graph interaction through dynamics-based skeletal animation · graph interaction through...

Graph Interaction through

Dynamics-Based Skeletal Animation

COLIN MURRAY

A dissertation submitted in partial fulfillment of

the requirements for the degree of

Bachelor of Computer Science and Technology (Advanced) (Honours)

Supervisors:

Dr. Masahiro Takatsuka

Damian Merrick

School of Information Technologies

The University of Sydney

Australia

2 November 2003

Abstract

Skeletal animation is a concept that has been used in the areas of motion pictures and com-

puter games to create realistic motion for the animation of articulated characters. Recent work

has applied skeletal animation techniques from inverse kinematics to the field of graph interac-

tion. The previous work introduced an interesting idea suggesting a skeletal graph interaction

system would be intuitive and natural. However, due to problems with the skeletal animation

techniques used, the benefits of the system were limited.

This thesis investigates the use of a new dynamics-based technique previously used in the

area of skeletal animation for graph interaction. The motivation for this work was to improve

upon the previous work by providing a skeletal graph interaction system that reduces the diffi-

culty and time taken in navigating the graph and increases the amount of information that can

be interpreted and understood. We have adapted this technique and implemented it in a graph

interaction environment allowing the user to drag nodes of the skeleton in order to manipulate

the layout of the graph. The graph reacts to the user movement while maintaining the physical

constraints placed upon it by the skeletal metaphor. We have also conducted a user study, the

results of which confirm the usefulness of both this new technique and the skeletal metaphor in

aiding the user’s understanding of a graph.

ii

Acknowledgements

I would like to express my gratitude to:

• Damian Merrick, for providing a huge amount of support, advice and encouragement

at every stage throughout the year.

• Masahiro Takatsuka, for his ever-present ideas and advice.

• The rest of the Information Visualisation Research Group, for their suggestions and dis-

cussions about the project.

• All those who participated in the user study.

• Mum and Dad, for always supporting me.

• Brian, Cassie & Tanya for providing many distractions and keeping me sane.

• All the guys in G28, for making Computer Science Honours such an enjoyable and un-

forgettable experience.

iii

Conventions

Several conventions are utilised throughout this thesis.

The thesis is divided into chapters, numbered in sequential order starting from Chapter 1.

Each chapter is divided into sections. (e.g. Chapter 3 contains Section 3.2). The Contents part

of this thesis gives the page numbers and titles of each chapter and section.

Figures are numbered sequentially within each chapter, the second figure of Chapter 3 be-

ing Figure 3.2. A list of figures is given on the pages immediately following the Contents part

of the thesis.

Pseudocode is provided throughout the thesis.All pseudocode is presented

in this typeface.

iv

CONTENTS

Abstract ii

Acknowledgements iii

Conventions iv

List of Figures vii

Chapter 1. Introduction 1

1.1. Aim of the Project . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2. Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.3. Summary of Contribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.4. Structure of this Thesis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

Chapter 2. Literature Review 8

2.1. Graph Drawing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.2. Layout Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.3. Graph Interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.4. Physical Metaphors for Graph Interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.5. Skeletal Animation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.6. Skeletal Animation in Graph Interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

Chapter 3. A System for Force-Based Skeletal Graph Interaction 29

3.1. A Particle System for Skeletal Animation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.2. Implementation of the Particle System in a Graph Interaction Environment . . . . . . 36

3.3. Implementation of a Force-Directed Algorithm in a Graph Interaction

Environment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

3.4. Computational Complexity of the methods used . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

v

CONTENTS vi

Chapter 4. Evaluation of Skeletal Animation Techniques in Graph Interaction 48

4.1. User Study. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

4.2. Results & Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

4.3. Comparison of the Particle System with Previous Techniques . . . . . . . . . . . . . . . . . . 62

Chapter 5. Conclusion 70

5.1. Research Contribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

5.2. Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

References 74

Appendix A. System Screenshots 81

Appendix B. User Survey Forms 88

List of Figures

2.1 An Example of a Polyline Graph Drawing (Source: [Gutwenger98]). 9

2.2 An Example of a Straight-Line Graph Drawing (Source: [Eades99]). 9

2.3 An Example of an Orthogonal Graph Drawing (Source: [Biedl98a]). 9

2.4 An Example of an Upward Graph Drawing (Source: [Germano99]). 10

2.5 An Example of a Visibility Representation of a Graph (Source: [Kant97]). 10

2.6 The Evolution of a Force-Directed Graph Drawing (Source:

[Davidson96]). 12

2.7 Fisheye view of a regular grid of the plane (Source: [Herman00]). 17

2.8 Initial Configuration showing Desired End-Effector Position and a

Possible Final Configuration (Source: [Merrick02 ]). 24

2.9 An example screenshot from the first system developed for skeletal

animation in graph interaction (Source: [Merrick02 ]). 27

3.1 An Articulated Figure Representing a Character’s Skeleton (Source:

[Jakobsen01]). 30

3.2 Dragging an Articulated Figure around an environment in IO Interactive’s

"Hitman: Codename 47". (Source: http://www.ign.com). 31

3.3 Motion governed by Verlet integration within a box environment. (Source:

[Jakobsen01]). 33

3.4 The stick constraint is validated by moving the particles closer together or

further apart (Source: [Jakobsen01]). 33

3.5 Two rigid bodies connected by a pin joint to form an articulated body

(Source: [Jakobsen01]). 35

vii

L IST OF FIGURES viii

3.6 Two rigid bodies connected by a hinge joint to form an articulated body

(Source: [Jakobsen01]). 35

3.7 The length of the dotted line must exceed some threshold. This simulates

an angular constraint. (Source: [Jakobsen01]). 35

3.8 A simplified outline of the execution process of our implementation of the

particle system in a graph interaction environment. 36

3.9 Initial Configuration Showing Desired Position of End-Effector. 40

3.10 Movement of the Node to the Desired Position followed by the Constraint

Solver Produces the Final Configuration. 40

3.11 As neither node is fixed, the edge-length constraint is easily satisfied. 41

3.12 Both nodes are fixed. It is not possible to satisfy the edge-length

constraint, as moving the nodes closer together would violate the fixed

nodes constraint. 41

3.13 All constraints are satisfied. This is indicated by green edges of standard

width. 43

3.14 Some edge-length constraints are not satisfied. This is indicated by edges

of varying width and colour. 43

4.1 The graph used for part 1 of the user study. 49

4.2 The graph used for part 2 of the user study. 52

4.3 The web graph used for part 3 of the user study. 54

4.4 The hex game graph used for part 3 of the user study. 54

4.5 The results for part 1 of the user study. 64

4.6 The results for part 2 of the user study. 65

4.7 The results for the web graph in part 3 of the user study. 66

4.8 The results for the hex game graph in part 3 of the user study. 67

4.9 Comparing the results for web and hex graphs using the skeletal technique. 68

L IST OF FIGURES ix

4.10 Comparing the results for web and hex graphs using the skeletal technique

with angular constraints. 69

A.1 Dragging nodes using our dynamics-based skeletal graph interaction

system. By ensuring the constraints are satisfied, it quickly produces a

much better graph drawing and allows us to improve our understanding of

the graph structure. 81

A.2 Top: A highly stretched graph using the technique without a threshold.

Bottom: A graph stretched as far as possible using the technique with a

threshold. 82

A.3 The graph in part 2 of the study using the force-directed algorithm. 83

A.4 The graph in part 2 of the study using the standard skeletal technique. 84

A.5 The graph in part 2 of the study using the skeletal technique with angular

constraints. 85

A.6 The web graph in part 3 of the study. Top: using the standard skeletal

technique results in bunching up of nodes. Bottom: using the skeletal

technique with angular constraints gives a much better angular resolution. 86

A.7 The articulated figure from figure 3.1 represented by a graph in our graph

interaction environment. 87

CHAPTER 1

Introduction

1.1. Aim of the Project

A relational network or graph is commonly used to represent relations between objects. A

graph consists of a set of nodes and edges, where nodes represent objects and edges represent

the relationship between objects. Edges can be directed or undirected.

Graph drawing takes the set of nodes and edges of a graph and assigns coordinates to the nodes

so that they can be drawn. The aim of graph drawing is to produce a picture that allows the

viewer to easily understand the information that is represented by a graph. A lot of research has

been done in the area of graph drawing and this is given in more detail in Chapter 2 on page 8.

While a number of good graph drawing algorithms have been developed, there are always

situations, particularly in 3D, where the layout provided is difficult to understand. In these

situations it is desirable to have a mechanism to allow the viewer to modify both the graph

layout and their view of this layout. Graph interaction techniques are designed to provide this

mechanism.

Graph interaction is a field that aims to complement graph drawing by reducing the difficulty

and time taken in understanding the information that is being represented by a graph. Effective

1

1.1. AIM OF THE PROJECT 2

interaction techniques have the potential to reduce the time that an analyst needs in order to nav-

igate the presented information, and increase the amount of information that can be interpreted

and understood.

The aim of this project is to investigate graph interaction methods and provide a graph inter-

action system that allows the analyst to understand a graph’s structure more quickly and easily

than other graph interaction systems. This goal is achieved by using a skeleton as a physical

metaphor for a graph and using a skeletal animation technique that is very highly suited for this

purpose.

There are a large number of skeletal animation techniques available. This thesis describes

different skeletal animation techniques and how they can be adapted to provide an intuitive

graph interaction environment. It also evaluates these particular techniques in terms of their

usefulness in graph interaction, both against each other and against other techniques for graph

interaction. This evaluation aims to demonstrate that the skeletal metaphor is useful for graph

interaction, and also to show which skeletal animation techniques are best suited for this pur-

pose. The evaluation aims to affirm or reject the following hypotheses:

a) New techniques will have a positive effect on the simulation of the skeletal metaphor in

graph interaction.

b) The use of the skeletal metaphor will have a positive effect on the user’s ability to

understand the graph structure

Graphs are used in a variety of domains and the skeletal metaphor is potentially useful in many

graph drawing domains. This thesis will also briefly explore the areas of web graphs and the

hex game graph as domains in which skeletal graph interaction could be useful.

1.2. MOTIVATION 3

1.2. Motivation

Graph Visualisation

While graph visualisation has been an area of great interest for researchers, the basis on which

it has been built isn’t perfect. Graph drawing algorithms are generally designed to optimise

certain aesthetic criteria. Examples of aesthetic criteria include minimisation of crossings,

area, bends and maximisation of smallest angle and symmetries. The extent to which a graph

drawing satisfies this criteria is often used as a basis for evaluating its understandability.

A study by Purchase et al. has shown that aesthetic criteria are not always the best way of

drawing or measuring the understandability of a graph [Purchase00]. It argues that, because

graph drawing algorithms are often developed without a specific domain in mind, they are

often not useful once they are applied to domain-specific graphs. The characteristics of an

understandable graph drawing are shown to be highly dependent upon the domain in which

the graph is being drawn. Therefore the aesthetic criteria may not be a reliable measure of the

understandability of a graph. If these graph drawing algorithms are based upon the aesthetic

criteria, their ability to produce an understandable graph drawing is not guaranteed.

It is also reasonable to suggest that the level of readability of a graph is dependent upon the

viewer. Different analysts may gain different levels of understanding from the same graph

drawing. They may also respond differently to certain characteristics of a graph drawing.

In most cases, it seems somewhat naive to provide an analyst with only one drawing of a graph.

If they are unable to understand the graph based on this one drawing, they will have no further

avenue in which to comprehend the information. There may also be no individual drawing of

a graph that is easily understandable in its entirety. By viewing several different visualisations

of the same graph, the user is more likely to obtain an understanding.

1.2. MOTIVATION 4

By providing the user with a mechanism to change the graph drawing, we provide infinite ways

in which the graph may be represented. Not only will this allow changes to be made to an

incomprehensible graph, the changes will also be made by the analyst who is attempting to

understand the graph. By providing their own graph drawing, they are more likely to produce

a representation that they themselves are able to understand.

Graph Interaction

Simply providing the user with the ability to change the positions of nodes may not be enough.

We would like to assist the user in finding a good layout. Graph interaction allows the user to

make changes to a graph drawing with the aim of increasing the understanding of the graph

within a short time. Graph interaction techniques determine how the graph will respond to

input from the user, determining how quickly and easily they will find an understandable con-

figuration.

