different rotation system and the corre-sponding embedding on the toroid.

The graph rotation system also provides a concise and

simple internal data representation as a set of vertices and

a rotation order for each vertex. For example, the rotation

system of the graph in Figure 1 is given by the ordered lists:

$

8

$9 $ B $ D ( $ B

8

$D $

$9 (

$D

8

$

$B $ 9 ( $ 9

8

$

$D $ B (

and the rotation system of the graph in Figure 2 is given by

ordered lists:

$

8

$9 $ D $ B ( $ B

8

$D $

$9 (

$D

8

$9 $ B $

( $ 9

8

$

$D $ B I

Rotation systems can be easily implemented as a set of

linked lists. We should also point out that each edge of a

graph appears exactly twice in any of its rotation systems.

Therefore, the amount of computer memory used for this

representation is small.

Based on a rotation system, each face can be easily con-structed by traversing the face boundary, following the ro-

tation order of the vertices on the boundary, as shown in

Figure 3. By a face-cornerof a face, we refer to a vertex on

the face boundary, plus the two neighboring edges.

v1

v2

v3

v4

v1

v2

v3

v1

v2

v3

v4

v1

v2

v3

Rotation System Corresponding Polygon

Figure 3. Rotation order of faces is providedby rotation system.

4 Changing Topology with Graph Rotation

Systems

We have observed that graph rotation system provides

a great convenience in topology changes [2]. Only edge

insert and edge delete is enough to change the topology of

a 2-manifold mesh supported by graph rotation systems.


5/11


6/11

sualize this face-handle, however the structure of the handle

becomes apparent when the face is smoothed by subdivision

operations as represented in Figure 7.

If this handle goes through the inside of a 2-manifold,

it becomes a hole. If the handle goes outside of the 2-

manifold, it becomes a handle. The hole and the handle

are automatically created or deleted based on geometry andedge insertion/deletion rules we described in previous sec-

tion.

v1

v2

v3

v6

v4

v5

v4

v5

v6

v1

v2

v3

Subdivision

Figure 7. Handle becomes apparent after sub-division operations.

The handle that results from a subdivision operation in-

cludes an extraordinary vertex with a high valance. (The

term valance is used to denote the number of edges incident

on a vertex. Valance of an extraordinary vertex on a handle

is at least equal to X ). In most cases, the resulting subdivi-

sion surface will not be Y

continuous at this extraordinary

vertex because of the unusual structure of our face-handles.

Fortunately, it is possible to improve the quality of the

handle simply by inserting new edges. As shown in Figure 8

if a second edge is inserted to connect any two vertices of

the face-handle, the new edge separates the face-handle into

two faces without changing the topology of the mesh. On

the other hand, the quality of the handle is usually improved.

5 Implementation

As we have stated earlier, graph rotation is a mathemat-

ical concept. Although, it can be easily implemented as a

set of linked lists, it does not support computer graphics

implementations efficiently. One of the main problems us-ing such a graph in an interactive system is that it requires

construction of faces in each rendering step.

5.1 DLFL Structure

In our implementation, we use a Doubly Linked Face

List (DLFL), a data structure we have proposed theoreti-

cally earlier [2]. A DLFL structure always corresponds to

v6

A single facev1v2v3v1v4v5v6v4

v1

v2

v3

v6

v4

v5

Insertingan edge

between v3v5

Two facesv1v2v3v5v6v4

andv3v1v4v5

Insertingone more

edge betweenv6v2

Three facesv1v2v6v4,

v2v3v5v6

andv3v1v4v5

v1

v2

v3

v6

v4

v5

Subdivision

v1

v2

v3

v4

v5

Figure 8. Improving the quality of handles byinserting more than one edge.

a graph rotation system. Therefore, it always guarantees

valid topological consistency [2]. The main reason that we

use DLFL is that it supports logarithmic time edge inser-

tion and deletion operations with face construction. With

the DLFL data structure, subdivision operations can also be

implemented easily.

In DLFL structure, each face is given by a sequence of

vertices corresponding to a boundary traversing of the face.1

The vertex appearances in the sequence will be called ver-tex nodes. Note that two consecutive vertex nodes in the

sequence correspond to an edge side in the rotation system.

The sequence is represented by a cyclically concatenatable

data structure.