One such way in which we can control how movements by the user will result in changes to the

graph is through the use of a physical metaphor. The aim of a physical metaphor is to simulate

a physical structure that the user is familiar with. This will result in changes that feel natural

to the user. Any graph interaction technique should feel natural and intuitive, and a useful

physical metaphor is able to provide this.

This leads to the question of what physical metaphor would be suitable for graph interaction.

Such a metaphor must be easily mapped onto a graph structure. A physical metaphor with these

properties that was recently introduced [Merrick02 ] is the skeletal structure. A skeleton in this

case may be seen as a system of bones of fixed length connected by joints that may be rotated

at any angle.

The skeletal structure has many advantages. Firstly, it is easily mapped onto a graph. The nodes

of the graph are mapped to the joints of the skeleton and the edges of the graph are mapped

1.2. MOTIVATION 5

to the bones of the skeleton. As the user drags a node, other nodes will move and joints will

rotate. The interaction technique must ensure that edge-lengths remain constant.

The second major advantage of the skeletal metaphor is that a large amount of research has

been performed in the area of skeletal animation. The majority of skeletal animation research

has been driven by its need in robotics, computer games and motion pictures. These algorithms

can also be applied to the graph structure. As a result we have a large number of algorithms to

choose from and are likely to be able to find a technique that is suited to both skeletal animation

and graph interaction.

The skeleton is also a structure that is manipulated by humans on a regular basis and therefore,

it should feel natural and intuitive.

While Merrick introduced the skeletal metaphor, the skeletal animation techniques he used did

not provide as natural a feel as expected. As a result, there remains a need for an investigation

into other skeletal animation techniques, and an implementation of the skeletal metaphor in a

graph drawing environment that fully realises its potential in terms of its ability to increase the

user’s understanding of a graph.

Potential Applications

Merrick has mentioned domains in which the skeletal metaphor could be useful. The domains

mentioned were PERT diagrams, cluster trees and graphs containing an inherent hierarchy. Web

graphs are another area where the skeletal metaphor could be useful. The skeletal metaphor is

potentially useful in any application where it is crucial to visualize structural information as

graphs.

1.3. SUMMARY OF CONTRIBUTION 6

1.3. Summary of Contribution

This thesis makes several contributions to the field of graph interaction. We provide a brief

summary here and the remainder of this thesis will detail how they were achieved.

We have introduced a technique from skeletal animation as a method that can be used for graph

interaction tasks. We have also provided additional features designed to make it even more

suitable for graph interaction. This technique improves upon the previous techniques used for

skeletal animation in graph interaction.

We have demonstrated the usefulness of the skeletal metaphor through a comprehensive user

study into the benefits of the skeletal metaphor and the different methods used. The results

show that the skeletal techniques are much more effective at providing a mechanism for which

the user can easily understand the graph and provides solid evidence of the usefulness of the

skeletal metaphor. We have also provided some evaluation of the skeletal metaphor in the

domains of web graphs and the hex game graph to give further insight into how useful it is.

We have also added to the definition of the skeletal metaphor to further improve its utility. We

have added the condition of angular constraints. This adds even more realism to the structure

of the graph and further increases its usefulness for graph interaction.

This work adds to the field of graph interaction by providing a natural system of navigating a

graph with the ability to increase the user’s understandability of the graph structure.

1.4. STRUCTURE OF THISTHESIS 7

1.4. Structure of this Thesis

Chapter 2 provides a summary of previous research in the areas of Graph Drawing, Layout

Algorithms, Graph Interaction, Physical Metaphors for Graph Interaction, Skeletal Animation

and Skeletal Animation in Graph Interaction.

Chapter 3 provides a description of the methods that we have used for simulating skeletal

animation in a graph drawing environment. It also provides the details of our implementation.

Chapter 4 describes the process of evaluating the use of the skeletal metaphor through a usabil-

ity study.

Chapter 5 concludes with a review of the project’s goals and results, and details areas that are

seen as future research possibilities.

CHAPTER 2

Literature Review

This chapter provides an overview of prior research done in the areas of research that are

applicable to this project. In each section below, the focus is on the literature that is most

relevant to the research that is documented by this dissertation.

2.1. Graph Drawing

Graphs have been used for a long time in conveying information and as a result of this a lot of

research has been done in the area of graph drawing. Graph drawing or visualisation takes the

set of nodes and edges of a graph and assigns coordinates to the nodes so that they can be drawn.

This produces a drawing that provides a geometric representation for a set of relational data.

The aim of graph drawing is to produce a picture that allows the viewer to easily understand

the information that is represented by a graph [Battista94, Battista99].

A particular embedding of a graph drawing defines the ordering of edges around each node.

While a given graph has an infinite number of possible graph drawings it has a possibly very

large but finite number of embeddings.

In an attempt to increase graph comprehensibility, graphs have been drawn in many different

ways. Polyline [Gutwenger98] (See Figure 2.1), straight-line [Eades99] (See Figure 2.2),

orthogonal [Biedl98a] (See Figure 2.3), upward [Hutton96] (See Figure 2.4) and visibility

8

2.1. GRAPH DRAWING 9

representation [Kant97] (See Figure 2.5) are often used. A graph may be drawn in two or three

dimensions.

FIGURE 2.1. An Example of a Polyline Graph Drawing (Source: [Gutwenger98]).

FIGURE 2.2. An Example of a Straight-Line Graph Drawing (Source: [Eades99]).

FIGURE 2.3. An Example of an Orthogonal Graph Drawing (Source: [Biedl98a]).

Various aesthetic criteria have been discussed which attempt to measure the readability of a

graph drawing by general optimisation goals [Esposito88, Sugiyama87, Tamassia88]. Exam-

ples of aesthetic criteria include minimisation of crossings, area, bends and maximisation of

smallest angle and symmetries [Battista94]. In many cases, it is not possible to completely

2.1. GRAPH DRAWING 10

FIGURE 2.4. An Example of an Upward Graph Drawing (Source: [Germano99]).

FIGURE 2.5. An Example of a Visibility Representation of a Graph (Source: [Kant97]).

satisfy more than one aesthetic criteria simultaneously. It is also common to have an additional

criterion requiring that certain constraints are satisfied [Dengler93]. Examples of possible con-

straints are keeping a certain vertex in a fixed position or keeping a group of vertices close

together.

2.2. LAYOUT ALGORITHMS 11

2.2. Layout Algorithms

An algorithm for producing a graph drawing is known as a layout algorithm. As a result of the

large amount of research into graph drawing, many layout algorithms have been developed that

attempt to optimise specific aesthetic criteria. Di Battista et al. provide a good review of these

[Battista94, Battista99].

There are several types of graphs that require different layout algorithms. A planar graph

can be drawn in the plane such that no two edges cross. Such planar graph drawings are

easy to understand. Methods have been discussed that test for planarity and construct a planar

embedding in linear time [Hopcroft74, Lempel67]. However, these are difficult to implement

because they are quite complicated and hard to understand.

Directed graphs have the feature that each edge has a specified direction. The visualisation

of organisational hierarchies and biochemical pathways are areas where directed graphs have

been used [Schreiber02]. When graphs are directed we desire that all edges follow the same

direction and this is the goal of upward graph drawing for directed planar graphs. Upward graph

drawings are most effective when the graph is acyclic. The Sugiyama method [Sugiyama81]

is commonly used for both planar and non-planar graphs.

The focus in this thesis is on general undirected graphs. There are a few different paradigms on

how to draw general undirected graphs and within these ideas there are many methods.

The planarisation method draws a graph using existing algorithms for planar graphs. If the

graph is nonplanar, it is made planar by replacing edge crossings with dummy vertices. A

planar graph drawing algorithm is then used to draw the graph. An example of the use of this

method is the GIOTTO system [Tamassia88].


The orientation method draws a graph using existing algorithms for directed graphs. First

the edges of the graph are given a direction to form a directed graph. Then a directed graph

algorithm is used to draw the graph.

Force-directed methods define a system of forces acting on the vertices and edges. The graph

drawing then evolves to a minimum energy state. Force-directed methods are relatively easy to

implement, can easily be extended to three dimensions and the smooth evolution of the drawing

helps to preserve the user’s mental map as the layout changes (See figure 2.6).

FIGURE 2.6. The Evolution of a Force-Directed Graph Drawing (Source: [Davidson96]).

The first force-directed method was Tutte’s barycentre method [Tutte60, Tutte63] where each

node is positioned at the barycentre of its neighbours. It is equivalent to a system where edges

are represented as springs with natural length zero and the external face is fixed. The barycentre

method can also be used to draw planar graphs.

The spring embedder method [Eades84] represents edges as springs with unit natural length

and non-connected vertices in the graph are connected by springs in the system with infinite


natural length. As a result connected vertices are attracted to each other unless they are closer

than unit length together. In that case they will repel each other. Non-connected vertices will

repel each other. The system is then let go until a minimum energy state is reached.

In the work of Kamada et al., springs are placed between every pair of vertices. The natural

length of each spring is set to the graph-theoretic distance between the vertices [Kamada89].

Simulated annealing is used by Davidson and Harel [Davidson96] to reduce the energy of a

system. Simulated annealing is an optimisation technique that uses an energy function. The

energy function used by Davidson and Harel was based on the various aesthetic criteria and

weights could be used to emphasize particular criteria.

Force-directed methods do have problems. There are situations where a force-directed ap-

proach results in unfavourable drawings as the nodes can get stuck in a local minimum. This

can be due to poorly chosen initial layouts. It is also often difficult to know when a graph

has reached a minimum energy state, making stopping criteria hard to determine. This con-

tributes to a poor running time for force-directed methods as the number of nodes increases.

As a result, work has been done that uses heuristics to determine the quality of a graph drawing

[Behzadi99]. The results of these heuristics are used to generate an initial layout, move nodes

out of local minima and provide effective stopping criteria.

A modification of Eades’ work is presented by Fruchterman and Reingold [Fruchterman91].

It uses a heuristic that emphasises uniform edge lengths to provide an efficient force-directed

graph drawing algorithm.

Force-directed methods are widely used in the graph drawing field and many different force-

directed algorithms exist. This has produced a need for the evaluation of these methods.

Brandenburg et al. experimented on five graph drawing algorithms for general undirected

graphs. A wide collection of examples was used and different settings of parameters applied


[Brandenburg96]. This evaluation confirmed the good performance of force-directed meth-

ods.

Sugiyama and Misue [Sugiyama95] apply magnetic fields to force-directed graph drawing. A

magnetic field is pre-specified and the forces on each are calculated such that they are brought

into alignment with this magnetic field.

Bertault introduced a force directed graph drawing algorithm which preserves the embedding of

an initial drawing [Bertault00]. The algorithm works like a classical force-directed algorithm

with one main difference: for each node, the maximum amplitude of a move is restricted to stay

within a zone that is calculated such that edge-crossing properties are preserved. The algorithm

thus can improve symmetry and uniformity of edge lengths, while for example preserving the

planarity of a drawing.

Force-directed methods are used for a wide variety of graph drawing applications. Recent

work has applied force-directed graph drawing algorithms to the game of hex [Murray03 ].

There exists an intuitive mapping of the state of a Hex game onto an undirected graph. This

graph representation can increase the player’s ability to find good moves/strategies if a good

graph drawing is provided. Two force-directed methods are combined to produce a planar

graph drawing with uniform edge lengths. The edge-crossings are removed by the Tutte al-

gorithm [Tutte60, Tutte63]. The force-directed algorithm preserving edge crossing properties

[Bertault00] is then used to improve symmetry and uniformity of edge lengths without intro-

ducing any edge crossings.

Force-directed methods can easily be extended to three dimensions. Bruβ and Frick [Frick94,

Bruβ96] have developed a force-directed method and extended it to three dimensions. These

methods are designed to be very fast in order to be suitable for interaction.

Dwyer [Dwyer01] has applied three dimensional graph drawing to the area of UML diagrams.

He used a variety of force-directed methods including springs and magnetic fields.