Formally, let 7 % # 1 be a rotation system of a graph #

%b ( d 1

with face set f . A doubly-linked-face-list(DLFL)

for the rotation system 7 % # 1 is a triple g

Uh ( q ( t W ,

where the face list h consists of a set ofu f u

sequences,

each is given by a linked list and corresponds to the bound-

ary walk of a face in the rotation system 7% # 1

. Moreover,

these linked lists are connected by a circular doubly linked

list. The vertex array q hasu b u

items. Each qy $

is a linked

list of pointers to the vertex nodes of $ in the face linkedlists in h . The edge array t has

u d uitems. Each q

y $ also

includes the 3D position of the related vertex. Each ty

is

doubly linked to the first vertex nodes of the two edge sides

of the edge in the face linked list in h . Figure 9 gives an

illustration of the DLFL data structure for a tetrahedron.

1To simplify the implementation we use linked lists for the faces. To

further improve efficiency of our system, balanced trees for boundary

walks of faces can be used instead of linked lists [2].


7/11

v1

v2

v3

v4

e1e2

e3

e4

e5e6

v1 v2 v4v3

f2

v1 v4v2

f1

v1 v2v3

f3

v1 v3v4

f4

v2 v3

e1 e2 e4e3 e5 e6

v4

Figure 9. An illustration of the DLFL datastructure for a tetrahedron (Only details forvertex $ and edge

Rare shown).

5.2 Examples

To show the feasibility of our approach, we have created

various 3D models that show the connection of several 2-

manifolds and the creation of handles and holes. In all our

examples, we first convert irregular meshes into meshes that

consist of only quadrilaterals to obtain initial mesh. We,

then, smooth out this initial mesh by using Catmull-Clark

subdivision scheme [7].

T

Figure 10 shows the connection of four disconnected

tetrahedra. All faces of the tetrahedra are first sub-

divided to make each face a quadrilateral. Then, the

inside faces of every two neighboring tetrahedra are

connected by an infinitely small edge. It is interesting

to note that these 3D shapes may look non-manifold.

They are actually 3D representations of a rotation sys-

tem which is a representation of a 2-manifold. There-

fore, we can apply subdivision algorithms. On the

other hand, if they had been represented by set opera-

tions, even the internal representation would have been

non-manifold.

T

Figure 11 shows the connection of two disconnected

toroids. This figure also gives an example of handle

improvement by inserting additional edges. As can be

seen clearly in Figure 11, when only one edge is in-

serted it creates a Y

discontinuous extraordinary ver-

Figure 10. Connecting four tetrahedra.

tex. Inserting extra edges removes this Y

discontinu-

ity.

T

Figure 12 shows an example of the creation of a han-

dle for a cup and demonstrates handle improvement by

inserting additional edges.

T Figure 13 shows an example of the creation of a hole.

As it is clear in this figure, each additional edge im-

proves the quality of the hole in the cube.

More figures are provided in the color pages. The Fig-

ures 14, 15, 16 and 17 shows two examples of topologi-

cal changes. The Figures 14 and 15 shows the creation of

various handles for a small stellated dodecahedron [43] (a

regular polyhedron discovered by Kepler). The Figures 16

and 17 shows the creation of several holes in a great do-

decahedron [43] (A regular polyhedron discovered by De-Poinsot, 100 years after Kepler. It is the dual of small stel-

lated dodecahedron).

We offer a few implementation remarks before we close

this section.

The topological changes are implemented in our ap-

proach by connecting faces in a polygonal mesh. The type

of the topological changes is closely related to the visibility

of the corresponding faces. Generally speaking, if the faces


8/11

Figure 11. Connecting two toroids: the samecolored faces connected with edge insertion.

are visible from each other from the outside of the mesh,

then adding an edge to connect the two faces creates a han-

dle to the mesh. If the faces are visible from each other from

inside of the mesh, then adding an edge to connect the two

faces opens a hole in the mesh. When the two faces are not

visible from each other, either from outside or from inside,

adding a straight edge to connect the two faces still gives a

topologically correct mesh but the mesh will be geometri-cally self-intersected. Thus, the users can easily avoid such

self-intersections based on the face visibility properties.

Using more than one straight edge to connect two faces

creates an extraordinary point with very high valance in the

face handle since each edge adds to the valance. Such an

extraordinary point with a high valance will most likely be

Y

discontinuous. It is easy to avoid extra edges by extrud-

ing the faces before inserting an edge as shown in Figure 12

Figure 12. Creation of handle for a cup.

as an example of the creation of a handle.

6 Conclusion and Future Work

In this paper, we presented a new paradigm for changing

topology of 2-manifold meshes without using an implicit

approach. Our new paradigm guarantees the 2-manifold

property for meshes during these topological changes. The

new paradigm for changing topology is highly consistent

with the subdivision approach in a 3D modeling system.