While force-directed methods are useful, other methods have also been used for drawing graphs

in three dimensions. Orthogonal graph drawing is one such method. Minimising the number of

bends is an important aesthetic in orthogonal graph drawing. Eades et al. have presented two

algorithms for drawing three dimensional orthogonal graphs that are designed to achieve this

goal [Eades97].

Traditionally orthogonal graph drawings restricted the degree of the graph that could be drawn.

Two dimensional graphs were restricted to degree 4 and three dimensional graphs were re-

stricted to degree 6. Biedl et al. have introduced methods for drawing orthogonal graphs of any

degree in both two and three dimensions [Biedl98b, Biedl99].

Another common method of drawing graphs is the visibility representation. Bose et al. de-

signed a three dimensional visibility representation of graphs [Bose98]. In the three dimen-

sional visibility representation, vertices are mapped to rectangles parallel to the x,y-plane, with

edges represented by vertical lines of sight.

2.3. GRAPH INTERACTION 16

2.3. Graph Interaction

Graph interaction techniques determine how the graph reacts to actions by the user. The main

problem in graph interaction is providing a mechanism that will ensure that the graph becomes

and remains understandable as the user navigates the graph. It is desirable that the graph will

react in a way that is natural and intuitive [Merrick02 ]. Graph interaction techniques attempt

to provide solutions for this problem.

Evaluation of the benefits provided by the aesthetic criteria has produced mixed results [Batini85,

Ding90, Purchase97, Purchase00]. In any automated graph drawing system there will be situ-

ations where the initial graph drawing provided is not easy to understand. By providing the user

with a mechanism for changing the layout of the graph we facilitate a potential increase in the

users understanding of the graph. Studies into the effectiveness of graph interaction techniques

in increasing comprehensibility support this [Herman00].

A number of techniques have been used to aid the interaction of a user with a graph. Methods

that effectively only change the view of the graph include the operations of rotation, scaling

and translation. Scaling with the use of a zoom function has two variations. Simply increasing

or decreasing the size of the graph is referred to as geometric zooming. Semantic zooming is

a method that is often used for large and clustered graphs. This method actually changes the

information content by increasing or decreasing the level of detail.

A problem that can occur with zooming techniques is that a zooming in operation often results

in a loss of context. This means that while the user gains additional details, they cannot identify

the relationship between the current focused view and the rest of the graph. Focus+context

techniques such as fisheye distortion [Furnas86] attempt to solve this problem by allowing the

user to focus on some detail without losing the context. Fisheye views achieve this by zooming

in on the area of interest while showing other portions of the graph with successively less detail

(See figure 2.7). Many other distortion techniques have been used. Leung & Apperley provide

a good review of these [Leung94].


FIGURE 2.7. Fisheye view of a regular grid of the plane (Source: [Herman00]).

Often graphs are so huge that it is extremely difficult to handle the full graph at any one time.

Graphs which often fall into this category are World Wide Web graphs. For huge graphs,

incremental exploration techniques are often used [Eklund99, Huang98b, Huang98a]. With

these techniques, the system only displays a small part of the graph at a time. Different parts

of the graph are displayed as needed.

In addition to these aspects that control the view of a fixed graph drawing, there are meth-

ods [Merrick02 ] that allow the user control over the positions and orientation of the nodes

themselves, facilitating modification of the graph layout. These methods give the user greater

control over how the graph is drawn as the embedding of the graph can also be changed, effec-

tively allowing the user to arrange the graph in any way possible. It is these types of methods

for graph interaction that we will be focusing on in this thesis.

While we want to give the user a degree of freedom in navigating the graph, we also want to

maintain some control over the layout of the graph that will require certain properties remain

unchanged. Placing such constraints over the control of the graph will help to ensure that

we continually provide an aesthetically pleasing graph drawing and this will assist the user

in finding an understandable layout. In addition it may reduce the time taken for the user to

achieve such a layout.

When a graph is drawn in two dimensions it is easier for a user to navigate the graph. In terms

of changing the view of the graph, interaction in two dimensions is usually limited to scrolling

or zooming and this is only necessary for graphs that exceed the size of the screen. It is the


positions of the nodes and the way in which the edges are drawn that will determine how easily

the graph can be understood, as these factors alone will determine the aesthetic properties of

the graph. This is because there is no depth to the drawing. As a result, it is methods that

allow the user to change the graph layout that are most important for two dimensional graph

interaction. Mechanisms that allow layout changing modifications are usually only necessary if

the drawing provided by the original layout algorithm is incomprehensible. There will always

be some situations where this is the case, and therefore such mechanisms are important.

The benefits of providing interaction can be seen in the work of do Nascimento & Eades

[Nascimento02], where user actions were used to make improvements to two dimensional

graph drawings. Modifications to the graph layout provided by the user were used as additional

input to a genetic algorithm. These hints provided by the user provided a basis for the system

to improve the drawings. It also helped to provide drawings that were better than they would

have been without user interaction.

When a graph is drawn in three dimensions, navigating the graph is much more difficult. On

regular displays the user is only seeing a two dimensional projection of a three dimensional im-

age. In any given view, it is likely that part of the graph will be obscured due to the additional

depth of the drawing and lack of depth of the display device. The user will need to be able

to change their view of the graph in order to experience all three dimensions. In any number

of dimensions it is good to start with a good graph drawing. However, in three dimensions, in

addition to the positions of the nodes, an important aspect that will determine how easily a user

can understand the graph structure is their ability to modify their view of the graph. Graph In-

teraction in three dimensions is even more important than in two dimensions as there are many

additional factors that could lead to an original graph drawing being incomprehensible. This

also makes graph interaction in three dimensions a more difficult problem. Three dimensional

graph drawing mechanisms need to provide the ability to change both the layout and view of

a graph. Also, these mechanisms are more complicated in three dimensions. There are many

more ways in which the layout can be changed as node movement isn’t restricted to a single


plane, and when changing the view, rotation needs to be provided in addition to translation and

zooming.

Not only is graph interaction in three dimensions more difficult, it is also more necessary. As

drawing graphs in 3D is more complicated, there are likely to be even more situations where

the initial graph drawing is difficult to understand.

Graph interaction is a field that has become more prevalent due to a more widespread use of

three dimensions in graph drawing. There are a number of reasons why this is the case. Us-

ability studies [Ware93, Ware94, Ware96, Cockburn01] have shown the benefits of three

dimensional drawings over two dimensional drawings. These studies have shown that three

dimensional drawings are more easily understood, that it is easier to follow the structure of

the graph and that it feels more realistic and natural. A wider availability of hardware capa-

ble of drawing graphs in three dimensions has also lead to three dimensional graph drawings

becoming more widespread.

While there have been studies showing the benefits of three dimensional drawings, there are

also studies claiming that it is better to draw graphs in two dimensions [Nielsen98]. Cases

where three dimensions have been shown to be not useful may be due to a poor interaction

model, as a good interaction model would have improved the understandability of the graph.

There are certainly cases where three dimensional graph drawings are useful and they can only

become more useful with a good interaction model [Cockburn01]. A good interaction model

will ensure that the benefits of three dimensions are fully realised. While graph drawing is a

field that has been thoroughly explored [Battista94, Battista99], graph interaction remains a

relatively new field and novel methods that aid the user’s understanding of the graph structure,

particularly in three dimensions, are still required.

While navigating the graph visualisation space, the user spends a reasonable amount of time

building a mental map [Gould86, Eades91, Misue95] of the structure of the graph. Being able

to maintain and add to this mental map as they navigate the graph is essential for developing


an understanding of the structure of the graph. The graph interaction mechanism must not

make drastic, unintuitive changes to the drawing of the graph. Graph animation [Friedrich02]

is a method that demonstrates the transition between drawings of a graph. An intuitive graph

interaction mechanism, with the aid of animation allows the user to easily follow the transition

from the initial graph drawing to the final graph drawing. Animation, when applied through

successive layouts of a graph that do not differ significantly, ensures that the mental map is

maintained.

The simplest way of providing graph interaction is to use an adaptive graph drawing algorithm.

An adaptive graph drawing algorithm produces a drawing that is dependent upon the initial

geometric properties given to the graph. These types of algorithms take a graph drawing as

input and their aim is to improve upon it. Adaptive graph drawing algorithms are useful for

graph interaction tasks as they are more likely to preserve the user’s mental map. The speed

requirements of an algorithm used for graph interaction are generally higher than that of general

graph drawing. Algorithms developed with graph interaction in mind [Bruβ96] are designed

to be very fast.

2.4. PHYSICAL METAPHORS FORGRAPH INTERACTION 21

2.4. Physical Metaphors for Graph Interaction

One method used in graph interaction involves placing a physical metaphor on the structure

of the graph such that the graph responds to movements of nodes in a way corresponding

to the physical metaphor. A common physical metaphor in undirected graph drawing is the

force-directed paradigm and this idea is easily extended to graph interaction as force-directed

algorithms are adaptive. Forces are recalculated as the user moves a node and the position of

the nodes are updated accordingly. The aim of a physical metaphor is to provide behaviour that

is naturally experienced in everyday life and therefore easy to understand, follow and predict

when applied to graph interaction.

Skeletal animation is a concept that has been applied to the creation of realistic motion for the

lifelike computer-generated characters seen in today’s motion pictures and computer games.

Recent work has applied this physical metaphor to the field of graph interaction [Merrick02 ].

The skeletal metaphor maps nodes of a graph to joints of a skeleton and edges of a graph to

the bones of a skeleton. A simplified view of a skeleton is that joints can be rotated and bones

remain at a constant length. Merrick has used skeletal animation to smoothly animate the graph

layout according to the movement applied by the user as well as various underlying constraints

forced on the graph’s skeleton. By applying these constraints on a graph he provided a physical

metaphor that feels natural to the user. It produces a graph interaction environment such that

the structure of the graph is easy to understand and the time taken in navigating the graph is

reduced. Merrick has introduced the skeleton as a physical metaphor for graph interaction and

has opened up many possibilities for future work in graph interaction.

2.4. PHYSICAL METAPHORS FORGRAPH INTERACTION 22

The definition of a skeleton provided by Merrick is:

A skeleton is a set of joints (J) and a set of bones (B). Each jointj ∈ J is connected by a set

of bonesBj ⊆ B to set of jointsjADJ ⊆ J, |jADJ | > 0.

There exist constraints on the skeleton that should be preserved:

- The lengthlb of each boneb ∈ B should not change.

- Given a set of jointsF ⊆ J , termed the fixed point set, each jointj ∈ F should remain

at its initial location in Euclidean space.

2.5. SKELETAL ANIMATION 23

2.5. Skeletal Animation

Skeletal animation is an area that is commonly used in the areas of computer games and mo-

tion pictures. As a result a large number of techniques have been developed that control the

movement of a skeleton in a way that is designed to appear natural to the viewer.

Skeletal animation is also used in the area of robotics where it is referred to as the study of

the motion of articulated bodies. Many of the methods used for skeletal animation were first

developed for use in robotics, especially in the area of inverse kinematics.

Throughout this thesis we are primarily concerned with dynamics-based modelling of artic-

ulated structures, and thus do not cover in detail the issues involved with kinematics-based

approaches to skeletal animation. We will however give a brief summary of the most important

literature in this area.

Inverse kinematic methods move a joint to a desired position while controlling the movement

of several other joints and taking into account positions of fixed joints, bone lengths and joint

angles. Korein and Badler referred to this concept as "goal-directed motion" [Korein82]. An

example of this can be seen in figure 2.8 where the red node is the end-effector and the top node

is fixed. There are potentially an infinite number of such configurations which result in the end-

effector moving to the desired point in space. There are also situations where it is impossible

to move the node to the desired location. In this case, the aim is to get it as close as possible to

the goal location.

Situations where there are a finite number of final configurations are more likely to occur in

the field of robotics where articulated bodies typically have fewer joints and are less complex.

These situations have been solved using a system of nonlinear equations [Paul81, Brady82].

The use of these methods has been limited by the difficulty in solving the equations and the

small likelihood that there will be a finite number of final configurations.


FIGURE 2.8. Initial Configuration showing Desired End-Effector Position and

a Possible Final Configuration (Source: [Merrick02 ]).