Our approach can be used for modeling initial 2-manifold

polygonal meshes, as well as for changing the topology

of 2-manifold polygonal meshes that are smoothed by

Catmull-Clark subdivision operations. Note that smooth-ing operations provided by subdivision approach are essen-

tial for improving the quality of 2-manifolds. Thus, the

topological change and subdivision operations performed

alternatively during model construction provide a powerful

shape modeling paradigm that supports a hierarchy of topol-

ogy changes and quality improvement in different levels of

details.

Our approach can also be extended to include topologi-


9/11

Figure 13. Creation of a hole in a cube.

cal changes into progressive meshes [26] and multiresolu-

tion representations of meshes [15, 11]. Two major opera-

tions in progressive meshes technique are vertex-split and

edge-collapse. It can be shown that these operations do

not change the topology [20]. The same can also be eas-

ily shown for multiresolution representations of meshes. In

fact, one of the major problem in multiresolution mesh mor-

phing is that source and target must share the same topol-

ogy [27]. By including the edge insertion and edge deletion

operations, it is possible to change the topology for pro-

gressive meshes and multiresolution representations. Thus,

this approach may potentially add new power to the existingmorphing, compression and simplification schemes.

References

[1] A. V. Aho, J. E. Hopcroft, and J. D. Ullman, The De-

sign and Analysis of Computer Algorithms, (Addison-

Wesley, Reading, MA, 1974).

[2] E. Akleman and J. Chen, Guaranteeing the 2-

Manifold Property for meshes with Doubly Linked

Face List, International Journal of Shape Modeling

Volume 5, No 2, pp. 149-177.

[3] G. Barequet and S. Kumar, Repairing CAD mod-

els, in Proceedings of IEEE Visualization97, (Octo-

ber 1997) pp. 363-370.

[4] R. H. Bartels, J. C. Beatty, and B. A. Barsky, An Intro-

duction to Splines for Use in Computer Graphics and

Geometric Modeling, (Morgan Kaufmann Publishers,

Los Altos, CA, 1987).

[5] B. J. Baumgart, Winged-edge polyhedron representa-

tion, Technical Report CS-320, Stanford University,

1972.

[6] B. J. Baumgart, A polyhedron representation for

computer vision, in 44th AFIPS National Computer

Conference, (1975) pp. 589-596.

[7] E. Catmull and J. Clark, Recursively Generated B-

spline Surfaces on Arbitrary Topological Meshes,

Computer Aided Design, 10 (September 1978) 350-

355.

[8] J. Chen, D. Archdeacon, and J. Gross, Maximum

Genus and Connectivity, Discrete Mathematics, 149

(1996) 19-29.

[9] J. Chen, J. Gross and R. Riper, Overlap Matrices and

Imbedding Distributions, Discrete Mathematics, 128

(1994) 73-94.

[10] J. Chen, Algorithmic Graph Embeddings, Theoreti-cal Computer Science, 181 (1997) 247-266.

[11] A. Certain, J. Popovic, T. DeRose., T. Duchamp, D.

Salesin, and W. Stuetzle, Interactive Multiresolution

Surface Viewing, Computer Graphics, 30 (August

1996) 91-98.

[12] T. DeRose. M. Kass and T. Truong, Subdivision Sur-

faces in Character Animation, Computer Graphics,

32 (August 1998) 85-94.

[13] D. Doo and M. Sabin, Behavior of Recursive Subdi-

vision Surfaces Near Extraordinary Points, Computer

Aided Design, 10 (September 1978) 356-360.

[14] J. Edmonds, A Combinatorial Representation for

Polyhedral Surfaces, Notices American Mathematics

Society, 7 (1960) 646.

[15] M. Eck, T. DeRose. T. Duchamp, H. Hoppe, M.

Lounsbery and W. Stuetzle, Multiresolution Analysis

of Arbitrary Meshes, Computer Graphics, 29 (Au-

gust 1995) 173-182.


10/11

[16] G. Farin, Curves and Surfaces for Computer Aided ge-

ometric Design, A Practical Guide, (Academic Press,

Inc, London, 1988).

[17] H. Ferguson, A. Rockwood and J. Cox, Topological

Design of Sculptured Surfaces, Computer Graphics,

26 (August 1992) 149-156.

[18] P. A. Firby and C. F. Gardiner, Surface Topology,

(John Wiley & Sons, 1982).