The method of resolved motion rate control [Whitney69] can be used in situations where there

are an infinite number of possible final configurations. This technique calculates the inverse

of the Jacobian matrix in order to implement an iterative algorithm where each step moves

the end-effector closer to the desired end-effector position. The use of this method is limited,

however, as the Jacobian matrix is not always invertible. In this case, the pseudoinverse can be

used [Klein83, Girard85], but it can become unstable. It can be made more stable by using

Singular Value Decomposition (SVD) [Maciejewski90] but it does result in relatively high

additional runtime cost. The Jacobian transpose has proved useful [Wolovich84, Sciavicco87,

Welman93] in situations where high accuracy is not required.

An alternative to the methods based on the Jacobian matrix is a heuristic-based approach re-

ferred to as Cyclic Coordinate Descent (CCD) [Wang91]. CCD is an iterative approach that

involves visiting each joint in an articulated chain in turn. By visiting a single-joint at at time,

the problem is reduced to a simpler optimisation problem.

The inverse kinematics problem can also be approached from a nonlinear optimisation point-

of-view [Badler87, Zhao94]. This approach has been widely used for computer animation

involving articulated structures.

While inverse kinematics techniques operate on the geometry of articulated structures, with

dynamics, all interactions with the system are modelled as forces acting upon rigid and articu-

lated bodies. Inverse kinematics techniques have been useful for robotics, and as a result, it has

been the topic of the majority of research in skeletal animation. However, there has also been


some development of dynamics-based techniques, a lot of which has been driven by the need

for skeletal animation techniques in computer games, where the main goals are believability

and speed of execution.

In order to achieve believability, dynamics-based techniques focuses on emulating motion as

seen in real environments. These methods are less focused on goal directed motion. These

methods use an iterative approach where, at each iteration, the forces upon the particles that

make up a body, are summed in order to determine their following positions. This process is

repeated in order to generate motion.

The dynamics-based algorithms are described further in Chapter 3 on page 29 of this thesis,

where they are used as a basis for the implementation of a graph interaction environment using

skeletal animation. We give a brief overview here.

The traditional methods [Witkin97 ] for dynamics-based skeletal animation use Euler integra-

tion using Newton’s law for particle simulation. Bones connecting particles are simulated by

springs. The major problem with this approach is its speed as well as the difficulty in choosing

parameters that will make the springs appear stiff while remaining stable.

Dynamics-based algorithms were developed by Jakobsen [Jakobsen01] to control the physics

systems in computer games. These algorithms were designed to be well suited to interactive

use and improve upon traditional approaches by providing a faster and more stable system.

2.6. SKELETAL ANIMATION IN GRAPH INTERACTION 26

2.6. Skeletal Animation in Graph Interaction

Merrick was the first to apply skeletal animation techniques to graph interaction (See figure

2.9) [Merrick02 ]. These methods were previously used in the fields of inverse kinematics and

dynamics.

The inverse kinematics methods used were based on the CCD [Wang91] method and the NLP

[Badler87, Zhao94] method. The first method used a multiple chain version of the CCD

method to simulate a skeletal structure. The second method involved the formulation of the IK

problem as input to a nonlinear programming module, including the addition and implementa-

tion of various constraints. A naive dynamics-based approach was explored as a third method,

which was based on a force-directed graph drawing algorithm. The parameters were chosen

in an attempt to create extremely stiff springs with limited success in terms of it’s ability to

simulate the skeletal metaphor. Obviously, the focus of the previous work was on inverse kine-

matics. There remains scope for a study into dynamics-based techniques for skeletal animation

applied to graph interaction.

As inverse kinematics techniques were mostly developed for use in robotics, they are designed

to be very realistic. Dynamics-based techniques have typically been developed for computer

games and don’t need to be entirely realistic as long as they give the illusion of realism. We

have similar goals for graph interaction. Therefore, dynamics-based techniques may be more

suited for graph interaction purposes. This further supports investigation into dynamics-based

skeletal animation techniques for graph interaction.

The previous approaches used to implement the skeletal metaphor in a graph interaction system

provided a natural way for the user to navigate a graph. However, these methods could be im-

proved upon in terms of the satisfaction of the constraints defining the skeleton, the naturalness

of the motion and the ability to preserve a user’s mental map between successive configurations.


FIGURE 2.9. An example screenshot from the first system developed for skele-

tal animation in graph interaction (Source: [Merrick02 ]).

In the previous work, the CCD & force-directed approaches resulted in a stretchiness of bones.

This violates the prescribed skeletal metaphor and damages the perception that a physical entity

is being manipulated.

The CCD method became very slow in some cases in the previous system. For a graph inter-

action system to be useful it must be useable in real-time, which was not always true of this

approach.

The motion should also remain smooth and natural as is the intention of the skeletal metaphor.

The NLP method provided jumpy motion and this limited its usefulness significantly.

Methods used in the previous work all have advantages and disadvantages both in terms of their

satisfaction of the skeletal metaphor and in their usefulness in graph interaction. It would be

desirable to use a technique that achieved and combined the advantages of the previous methods


without the disadvantages. There are many techniques used for skeletal animation that could

achieve this. There remains a large gap in the research for experimentation with new techniques

for skeletal animation in this area. However, it is also important to find those that will be best

suited to graph interaction. An ideal technique would both provide simulation of the skeletal

metaphor and at the same time will be useful for graph interaction tasks. There is large potential

for improvement upon the existing work by providing a highly responsive system that helps to

increase the user’s understanding of the graph by staying true to the skeletal metaphor.

The previous work in this area has mentioned domains in which the skeletal metaphor could

be useful. The domains mentioned were PERT diagrams, cluster trees and graphs containing

an inherent hierarchy. It is still necessary to evaluate the usefulness of the skeletal metaphor in

these and other domains as well as in the general case.

There is still a significant need for empirical results indicating the advantages and disadvan-

tages, if any, of using a skeletal metaphor over other types of interaction with graph visualisa-

tions. Previous evaluation of the usefulness of the skeletal metaphor was inconclusive due to

the limited scope of the survey undertaken. There remains a gap in the research that could be

filled with a more comprehensive user study in order to produce solid evidence of the useful-

ness of the skeletal metaphor. It is also important to evaluate the usefulness of each technique

in simulating the skeletal metaphor.

Another area in which the skeletal metaphor in graph interaction has been explored is in apply-

ing it to large web graphs using clustering methods [Belista03]. Clustering methods are used

reduce the size of the graph to be displayed and to combat the massive information problem.

CHAPTER 3

A System for Force-Based Skeletal Graph Interaction

This chapter describes the methods used during the course of this research project, as well as

implementation details for the system used for the evaluation given in Chapter 4. Section 3.1

details the method described by Jakobsen [Jakobsen01] and also describes the nature of many

dynamics-based methods used for computer games [Witkin97 ]. Section 3.2 describes how we

have adapted this technique to create a graph interaction environment. It also details several

additional aspects that needed to be added in order for the method to be specifically designed

to be useful for graph interaction.

3.1. A Particle System for Skeletal Animation

With the graph modelled as a skeleton we need an algorithm to ensure that the skeletal con-

straints are maintained as the user modifies the positions of the nodes. The skeletal animation

algorithms [Jakobsen01] that we have used form a dynamics-based approach. These algo-

rithms were adapted from the work of Jakobsen. Jakobsen’s work was designed to control the

physics systems in computer games, specifically IO Interactive’s "Hitman: Codename 47" (See

figure 3.2), where the major concerns are believability and speed of execution. One respon-

sibility of the physics system is the simulation of articulated bodies. An articulated body is

composed of solids and joints which define the range of motion allowed by the different parts

of the body. The result is a structure that is very similar to that of a graph. Articulated bodies are

often used to represent the skeleton structure of computer games characters (See Figure 3.1).

As the physics system was designed for computer games, it provides simulation of a skeleton

29

3.1. A PARTICLE SYSTEM FORSKELETAL ANIMATION 30

that appears natural to the viewer, and is fast, stable and well suited for interactive use. One

way in which it has been used is for the dragging of articulated bodies around an environment

(See figure 3.2). In many ways this is very similar to graph interaction. These algorithms are

highly suitable for skeletal graph interaction because they have been specifically designed with

interactive dragging of articulated bodies in mind.

FIGURE 3.1. An Articulated Figure Representing a Character’s Skeleton

(Source: [Jakobsen01]).

The physics system created by Jakobsen (2001) consists of these features:

• a Verlet integration scheme

• handling collisions and penetrations by projection

• a constraint solver

• a square root approximation that reduces the processing time

• modelling rigid and articulated bodies as particles with constraints


FIGURE 3.2. Dragging an Articulated Figure around an environment in IO In-

teractive’s "Hitman: Codename 47". (Source: http://www.ign.com).

Particle simulation is an important part of a physics system. The most commonly used method

for particle simulation is Euler integration using Newton’s law. In Euler integration each par-

ticle has two variables, its positionx and its velocityv. In each time step the position and

velocity values of the particle are updated. The new valuesx′ andv′ are given by:

x′ = x + v ·∆t

v′ = v + a ·∆t

where∆t is the time step, anda is the acceleration computed using Newton’s lawf = ma.

The forcef is the accumulated force acting on the particle by elements in the environment such

as gravity.

The physics system [Jakobsen01] uses a Verlet integration scheme [Verlet67] for its particle

simulation. Verlet integration schemes use a velocity-less representation. The information


stored consists of a particle’s current positionx and it’s previous positionx∗. The new values

using Verlet integration are given by:

x′ = 2x− x∗ + a ·∆t2

x∗ = x

This works similarly to Euler integration as2x − x∗ = x + (x − x∗) andx − x∗ provides an

approximation of the velocity of the particle. Verlet integration is faster and more stable than

Euler integration because it is harder for the velocity and position to come out of synchronisa-

tion. This has allowed it to be useful for simulating molecular dynamics.

Common methods for handling collisions and contact of particles with the environment have

a number of disadvantages that can make them unsuitable for interactive use. One method

handles collisions by inserting springs at the penetration points. In this case, it is difficult

to choose spring constants such that the system remains stable and accuracy is maintained.

Another method goes back in time to the exact point of collision, where it handles the collision

analytically before restarting the simulation. This method is likely to run very slowly when

there are a lot of collisions making it impractical for interactive applications.

In the physics simulation collisions are handled by projection. This means moving the point

perpendicularly out from the collision surface as little as possible until it is free of the obstacle,

ensuring particles are kept inside the valid environment. This works well with the Verlet inte-

gration scheme as corresponding changes in velocity will be handled automatically and there is

no need to directly cancel the velocity in the normal direction. The motion of particles within a

box environment using Verlet integration can be seen in figure 3.3. This method is also fast and

simple to implement making it suitable for interactive use. Also, as we will see in the following

sections, it works particularly well with constraint satisfaction for rigid or articulated bodies.

In a physics system particles are often connected by springs to model rigid bodies or cloth.

The problems with this approach are once again associated with the choice of parameters. It is


FIGURE 3.3. Motion governed by Verlet integration within a box environment.

(Source: [Jakobsen01]).

difficult to find a balance between weak springs that appear too elastic and strong springs that

cause instability in the equations. It is also difficult to solve the differential equations used in

this method.

FIGURE 3.4. The stick constraint is validated by moving the particles closer

together or further apart (Source: [Jakobsen01]).

The algorithms of the physics system model the connections between particles as infinitely stiff

springs. These springs will instantly reach their rest length and are effectively more like sticks

than springs. This makes them highly suitable for modelling bones of a skeleton. For each stick

connecting two particles, the particles must be placed such that the distance between them is

equal to the desired length of the stick. After the movement of a particle takes place due to a

Verlet integration step, the stick length may have become invalid. This constraint is validated

by moving the particles away from each other or pulling them closer together, depending on

whether the length of the stick joining them, is too large or too small (See figure 3.4). This is

highly suited to an interactive application as it is a very fast and simple approach.


The pseudo-code for satisfying the stick-length constraint is:

delta = x2-x1

deltalength = sqrt(delta · delta)

diff = (deltalength-restlength)/deltalength

x1 += delta*0.5*diff

x2 -= delta*0.5*diff

There are two constraints governing the system that must be satisfied. These are keeping par-

ticles within the environment and maintaining ideal stick lengths. The satisfaction of either of

these constraints can lead to the invalidation of the other. To ensure that length and collision

constraints are both satisfied, local iteration is performed, where the two constraint solvers are

repeatedly run after each other. After a number of iterations the positions of the particles con-

verge to a valid solution. This iterative method can be stopped at any time depending on the

level of accuracy required, allowing for a time/accuracy trade-off.

This physics system also has a further speed up mechanism if required making it even more suit-

able for interactive use. The costly square root operation that takes place in the stick constraint

satisfier can be approximated. This approximation uses the fact that as long as the constraints

are satisfied, or close to it, the result of the square root operation should be the rest length of the

stick. Also, if the distance between the particles is already correct then no change is necessary.

The square root function is approximated by its 1st order Taylor-expansion at a neighborhood

of the squared rest length. The pseudo-code for satisfying the stick-length constraint becomes:

delta = x2-x1

delta *= restlength*restlength/

(delta · delta+restlength*restlength)-0.5

x1 += delta

x2 -= delta


Rigid bodies are modelled as particles connected by sticks governed by constraints. Articulated

bodies are modelled by connecting rigid bodies. It is possible to connect multiple rigid bodies

by hinges and pin joints. Simply let two rigid bodies share a particle, and they will be connected

by a pin joint (See figure 3.5). Share two particles, and they are connected by a hinge (See figure

3.6). This provides an intuitive mapping from the particle system to a skeleton where particles

are mapped to joints and sticks are mapped to bones.

FIGURE 3.5. Two rigid bodies connected by a pin joint to form an articulated

body (Source: [Jakobsen01]).

FIGURE 3.6. Two rigid bodies connected by a hinge joint to form an articulated

body (Source: [Jakobsen01]).

Angular constraints can add further realism to the particle system. One method given by Jakob-

sen uses additional constraints that are defined for each joint connecting two sticks (See figure

3.7). The constraints are that the distance between the unconnected particles of the sticks form-

ing the joint cannot fall below some threshold. These constraints are enforced in the same way

as the stick constraints in the case where the stick length is less than it’s ideal length. This

method effectively prevents some particles from coming too close together.

FIGURE 3.7. The length of the dotted line must exceed some threshold. This

simulates an angular constraint. (Source: [Jakobsen01]).

3.2. IMPLEMENTATION OF THE PARTICLE SYSTEM IN A GRAPH INTERACTION ENVIRONMENT 36

3.2. Implementation of the Particle System in a Graph Interaction Environment

We have implemented this particle system into a graph interaction environment in Java, using

Java3D for its graphics routines. The process that is followed is illustrated in the flowchart in

figure 3.8.

FIGURE 3.8. A simplified outline of the execution process of our implementa-

tion of the particle system in a graph interaction environment.


In our system the graph is modelled as an articulated body with joints representing nodes and

bones representing edges. Our implementation allows the user to drag the nodes of the graph

around and the graph will respond according to the underlying physics engine.

Merrick applied the skeleton to only part of the graph [Merrick02 ]. Some of the techniques

used by Merrick could only be applied to limited cases. As a result, he used a preprocessing

step to determine the subgraph that would represent the skeleton. We have modelled the entire

graph as a skeleton because the technique that we have used can be applied to any graph.

Verlet Integration

As a node is dragged around the environment momentum is generated in the other nodes. The

Verlet integration scheme is used to determine the positioning of the nodes. There was a prob-

lem that no friction existed in the system and the graph would tend to float around endlessly,

never settling. The equations governing the Verlet integration scheme were modified to intro-

duce drag into the system. Jakobsen suggested the equationx′ = 1.99x − 0.99x∗ + a · ∆t2 to

introduce drag. However, this provided no noticeable difference. After some experimentation

we used the equationx′ = 1.75x − 0.75x∗ + a · ∆t2 which reduced the floating sensation and

ensured the graph would settle quickly. This also gave the graph the impression of a rigid,

physical structure as is the nature of a skeleton. We also don’t have any need for gravity in our

system, so the equation simplifies tox′ = 1.75x− 0.75x∗.


The pseudocode for the Verlet integration step is:

FOR each particle p

IF p’s location is not fixed THEN

store p’s current location in temp

p.currentX = 1.75*p.currentX - 0.75*p.oldX

p.currentY = 1.75*p.currentY - 0.75*p.oldY

p.currentZ = 1.75*p.currentZ - 0.75*p.oldZ

p’s old location = temp

END IF

END FOR

Bounding Environment Constraints

A cube is used as the environment in which the graph must remain. Collisions and penetrations

are projected onto the inside of the cube and when dragging a node the user cannot move this

node outside the cube. Other nodes which move due to the momentum created by this drag

and the stick constraints, will also stay within the environment. This allows us to restrict the

movement of the graph within a small area and keeps the user’s focus on this area. The use of

this cube is optional and it when it is used only the outline is visible. This ensures that the user

will not be not be distracted by it.

Edge Length Constraints

As a node is dragged around, the edge length constraints are enforced by the constraint solver

(See Figures 3.9, 3.10). This ensures that as a node is dragged surrounding nodes will move

closer or further away from the dragged nodes in order to maintain the constraint. This creates

a natural feel in the system as the edges remain at constant length, similarly to the bones of a

skeleton.


Iterative Constraint Solver

For each frame the iterative constraint solver is run to ensure collision and stick length con-

straints are satisfied. It also helps to preserve the user’s mental map by providing a smooth

animation from one configuration of the graph to the next. In practice we found that only a

very small number such as 5 iterations of the constraint solver were necessary. This is enough

to reduce any large violations of the constraints. Any remaining violations in the constraints

will be small and will not be noticed by the user. Also, the satisfaction of constraints will

converge further over subsequent frames. Increasing the number of iterations does not provide

significant improvement in terms of constraint satisfaction believability. With a large increase

in the number of iterations performance can become an issue for large graphs, actually reduc-

ing the natural feel of the structure rather than increasing it. The pseudocode for the constraint

solver is:

FOR each iteration

FOR each particle p with current location x

x = componentWiseMin(componentWiseMax(x,minBox),maxBox)

END FOR

FOR each edge connecting particles x1 & x2

delta = x2-x1

deltalength = sqrt(delta · delta)

diff = (deltalength-restlength)/deltalength

IF x1’s location is not fixed THEN

x1 += delta*0.5*diff

END IF

IF x2’s location is not fixed THEN

x2 -= delta*0.5*diff

END IF

END FOR

END FOR


FIGURE 3.9. Initial Configuration Showing Desired Position of End-Effector.

FIGURE 3.10. Movement of the Node to the Desired Position followed by the

Constraint Solver Produces the Final Configuration.


Fixed Node Constraints

An additional aspect necessary for satisfying the definition of a skeleton but not handled in

the original physics system is that of fixed joints. In any graph interaction environment it is

desirable for the user to be able to fix nodes so that they will remain in the same location.

The user can select nodes to be fixed or can unfix nodes at any time. We have implemented

this in our system by adding the constraint that the subset of nodes that is fixed cannot move.

This is a hard constraint and is given priority over the other constraints so that the fixed nodes

will not move from the set position at all. This introduces a problem in ensuring that the edge

length constraints are satisfied, for example, if two nodes connected by a stick are fixed. If

the user drags one of these nodes away the edge will increase in length and the constraint will

be violated as the system cannot move either node in an attempt to satisfy the constraint (See

Figures 3.11, 3.12).

FIGURE 3.11. As neither node is fixed, the edge-length constraint is easily satisfied.

FIGURE 3.12. Both nodes are fixed. It is not possible to satisfy the edge-length

constraint, as moving the nodes closer together would violate the fixed nodes

constraint.


Error Threshold

To remedy this problem we have placed restrictions on how the user can drag nodes around.

Movements of nodes that will violate an edge length constraint significantly are not allowed.

This allows us to maintain some freedom of control for the user while still providing the nat-

ural feel of the skeletal structure. A threshold that measures the sum of violations across the

entire graph is used to determine if the extent of a violation is acceptable. Once the maximum

threshold is reached, movement will be limited to actions that reduce or maintain the current

extent of violations. After reducing the extent of the violations, the user will then once again be

able to make movements freely. This provides restrictions on the movement that more closely

resemble a physical structure as is the nature of the skeleton. As a result, it is expected to

provide a more natural feel and be more effective at maintaining the skeletal structure. In our

system, we have provided a mechanism for navigating the graph with or without the threshold

restrictions. These two techniques are used as a basis for part of the usability study in Section

4.1.

Our implementation can be used with or without the square root approximation mentioned in

Section 3.1. However, because of the addition of fixed nodes in space, the constraints of the

system are less likely to be satisfied. When using the square root operation it is important to

also use a low threshold to ensure that the constraints are at least close to being satisfied. The

square root approximation does not seem necessary on smaller graphs but as the size of graphs

increases it could be used to reduce the processing time helping to ensure a fast response to the

user actions.

Visual Cues

We also provide visual cues as to where and by how much violations are occurring. A signifi-

cant edge length violation will result in a change in colour of the edge to red. Also, the width

of the edge will decrease if the edge is stretched and will increase if the edge is compressed


(See Figures 3.13, 3.14). The visual cue of highlighting fixed nodes in pink is also provided to

ensure the user is aware of the subset of fixed nodes.

FIGURE 3.13. All constraints are satisfied. This is indicated by green edges of

standard width.

FIGURE 3.14. Some edge-length constraints are not satisfied. This is indicated

by edges of varying width and colour.


Determining the Graph Structure and Initial Layout

When drawing a graph in our system we not only need to know the set of nodes and edges but

also what the ideal length of each edge will be. In some domains, graph edges have associated

weights and these can be used directly to determine the ideal edge lengths. In other cases

determining the edge lengths may not be so straight-forward. A common aesthetic property

desired in graph drawings is that of uniform edge-lengths, where all edges in the drawing have

approximately equal lengths. In our system, the edge-lengths are provided as input along with

the set of nodes and edges. The initial layout of nodes is produced randomly. The constraint

solver then improves the drawing. The drawing can then be further improved by the user, as is

the aim of our system.

The Addition of Angular Constraints

Our original definition of a skeleton does not specify any angular constraints. However, in

terms of graph navigation, there is a possible benefit of implementing angular constraints as it

would keep nodes from getting too close together. We have implemented angular constraints

in our system by a similar method to that suggested by Jakobsen. We have applied forces

between unconnected nodes. Unconnected nodes whose distance is less than a threshold are

repelled from each other. The forces on each node are summed in a way similar to force-

directed methods and nodes are moved in the direction of the force. The calculation of forces

and movement of nodes is performed within the constraint satisfaction loop. While this is not

always an accurate way of providing angular constraints, it is fast and simple and provides a

realistic feel most of the time. This method not only adds realism to the skeletal structure, it

also provides aesthetic benefits. It prevents unconnected nodes from getting too close together

and this results in good angular resolution. Edges emanating from a node will spread evenly

around a node. Because angular constraints are not part of our original definition of a skeleton,

we provide the option of whether to use angular constraints or not, and this is also used as a

basis for comparison in part of our user study in Section 4.1.

3.3. IMPLEMENTATION OF A FORCE-DIRECTED ALGORITHM IN A GRAPH INTERACTION ENVIRONMENT 45

3.3. Implementation of a Force-Directed Algorithm in a Graph Interaction Environment

In addition to the particle system, we also implemented a force-directed algorithm into the

graph interaction environment. The reason for implementing the force-directed algorithm is

for a comparative evaluation of skeletal and non-skeletal techniques as detailed in Section 4.1.

The force directed method that we implemented consisted of the following features. Connected

nodes are attracted to each other if their length is longer than desired and repelled if their length

is shorter. Disconnected nodes are repelled from each other if they are within a certain distance.

Several iterations of the force-directed algorithm are run at each frame. As nodes are dragged

around the force directed algorithm will recalculate the forces and update the positions of the

nodes accordingly. This helps to ensure that edge lengths are as desired. However, the force-

directed algorithm is more forgiving of differences between desired and actual edge lengths

than the particle system. It terms of satisfying box and fixed node constraints, the force-directed

algorithm works in exactly the same way as the particle system.