[19] A. T. Fomenko and T. L. Kunii, Topological Modeling

for Visualization, (Springer-Verlag, New York, 1997).

[20] J. L. Gross and T. W. Tucker, Topological Graph The-

ory, (Wiley Interscience, New York, 1987).

[21] L. Guibas, J. Stolfi, Primitives for the manipulation

of general subdivisions and computation of Voronoi

diagrams, ACM Transaction on Graphics, 4 (1985)

74-123.

[22] M. Halstead, M. Kass, and T. DeRose, Efficient, Fair

Interpolation using Catmull-Clark Surfaces, Com-

puter Graphics, 27 (August 1993) 35-44.

[23] L. Heffter, Uber das Problem der Nachbargebiete,

Math. Annalen, 38 (1891) 477-508.

[24] C. M. Hoffmann, Geometric & Solid Modeling, An

Introduction, (Morgan Kaufman Publishers, Inc., San

Mateo, Ca., 1989).

[25] C. M. Hoffmann and G. Vanecek, Fundamental tech-niques for geometric and solid modeling, Manufac-

turing and Automation Systems: Techniques and Tech-

nologies, 48 (1990) 347-356.

[26] H. Hoppes, Progressive Meshes, Computer Graph-

ics, 30 (August 1996) 99-108.

[27] A. W. F. Lee, D. Dobkin, W. Sweldens and P.

Schroder, Multiresolution Mesh Morphing, Com-


[28] C. Loop, Smooth Subdivision Surfaces Based on Tri-

angles, Masters Thesis, Department of Mathematics,

University of Utah (1987).

[29] C. Loop and T. DeRose, Generalized B-spline Sur-

faces with Arbitrary Topology, Computer Graphics,

24 (August 1991) 101-165.

[30] C. Loop, Smooth Spline Surfaces over Irregular

Meshes, Computer Graphics, 28 (August 1994) 303-

310.

[31] M. Karasick, On the representation and manipula-

tion of rigid solids, Ph.D. Thesis, McGill University,

1988.

[32] M. Mantyla, Boolean operations on 2-manifolds

through vertex neighborhood classification, ACM

Transaction on Graphics, 5 (January 1986) 1-29.

[33] T. M. Murali and T. A. Funkhouser, Consistent solid

and boundary representations from arbitrary polygo-

nal data, in Proceedings of 1997 Symposium on In-

teractive 3D Graphics , (1997) pp. 155-162.

[34] F. P. Preparata and M. I. Shamos, Computational Ge-

ometry: An Introduction, (Springer-Verlag, 1985).

[35] J. Stam, Exact Evaluation of Catmull-Clark Subdi-

vision Surfaces at Arbitrary Parameter Values, Com-


[36] T. W. Sederberg, D. Sewell, and M. Sabin, Non-

Uniform recursive Subdivision Surfaces, Computer

Graphics, 32 (August 1998) 387-394.

[37] B. T. Stander and J. C. Hart, Guaranteeing the Topol-

ogy of an Implicit Surface Polygonization for Inter-

active Modeling, Computer Graphics, 31 (August

1997) 279-286.

[38] S. Takahashi, Y. Shinagawa and T. L. Kunii, A

Feature-Based Approach for Smooth Surfaces, in

Proceedings of Fourth Symposium on Solid Modeling,

(1997) pp. 97-110.

[39] . G. Vanecek, Set operations on polyhedra using de-

composition methods, Ph.D. Thesis, University of

Maryland, 1989.

[40] K. Weiler Polygon comparison using a graph repre-

sentation, Computer Graphics, 13 (August 1980) 10-

18.

[41] K. Weiler Edge-based data structures for solid mod-

eling in curved-surface environments , IEEE Com-

puter Graphics and Applications, (January 1985) 21-

40.

[42] K. Weiler, Topological structures for geometric mod-

eling, Ph.D. Thesis, Rensselaer Polytechnic Institute,N. Y., August 1986.

[43] R. Williams, The Geometrical Foundation of Natural

Structures, (Dover Publications, Inc., 1972).

[44] D. Zorin and P. Schroder, co-editors, Subdivision

for Modeling and Animation, ACM SIGGRAPH99

Course Notes no. 37, August, 1999.


11/11

Figure 14. Creation of various handles forsmall stellated dodecahedron.

Figure 15. Handles improvement for for small

stellated dodecahedron.

Figure 16. Creation of various holes for greatdodecahedron.

Figure 17. Hole improvement for great dodec-

ahedron.

ergun akleman, jianer chen and vinod srinivasan- a new paradigm for changing topology during...

Documents