It terms of the feel of the system the force-directed method provides a more stretchy and less

rigid feel than the more rigid particle system. In terms of satisfying the edge-length constraints,

the force-directed algorithm usually quickly finds an equilibrium state. However, there are

some situations where it gets stuck and cannot find an equilibrium state. Also, it is difficult to

find parameters that are suitable for a wide variety of graphs.

3.3. IMPLEMENTATION OF A FORCE-DIRECTED ALGORITHM IN A GRAPH INTERACTION ENVIRONMENT 46

The pseudocode for the calculation of forces is:

FOR each node n1

n1.force = (0, 0, 0)

IF n1 is not fixed

FOR each node n2

IF n1 != n2 THEN

distance = Euclidean distance between n1 and n2

IF n1 and n2 are connected by an edge THEN

ideal = ideal distance between n1 and n2

attraction = (distance-ideal)/ideal

n1.force += -attraction*(n1.pos - n2.pos)

ELSE

IF distance < some threshold t THEN

repulsion = (t*t)/(distance*distance)

n1.force += repulsion*(n1.pos - n2.pos)

END IF

END IF

END IF

END FOR

END IF

END FOR

FOR each node n

move n in direction of force

END FOR

3.4. COMPUTATIONAL COMPLEXITY OF THE METHODS USED 47

3.4. Computational Complexity of the methods used

An important factor when comparing different graph interaction techniques is the computa-

tional complexity. The computational complexity largely determines the size of graphs that can

be effectively used with the technique.

It can be seen from the pseudocode that the Verlet integration step can be performed in time

linear in the number of nodes. This is because each node must be visited once. The pseudocode

for the constraint solver shows that in order to satisfy the box constraint each node must be

visited once, and for the edge-length constraint, each edge must be visited once. Therefore

the standard skeletal technique is linear in the larger of the number of nodes or edges. The

computational complexity for the standard skeletal technique isO(|V |+ |E|).

The skeletal technique with angular constraints has the additional operation of calculating the

forces on each node. Each node must determine the force exerted upon it by possibly every

other node. Therefore the time complexity is quadratic in the number of nodes. The compu-

tational complexity for the skeletal technique with angular constraints isO(|V |2). This is the

main disadvantage of the method with angular constraints over the standard skeletal method.

The force-directed algorithm also needs to visit every other node to calculate the force on each

node. The time complexity of the force-directed algorithm is thereforeO(|V |2).

It can be seen from these figures that all techniques run in polynomial time allowing them to

be useful for many situations. However, the performance of the force-directed and angular

constraints methods could deteriorate on large graphs. The standard skeletal technique would

perform better on large graphs in most cases. In cases where every node is connected to every

other node (complete graphs), the time-complexity would be the same.

CHAPTER 4

Evaluation of Skeletal Animation Techniques in Graph Interaction

Chapter 3 introduced the particle system as a technique for simulating the skeletal metaphor

in a graph interaction environment. We would like to evaluate this technique both in terms of

its ability to simulate the skeletal metaphor and in terms of its usefulness for graph interac-

tion. This chapter presents an evaluation of skeletal animation techniques when used in graph

interaction in order to determine the value of each technique and also the merit of the skeletal

metaphor. The results are provided in the form of column graphs at the end of this chapter.

4.1. User Study

Aims of the User Study

A usability study is a reliable way of measuring the usefulness of graph interaction techniques,

as feedback from users is a strong indication of how easy and natural the system is to use. It

also provides an indication of how enjoyable it is to use the system.

A study was performed as a part of this project. This study aims to provide answers to the

following research questions:

a) Is the particle system useful for simulating skeletal animation in graph interaction?

b) Is the skeletal metaphor useful for graph interaction in the general case?

c) Is the skeletal metaphor useful for graph interaction in specific domains?

48

4.1. USERSTUDY 49

d) What domains are most suited to the skeletal metaphor?

In answering these questions, the study aims to affirm or reject the following hypotheses:

a) New techniques will have a positive effect on the simulation of the skeletal metaphor in

graph interaction.

b) The use of the skeletal metaphor will have a positive effect on the user’s ability to

understand the graph structure

Description of the Process and Survey Questions

Participants of the user study used the system developed in Sections 3.2 & 3.3 to perform graph

interaction tasks. They were asked to answer survey questions based on their experience with

the system. The survey is included in Appendix B on page 88. We had 16 participants take

part in our user study. They were postgraduate and Honours students in the field of computer

science.

FIGURE 4.1. The graph used for part 1 of the user study.

Part 1 of the survey involved using the system on the graph given in figure 4.1 using the particle

system both with and without the threshold. As described in Section 3.2 the threshold restricts

4.1. USERSTUDY 50

the movement of the user in order to better maintain the skeletal constraints. The threshold

used for the study was determined experimentally. We used a threshold that provided a bal-

ance between restricting the user’s movements too much and allowing excessive violation of

constraints.

The purpose of part 1 of the survey was to evaluate the particle system in terms of its ability

to simulate the skeletal metaphor. The aim of this part is to answer research question (a) while

giving a small insight into question (b). This part was also designed to compare the particle

system using the threshold to the particle system not using the threshold.

The participants of the study were asked to provide a rating for each technique on an integer

scale of 1 (lowest) to 5 (highest), for the following four questions:

A. How well does the system preserve the structure of the skeleton? (Structure is pre-

served when the lengths of "bones" are at their original length)

B. When dragging around a node, did the bones react in the way you expected (How

intuitive is the system)?

C. While moving a node was it easy to maintain your understanding of the structure of

the graph?

D. Was the system fun to use?

The participants were advised to give ratings in such a way that the techniques could be mea-

sured individually against the scale as well as comparatively. There was also space for addi-

tional comments after each question so that the user could further explain their choice if desired.

For the remaining questions in this part, the participants were asked to give a written response

explaining their answer:

E. Which aspect of each system did you like the most, Why?

F. Which aspect of each system did you like the least, Why?

G. Based on your experience with the two systems what do you consider to be most

4.1. USERSTUDY 51

important, maintaining the skeletal structure or allowing freedom of movement? Explain.

H. Do you have any further comments to make on any aspect of these systems or this

section?

Question A is aimed at measuring and comparing the ability of each technique to satisfy the

bone-length constraints of the skeleton. This is the most important factor in terms of measuring

a techniques ability to simulate a skeleton. It is the nature of the skeleton that bone-lengths

remain constant. If this is not the case then it could be argued that the system is only loosely

simulating a skeleton. The visual cues provided in our system mean that the user will always

know if the bone-length constraints are satisfied. This will ensure that the answers provided

by participants in this question will be an accurate representation of the level of constraint

satisfaction.

Question B tries to identify which technique users find most natural or intuitive. Due to the fact

that a skeleton is a natural structure that humans are used to manipulating, it is expected that a

system that simulates the skeleton effectively would be natural and intuitive. If the system is

found not to be intuitive it could mean that it is not simulating the skeleton very well. However,

it could also mean that the skeletal metaphor is not as natural and intuitive as expected when

applied to graph interaction.

Question C is present to measure the success of each algorithm in preserving the user’s mental

map. The mental map is likely to be maintained if only small, intuitive changes are made. If

large, drastic changes are made then the user will be unable to identify how the current drawing

relates to previous drawings and the benefits of interaction are reduced. A system that simulates

the skeleton well is expected to maintain the mental map as changes are restricted to maintain

similar aesthetic properties.

Question D has been included as an informal way to ascertain the algorithm of general prefer-

ence to the participants and also to determine how enjoyable each technique is to use. Questions

E & F have been included to record what the users consider to be the strengths and weaknesses

4.1. USERSTUDY 52

of the two methods. The purpose of Question G is to determine whether restrictions placed

upon the movement of the graph in order to better satisfy the skeletal constraints, helps or hin-

ders their ability to understand the graph structure. Question H is for any extra comments the

participant may have.

FIGURE 4.2. The graph used for part 2 of the user study.

Part 2 of the survey involved using the system on the graph given in figure 4.2 using the parti-

cle system, the force-based technique and the particle system with angular constraints. While

part 1 was designed to give some insight into the usefulness of the skeletal metaphor, only

by comparing the particle system with a non-skeletal technique can we completely identify

it’s strengths and weaknesses. Evaluating the new skeletal technique against a traditional non-

skeletal technique is the purpose of part 2 of the survey. The particle system simulates the

skeleton while the force-based method is non-skeletal. The particle system with angular con-

straints is a skeletal animation technique but also has force-based elements. These techniques

will be evaluated in terms of the users understanding of the graph structure. The aim of this

part is to answer research question (b).

Once again, the participants of the study were asked to provide a rating for each technique on a

scale of 1 to 5, for the first four questions. Space for additional comments was again provided.

4.1. USERSTUDY 53

The questions were:

A. How easy was it to obtain a picture in which you could easily understand the structure

of the graph?

B. While moving the structure around, did the graph react in the way you expected (How

intuitive is the system)?

C. While moving a node was it easy to maintain your understanding of the structure of

the graph?

D. Was the system fun to use?

The questions requiring a written response were:

E. Which aspect of each system did you like the most, Why?

F. Which aspect of each system did you like the least, Why?

G. Do you have any further comments to make on any aspect of these systems or this

section?

Question A attempts to capture a general notion of the skeletal metaphor’s effect on a user’s

ability to manipulate the graph into a readable form. The main aim of graph interaction is to

allow the user to easily produce a drawing that they can understand. Therefore, the performance

of a technique in this question give a strong indication of how useful it is for interaction tasks.

The remaining questions have a similar purpose to those in part 1, this time comparing skeletal

and non-skeletal techniques.

The purpose of part 3 of the survey is to evaluate the new skeletal technique against a non-

skeletal technique using graphs from specific domains. These will be evaluated similarly to

part 2. The aim of this part is to answer research questions (c) and (d).

4.1. USERSTUDY 54

FIGURE 4.3. The web graph used for part 3 of the user study.

FIGURE 4.4. The hex game graph used for part 3 of the user study.

The questions in this section were very similar to those in part 2, this time comparing the tech-

niques in the two domains and also comparing the two domains (See Appendix B on page 88).

The first graph that was used is shown in figure 4.3. It is a graph of high degree similar to a

possible web graph. High performance of a technique on this graph gives a small indication

that the technique will be useful for high degree graphs such as web graphs.

The second graph used is that of a hex game graph [Rijswijck02, Murray03 ]. It is shown in

figure 4.4.

4.2. RESULTS& D ISCUSSION 55

4.2. Results & Discussion

Part 1 - Comparison of Skeletal Techniques

The results from part 1 of the user study are shown as a column graph in figure 4.5. To obtain

the value for each method and question we took the average of the responses provided. From

the graph we can easily see the performance of each method against the scale and we can also

easily compare the methods.

From the results in question A it can be seen that the threshold method is more effective at

preserving edge lengths than the non-threshold method. This strong trend towards the threshold

method as a better preserver of edge-lengths was to be expected, as the non-threshold method

allows the user total freedom in stretching bones. However, the scores were still towards the

high end of the scale, indicating that edge lengths were well preserved. This is because it

is only in cases were the fixed nodes are chosen such that it is impossible to satisfy edge-

lengths, that violations will occur. In normal cases, all constraints will be satisfied. As the

threshold method prevents some cases where it is impossible to satisfy edge constraints, it is

better at preserving edges lengths. Some participants did not indicate any difference between

the threshold and non-threshold methods. This may be because they did not create or attempt

to create configurations in which the violation threshold would be exceeded. The threshold

chosen still allowed some violations. Perhaps with a lower threshold, the difference between

the methods would have been more obvious. However, this would have limited the usefulness of

the method, as movement would feel too restrictive. In hindsight, it would have been beneficial

to construct more study cases with a wider variety of circumstances. Based on the results

achieved, it seems fair to claim that both methods are very effective in satisfying the skeletal

constraints and therefore in simulating the skeletal metaphor.

The results in question B indicate that the method without a threshold is more intuitive. This

may be due to participants finding the threshold method restrictive. Also, it may not have been

obvious from the visualisation when the threshold was reached. Even though the threshold


method achieved a lower score, it still achieved a fairly high rating on the scale. Therefore, it is

reasonable to suggest that both methods are highly intuitive and feel natural to use. This shows

that these methods are effective in aiding the understanding of the graph and also shows that

the skeletal metaphor is useful for graph interaction.

From the results in Question C we can see that the threshold method received a slightly better

rating for preserving the user’s mental map. This is likely due to the fact that the movement is

more restrictive and therefore it is more difficult to make drastic changes to the graph. However,

the difference is fairly small and it would be irresponsible to make any drastic conclusions about

the difference between the techniques at maintaining the mental map. Of some concern is that

both methods are somewhat low on the scale. This indicates that too many changes could be

occurring to the graph as nodes are dragged around making it difficult to see how the current

state of the graph relates to previous drawings. While the ability of the skeletal metaphor to

preserve the mental map is not terrible, it does appear to be its weakest aspect.

As can be seen from the results for Question D, both techniques are quite fun to use. The results

also indicate that the users did not have a preference for either system.

The remaining questions produced a wide range of comments. Maintaining the skeletal struc-

ture was considered by the majority of the participants to be the more important factor in aiding

understanding of the graph. Many argued that by maintaining the skeletal structure we are more

likely to produce an aesthetically pleasing drawing, therefore making it easier to understand the

graph structure. Also, if the lengths of edges were representative of edge weights specific to the

graph, this is especially important for understanding the graph structure. Some also suggested

that when the graph was released from a highly stretched state by releasing nodes, the graph

would change drastically. This would result in a loss of the users mental map. If we maintain

the skeletal structure these highly stretched states are less likely to occur. Also, highly stretched

states are more likely to have poor aesthetic properties. Some commented that the threshold

added more realism to the skeleton structure as it would not be possible to stretch an actual

skeleton.


Some of the participants preferred freedom of movement as a more important aspect. This al-

lows them to place the graph in any possible configuration and by unfixing nodes, they can also

ensure that the skeletal structure is satisfied. One user commented that relaxing the constraints

is beneficial in an interactive environment. This makes sense, as for a single graph drawing

we do want all constraints to be satisfied. However, in an interactive situation, it seems accept-

able to temporarily allow some violations while a configuration is being formed. Once we have

achieved a configuration which we can understand, satisfying any small violations will increase

the aesthetics of the drawing and increase our understanding of the graph even further. Also,

for graph drawings subject to edge length constraints, there may be no embedding that satisfies

the constraints [Borcea02]. So we must allow some violation of constraints or we will limit the

areas in which our techniques will be useful.

Based on these comments, we conclude that maintaining the skeletal structure is the most

important characteristic as it helps to provide an aesthetically pleasing drawing that is easy

to understand. However, we also believe that the skeletal structure can be maintained in an

environment allowing complete freedom of movement, as long as the user makes an attempt

to avoid configurations that highly stretch the graph, by paying attention to the visual cues and

also keeping the subset of fixed nodes fairly minimal. Using this technique with that mind-

set would allow the user to configure the graph in any possible way while maintaining the

skeletal structure and therefore continually achieving an aesthetically pleasing drawing, that

feels realistic and natural, therefore increasing the users understanding of the graph.

Both systems have shown to be useful for simulating the skeletal metaphor. Whether this aids

the users understanding of the graph will be evaluated further in part 2.

One respondent commented that the structure responded quickly to user input in both tech-

niques. This further supports our claim that the particle system is fast and helps to provide

smooth, natural motion.


Part 2 - Comparison of Skeletal and Non-Skeletal Techniques

The results from part 2 of the user study are shown as a column graph in figure 4.6. To obtain

the value for each method and question we took the average of the responses provided. From

the graph we can easily see the performance of each method against the scale and we can also

easily compare the methods.

The first thing that we notice from the graph is that the skeletal technique with angular con-

straints outperforms the other techniques significantly in all questions. This indicates that the

skeletal technique with angular constraints is the best at aiding a user’s understanding of a

graph.

For question A, the skeletal technique slightly outperforms the force-directed technique, but

both techniques have high enough ratings to indicate that they are useful for quickly and easily

obtaining a readable graph. The rating of the skeletal technique with angular constraints is very

high showing that it is extremely useful for this purpose.

The results for question B indicate that the force-directed method is the least intuitive. Both

of the skeletal techniques are more intuitive. This supports our claim that the natural motion

provided by the skeletal techniques would be intuitive because people regularly manipulate a

skeletal structure. The skeletal structure with angular constraints is shown to be even more

intuitive. This is due to the fact that it provides added realism as the movement of joints in a

normal skeleton have restrictions on the range of movement.

In question C, the force-directed algorithm slightly outperforms the standard skeletal technique

but not by a significant amount. Both methods are shown to be fairly effective in preserving the

users mental map.

The users also have a preference for the angular constraints method as shown in question D.

They are more likely to enjoy using the system if it helps them to understand the graph.


The remainder of the questions requiring descriptive answers further supported the results ob-

tained previously but failed to provide much further insight into the usefulness of each tech-

nique.

In summary, the results show that the skeletal technique is better at assisting a user to under-

stand the graph structure than the non-skeletal technique. Adding angular constraints to the

skeletal metaphor has the effect of providing good angular resolution which increases its use-

fulness significantly. Therefore, we propose the following revised version of the definition of a

skeleton:

A skeleton is a set of joints (J) and a set of bones (B). Each jointj ∈ J is connected by a set

of bonesBj ⊆ B to set of jointsjADJ ⊆ J, |jADJ | > 0.

There exist constraints on the skeleton that should be preserved:

• The lengthlb of each boneb ∈ B should not change.

• Given a set of jointsF ⊆ J , termed the fixed point set, each jointj ∈ F should remain

at its initial location in Euclidean space.

• For each pair of jointsji, jj ∈ J not connected by a boneb ∈ B, the Euclidean distance

betweenji andjj must remain greater than some thresholdt.

The skeletal method with angular constraints does have the disadvantage that the extra work in

enforcing the angular constraints makes this method a bit slower.

Our results show that the skeletal metaphor is useful for graph interaction as it significantly

helps the user to comprehend the graph.


Part 3 - Specific Domain Graphs

The results for the web graph are given in figure 4.7. The standard skeletal technique performs

poorly on this graph. This is due to the fact that the graph was of high degree and edges tended

to bunch together around the node from which they emanate. With the angular constraints in

place however, we once again get very good results.

The results for the hex graph are given in figure 4.8. The standard skeletal technique performs

better on this graph. However, it is still slightly below the performance of the force-directed

algorithm.

Based on the ratings, all techniques perform slightly lower on these graphs than on the general

graphs. This could be because these graphs are more difficult to understand or because these

domains are less suited to the skeletal metaphor.

Figures 4.9 & 4.10 show a direct comparison between the web and hex graphs on the two

skeletal techniques. For the standard skeletal technique, the hex graph performs better largely

due to the poor performance on the web graph as described above. For the skeletal technique

with angular constraints, the web graph performs better because with good angular resolution

it is an easier graph to understand.

Our results indicate that the skeletal metaphor could be useful for graph interaction in specific

domains and that web-graphs are more highly suited to the skeletal metaphor than the hex game

graph. However, this part of our evaluation was somewhat limited. The evaluation of graph in-

teraction techniques in specific domains would give greater understanding by having expert

analysts complete specific graph interaction tasks with labelled graphs representing real data.

There are also many more domains in which these techniques could be evaluated. The skeletal

metaphor is potentially useful in any application where it is crucial to visualize structural in-

formation as graphs. Further evaluation of the use of the skeletal metaphor in specific domains

remains an area for future work.


Summary

From the results of our study we were able to confirm our hypotheses that new techniques will

have a positive effect on the simulation of the skeletal metaphor and that the use of the skeletal

metaphor will have a positive effect on the user’s ability to understand the graph structure.

4.3. COMPARISON OF THEPARTICLE SYSTEM WITH PREVIOUS TECHNIQUES 62

4.3. Comparison of the Particle System with Previous Techniques

Part of the motivation for this thesis was that the techniques used by Merrick were not highly

successful in simulating the skeletal metaphor or increasing the user’s understanding of the

graph structure [Merrick02 ]. Therefore we will demonstrate that we have improved upon the

previous work by showing that the particle system is more effective in achieving these aims.

The CCD & force-directed approaches resulted in a stretchiness of bones. This violates the

prescribed skeletal metaphor and damages the perception that a physical entity is being manip-

ulated. The CCD method lead to a stretch in bones as a result of a cumulative error. In the

particle system, any small errors will converge over subsequent frames while more iterations

are applied. The force-directed technique resulted in very stretchy bones as a result of the naive

approach in simulating the skeleton. The parameters were chosen in an attempt to create ex-

tremely stiff springs but it was not as effective as expected, as it is still possible for the springs

to stretch. In the particle system, the edges are modelled as infinitely stiff springs, so their ideal

length is reached instantly, minimising stretching.

Generally, in the particle system, the only time when edge-length constraints remain unsatis-

fied, is when the set of fixed-nodes make it impossible to satisfy both fixed node constraints

and edge-length constraints, as described in Section 3.2 (See figure 3.12). The constraint solver

ensures that the edge length constraint is always satisfied provided the set of fixed joints allows

this to be possible. Edges that have violated the length constraint due to the nature of the

fixed joints, will return to their original length once possible. The violation of bone-length

constraints can be further reduced by using the threshold mechanism.

The CCD method became very slow in some cases in the previous system. For a graph inter-

action system to be useful it must be useable in real-time, which was not always true of this

approach. The particle system is designed to be very fast in order to be useful for interactive

purpose. Verlet integration is a fast method of simulating particle motion. Satisfying environ-

mental constraints by projection is also very fast as is satisfying edge-length constraints. It also


allows for a speed versus accuracy trade off by adjusting the number of iterations per frame,

without accumulating visually obvious errors. The square-root approximation can also be used

to further speed up the system’s response to user actions.

The motion should also remain smooth and natural as is the intention of the skeletal metaphor.

The NLP method provided jumpy motion and this limited its usefulness significantly. Con-

versely, the motion provided by the physics system is smooth and natural, helping the user to

quickly and easily understand the graph structure.

None of the previous methods implemented any form of angular constraints. Angular con-

straints not only add a greater level of realism to the skeletal structure, they also greatly improve

the angular resolution of the resultant graph drawing. The results of our user study support this

and also show that the angular constraints significantly improve the users ability to understand

the graph. By providing a technique with angular constraints we provide our system with an

additional advantage over the previous techniques.

While Merrick applied the skeleton to only part of the graph, we have modelled the entire graph

as a skeleton. As a result our entire graph represents a physical structure while in Merrick’s

system this was only partially the case. Also, Merrick required preprocessing to determine the

skeleton and also required additional processing to determine the movements of non-skeletal

nodes.

The physics system provides the advantages of all three previous methods while also providing

some additional advantages. It also has none of the disadvantages of the previous techniques.

It provides satisfaction of the constraints of a skeleton while ensuring the naturalness of the

skeletal motion and fast speed of execution. The particle system provides a fast, accurate

simulation of the skeletal metaphor that increases the users ability to understand the graph

structure.


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CHAPTER 5

Conclusion

This chapter concludes this report by summarising the original goals, how they were achieved,

and what conclusions can be drawn from them. It also covers some directions for future re-

search work.

5.1. Research Contribution

In Chapter 1 of this thesis, we raised the issue that, although much research has been done in

the area of graph visualisation, there remains a need for investigation into techniques for graph

interaction that will complement the visualisation and increase the user’s understanding of the

graph.

We have introduced the particle system as a method for skeletal animation that can be used

for graph interaction tasks. As the skeleton is a natural structure, and this method was de-

signed for interactive use with the dragging of joints in mind, it is quite suitable for providing a

graph interaction environment that feels natural and intuitive. We made the method even more

more suitable for this purpose by adding features specifically designed to aid graph interaction

such as fixed node constraints, an error threshold and visual cues. The adapted particle sys-

tem improves upon the previous techniques used for skeletal animation in graph interaction.

This shows that our new technique has had a positive effect on the simulation of the skeletal

metaphor in graph interaction.

70

5.1. RESEARCHCONTRIBUTION 71

Previous evaluation of the usefulness of the skeletal metaphor was inconclusive due to the

limited scope of the survey undertaken. We have demonstrated it’s usefulness through a more

comprehensive user study into the benefits of the skeletal metaphor and the different methods

used. The results clearly show that the skeletal techniques, especially with angular constraints,

are much more effective at providing a mechanism for which the user can easily understand the

graph. The more comprehensive user study that we have undertaken produces solid evidence

of the usefulness of the skeletal metaphor.

We have also provided some evaluation of the skeletal metaphor in the domains of web graphs

and the hex game graph. Graph drawing and interaction techniques designed solely for general

graphs are often not useful once they are applied to domain-specific graphs [Purchase00]. By

evaluating the particle system and the skeletal metaphor in these domains we give a greater

insight into how useful it truly is.

We have also added to the definition of the skeletal metaphor to further improve its utility. We

have added the condition of angular constraints. Our user study confirms that this is advanta-

geous.

The satisfaction of the constraints of a skeleton while ensuring the naturalness of the skeletal

motion and fast speed of execution, provides a fast, accurate simulation of the skeletal metaphor

that increases the user’s ability to understand the graph structure.

This work adds to the field of graph interaction by providing a natural system of navigating a

graph with the ability to increase the user’s understandability of the graph structure, much more

so than alternative methods such as a force-directed algorithm. This shows that the use of the

skeletal metaphor has a positive effect on the user’s ability to understand the graph structure.

5.2. FUTURE WORK 72

5.2. Future Work

The only method of interaction with the system presented in this thesis, and the system de-

scribed by Merrick [Merrick02 ] was through the use of a standard two dimensional mouse.

Other devices for interaction with three dimensional displays could provide a more natural in-

terface between the user and the system, complementing the skeletal metaphor. In particular,

haptic interaction devices not only provide a method for interaction in three dimensions but use

force feedback techniques to give the user an actual physical feel of what they are interacting

with. The benefits of haptic devices have been explored [Hinckley96] and these devices have

been shown to be quite useful for a range of applications. It remains to be determined, when

using a graph interaction system utilising the skeletal metaphor, what additional benefits would

be provided by using three dimensional input devices such as haptic devices. One of many

possible benefits would be to allow the user to more easily determine when a threshold has

been reached through the use of force-feedback. We could not answer this question here, as the

complexities involved in using haptic devices, and their high cost, placed this task outside the

scope of an Honours thesis, unless it was the primary focus.

The work of Ware and Franck has shown [Ware94] that many of the benefits of three dimen-

sional visualisations are further realised when 3D display devices such as a stereoscopic, hand-

tracked 3D display are used. The use of the skeleton metaphor with such devices is therefore

likely to see even more benefits.

Our evaluation of the different techniques used gave us a strong indication of how useful the

skeletal metaphor is for general graphs. However, our evaluation in specific domains was some-

what limited as it didn’t involve completion of specific graph interaction tasks with labelled

graphs representing real data. It would also be beneficial to have expert analysts from that

field using the system. There are also many more domains in which these techniques could be

evaluated. The skeletal metaphor is potentially useful in any application where it is crucial to

visualize structural information as graphs. An evaluation of the skeletal metaphor in any of the

areas would provide a significant research contribution.

5.2. FUTURE WORK 73

An alternative to a graph interaction environment allowing the user to make changes, would be

software that animates a drawing of a graph. This animation would show many different views

of the graph independent of user input. The aim of this approach would be to smoothly animate

between configurations of the graph while maintaining aesthetic properties in order to increase

the users understanding of the graph while preserving the user’s mental map. This approach

could achieve all the advantages of graph interaction without relying on actions from the user

to produce readable drawings. It would also be possible to use skeletal animation techniques

for this approach towards aiding a user’s understanding of a graph.

Another possibly would be to use additional data associated with the graph to determine phys-

ical aspects of the graph. Some parameters that could depend on the input graph could be bone

thickness, length and stretching threshold. These factors would provide a physical structure

that more closely resembles the graph data. It remains to be seen whether this would increase

the user’s understanding of the graph but is definitely an area worth exploring.

In our system it was possible for nodes and edges to pass through each other. Collision detection

involves determining if two objects intersect each other. Adding collision detection to a skeletal

graph interaction system would provide further realism, as it could be used to prevent nodes

and edges from intersecting. This may increase the ability of the user to understand the graph,

and is therefore an area worth investigating.

This thesis has not only contributed to the field of graph interaction but have also opened up

many possibilities for future work in this area. Such research is likely to provide an important

contribution in the fields of both graph visualisation and human computer interaction in the

future.

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APPENDIX A

System Screenshots

FIGURE A.1. Dragging nodes using our dynamics-based skeletal graph inter-

action system. By ensuring the constraints are satisfied, it quickly produces a

much better graph drawing and allows us to improve our understanding of the

graph structure.

81

A. SYSTEM SCREENSHOTS 82

FIGURE A.2. Top: A highly stretched graph using the technique without a

threshold. Bottom: A graph stretched as far as possible using the technique

with a threshold.


FIGURE A.3. The graph in part 2 of the study using the force-directed algorithm.


FIGURE A.4. The graph in part 2 of the study using the standard skeletal technique.


FIGURE A.5. The graph in part 2 of the study using the skeletal technique with

angular constraints.


FIGURE A.6. The web graph in part 3 of the study. Top: using the standard

skeletal technique results in bunching up of nodes. Bottom: using the skeletal

technique with angular constraints gives a much better angular resolution.


FIGURE A.7. The articulated figure from figure 3.1 represented by a graph in

our graph interaction environment.

APPENDIX B

User Survey Forms

The following seven pages contain the forms given to participants of the surveys detailed in

Section 4.1 on page 48 of this thesis.

Pages 89 - 91 comprise the first part of the survey. It compares the skeletal technique with a

threshold to the skeletal technique without a threshold in terms of their ability to simulate the

skeletal metaphor.

Pages 92 - 93 comprise part 2 of the survey. It compares the force-directed algorithm, stan-

dard skeletal method and the skeletal technique with angular constraints (labelled as "hybrid").

These are evaluated in terms of their ability to increase the user’s understanding of the graph.

Pages 94 - 95 comprise part 3 of the survey, which compares the different techniques of part 2

on the specific domain graphs, including a web graph and the hex game graph.

88

89

School of Information Technologies

Faculty of Science College of Sciences and Technology

Madsen Building, F09 Telephone +61 2 9351 3776

Graph Interaction User Study Questionnaire Note: By answering these survey questions you are agreeing to participate in this study. PART 1 – Comparison of Skeletal Techniques using General Graphs For this part of the survey, we ask that you use the Skeletal Graph software and provide some brief feedback on your thoughts on two of the implemented techniques. Please run each of these commands in turn from the command line and evaluate each technique. 1 Skeletal System without thresholding (NO THRESHOLD) java -Xmx512m SkeletonGraphApp part1.cgf –s 2 Skeletal System with thresholding (THRESHOLD) java -Xmx512m SkeletonGraphApp part1.cgf –s -th A series of questions are provided below. Please circle on the given scale of 1 (LOW) to 5 (HIGH) how you feel the two systems rated. A section for extra comments is included after each question, as well as a few sections for general comments at the end of this section. A) How well does the system preserve the structure of the skeleton? (Structure is preserved when the lengths of “bones” are at their original length) NO THRESHOLD 1 2 3 4 5 THRESHOLD 1 2 3 4 5

(not well preserved) (well preserved) Additional comments:

90

B) When dragging around a node, did the bones react in the way you expected (How intuitive is the system)? NO THRESHOLD 1 2 3 4 5 THRESHOLD 1 2 3 4 5

(least intuitive) (most intuitive) Additional comments: C) While moving a node was it easy to maintain your understanding of the structure of the graph? NO THRESHOLD 1 2 3 4 5 THRESHOLD 1 2 3 4 5

(easiest) (hardest) Additional comments: D) Was the system fun to use? NO THRESHOLD 1 2 3 4 5 THRESHOLD 1 2 3 4 5

(least fun) (most fun) Additional comments: E) Which aspect of each system did you like the most, Why? NO THRESHOLD THRESHOLD F) Which aspect of each system did you like the least, Why? NO THRESHOLD THRESHOLD

91

G) Based on your experience with the two systems what do you consider to be most important, maintaining the skeletal structure or allowing freedom of movement? Explain. H) Do you have any further comments to make on any aspect of these systems or this section?

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PART 2 – Comparison of Skeletal and Non-Skeletal Techniques For this survey, we ask that you use the Skeletal Graph software and provide some brief feedback on your thoughts on three of the implemented techniques. Please run each of these commands in turn from the command line and evaluate each technique. 1 Force-Directed Approach (FORCE) java -Xmx512m SkeletonGraphApp part2.cgf –f 2 Skeletal Approach (SKELETAL) java -Xmx512m SkeletonGraphApp part2.cgf –s 3 A Combination of the two above approaches (HYBRID) java -Xmx512m SkeletonGraphApp part2.cgf –sf Four questions are provided below. Please circle on the given scale of 1 (LOW) to 5 (HIGH) how you feel each of the three systems rated. A section for extra comments is included after each question, as well as a larger section for general comments at the end of this section. A) How easy was it to obtain a picture in which you could easily understand the structure of the graph? FORCE 1 2 3 4 5 SKELETAL 1 2 3 4 5 HYBRID 1 2 3 4 5

(very hard) (very easy) Additional comments: B) While moving the structure around, did the graph react in the way you expected (How intuitive is the system)? FORCE 1 2 3 4 5 SKELETAL 1 2 3 4 5 HYBRID 1 2 3 4 5

(least intuitive) (most intuitive) Additional comments:

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C) While moving a node was it easy to maintain your understanding of the structure of the graph? FORCE 1 2 3 4 5 SKELETAL 1 2 3 4 5 HYBRID 1 2 3 4 5 (very hard) (very easy) Additional comments: D) Was the system fun to use? FORCE 1 2 3 4 5 SKELETAL 1 2 3 4 5 HYBRID 1 2 3 4 5

(not at all) (definitely) Additional comments: E) Which aspect of each system did you like the most, Why? FORCE SKELETAL HYBRID F) Which aspect of each system did you like the least, Why? FORCE SKELETAL HYBRID G) Do you have any further comments to make on any aspect of these systems or this section?

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PART 3 – Specific Domain Graphs For this survey, we ask that you use the Skeletal Graph software and provide some brief feedback on your thoughts on the three implemented techniques on specific domain graphs. Please run each of these commands in turn from the command line and evaluate each technique. 1 Web Graphs (WEB) java -Xmx512m SkeletonGraphApp web.cgf –f java -Xmx512m SkeletonGraphApp web.cgf –s java -Xmx512m SkeletonGraphApp web.cgf –sf 2 Hex Graph (HEX) java -Xmx512m SkeletonGraphApp hex.cgf –f java -Xmx512m SkeletonGraphApp hex.cgf –s java -Xmx512m SkeletonGraphApp hex.cgf –sf Four questions are provided below. Please circle on the given scale of 1 (LOW) to 5 (HIGH) how you feel each of the three systems rated. A section for extra comments is included after each question, as well as a larger section for general comments at the end of this section. A) How easy was it to obtain a picture in which you could easily understand the structure of the graph? WEB FORCE 1 2 3 4 5 SKELETAL 1 2 3 4 5 HYBRID 1 2 3 4 5 HEX FORCE 1 2 3 4 5 SKELETAL 1 2 3 4 5 HYBRID 1 2 3 4 5

(very hard) (very easy) Additional comments:

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B) While moving the structure around, did the graph react in the way you expected (How intuitive is the system)? WEB FORCE 1 2 3 4 5 SKELETAL 1 2 3 4 5 HYBRID 1 2 3 4 5 HEX FORCE 1 2 3 4 5 SKELETAL 1 2 3 4 5 HYBRID 1 2 3 4 5

(least intuitive) (most intuitive) Additional comments: C) While moving a node was it easy to maintain your understanding of the structure of the graph? WEB FORCE 1 2 3 4 5 SKELETAL 1 2 3 4 5 HYBRID 1 2 3 4 5 HEX FORCE 1 2 3 4 5 SKELETAL 1 2 3 4 5 HYBRID 1 2 3 4 5 (very hard) (very easy) Additional comments: D) Do you have any further comments to make on any aspect of these systems or this section?

graph interaction through dynamics-based skeletal animation · graph interaction through...

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