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Algorithmic Foundation of
Multi-Scale Spatial
Representation
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Algorithmic Foundation of
Multi-Scale Spatial
Representation
Zhilin Li
Department of Land Surveying and Geo-Informatics
The Hong Kong Polytechnic University
CRC Press is an imprint of the
Taylor & Francis Group, an informa business
Boca Raton London New York
9072_C000.fm Page iii Friday, September 8, 2006 11:20 AM

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Library of Congress Cataloging-in-Publication Data
Li, Zhilin.
Algorithmic foundation of multi-scale spatial representation / by Zhilin Li.
p. cm.
ISBN 0-8493-9072-9
1. Geographic information systems. 2. Algorithms. I. Title.
G70.212L487 2006
526.801’5181--dc22 2006045503
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9072_C000.fm Page iv Friday, September 8, 2006 11:20 AM

Preface
The representation of spatial data, or simply spatial representation, can be made in
various forms, that is, graphics or nongraphics. Graphic representation includes
maps, images, drawings, diagrams, movies, and animation, either 2-dimensional (2-
D) or 3-D. Nongraphical representation includes audio, text, digital numbers, and
so on. Graphic representation is still the more popular alternative in geo-information
science and thus will be the focus of this book. Spatial data can also be represented
at different scales, leading to the issues of multi-scale representation, which is the
topic of this book.
Multi-scale representation of spatial data is termed multi-scale spatial represen-
tation in this context. It is simply termed multiple representation in some literature.
Multi-scale representation is a traditional topic in cartography, geography, and all
other geo-sciences. It has also become one of the most important issues in geo-
information science.
Among the many forms of graphic representation of spatial data, maps are still
the most effective and popular means. The multi-scale issue related to map produc-
tion is the derivation of maps at a smaller scale than those at a larger scale, which
is a process traditionally termed generalization. Map generalization is also required
for real-time zooming in and out in a geographic information system. Therefore, the
emphasis of this book is on map generalization.
Generalization is a process of abstraction. The terrain features represented on
maps at smaller scales are the abstractive representations of those at larger scales.
For example, at 1:1,000,000, a city is represented by an abstractive symbol — a
point symbol — while at 1:1,000, every street is represented by double lines. Perhaps
the highest level of generalization was achieved by Sir Isaac Newton, who repre-
sented any planet by a point in his famous “Law of Gravitation.” This means that
generalization is not only an issue related to the representation of spatial data but
also to the modeling of spatial processes.
Although map generalization is a traditional topic, the generalization of maps
in a digital environment, simply digital map generalization, is a result of computer-
ization in recent decades. Although it is difficult to locate the exact date at which
research on digital map generalization started, three very important publications
appeared in 1966: Tobler (1966), Töpfer and Pillewizer (1966), and Perkal (1966).
From 1966 to the early 1980s, research on digital map generalization was rather
isolated. In the 1970s, efforts were made to develop algorithms for line features. In
the early 1980s, those line algorithms were evaluated and algorithms for area features
were investigated. In the later 1980s, more attempts were made to develop strategies
and rule-based systems.
As early as the 1980s, multi-scale representation was recognized as a funda-
mental issue in spatial data handling. In 1983, a small group of leading scientists in
9072_C000.fm Page v Friday, September 8, 2006 11:20 AM

the United States was gathered by NASA to define critical research areas in spatial
data handling, and multi-scale representation was identified as one of them (Marble,
1984). Since then, this topic has become part of the international research agenda
in spatial information sciences. Indeed, many researchers have advocated research
on this topic (Abler, 1987; Rhind, 1988; Müller, 1990). As a result, the topic has
attracted the attention of researchers in relevant disciplines, and a number of research
projects on this topic have been initiated internationally. The importance of multi-
scale representation was highlighted by an NCGIA (National Center for Geographic
Information and Analysis) initiative under the title “Multiple Representation” in
1989.
Since the early 1990s, multi-scale representation (or generalization) has been a
popular research topic, with the widespread use of the Geographic Information
System (GIS), which integrates multi-scale, multi-source spatial data. Multi-scale
representation has also become an important topic in computational geometry and
computing. The International Cartographic Association (ICA) established a working
group on automated map generalization in 1991, which became a commission in
2001. This commission, chaired by Professor Robert Weibel (University of Zurich)
until 2003, has made significant advancement. In 2000, the International Society for
Photogrammetry and Remote Sensing (ISPRS) also established a working group on
multi-scale representation. Special sessions on this topic have been organized at
conferences, and special issues have been published in journals. Indeed, multi-scale
representation gained a boost in the later 1990s with the funding of the AGENT
project by the European Union.
Forty years have passed since the publication of the first three important papers
in 1966. An age of 40 has special meaning in Chinese: enlightenment. It means one
will rarely be puzzled after this age. Accordingly, a discipline after 40 years of
development should be well established. This maturity is signified by the growing
number of publications in the discipline. We have seen two edited books (resulting
from expert meetings) (Buttenfield and McMaster, 1991; Müller et al., 1995) and a
Ph.D. thesis (Joa
~
o, 1998) published by formal publishers. We have also seen a
resource booklet (McMaster and Shea, 1992) published by The Association of
American Geographers. Unfortunately, no book has been authored to systematically
address different aspects of the discipline. The author of this book feels it is extremely
difficult to cover all the aspects of multi-scale spatial representation in a single book
at the present time. Therefore, a compromise is struck here to provide a book only
addressing the mathematical basis of the discipline, more precisely the algorithmic
foundation.
In multi-scale spatial representation, various types of transformations are under-
taken, that is, geometric, thematic, and topological transformations. In this book,
only geometric transformations are discussed. Indeed, this book covers the low-level
algorithms available for the multi-scale representations of different types of spatial
features, that is, point clusters, individual line features, a class of line features
(contours, transportation networks, and hydrographic networks), individual area
features, and a class of area features. In addition, algorithms for multi-scale repre-
sentation of 3-D surfaces and 3-D features are briefly discussed.
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This book consists of 12 chapters. Chapter 1 is the introduction, providing an
overview of the contents. Chapters 2 and 3 provide some mathematical and theo-
retical foundations to facilitate the discussions in the later chapters. Chapter 4
describes the algorithms for a class of point features (or point clusters). Chapters 5
through 7 are devoted to algorithms for individual line features — Chapter 5 for
reduction of data points, Chapter 6 for smoothing (filtering), and Chapter 7 for scale-
driven generalization. Chapter 8 discusses the algorithms for a set of line features,
namely, contour set, river network, and transportation network. Chapter 9 addresses
the algorithms for individual area features, and Chapter 10 discusses a set of area
features. Chapter 11 covers the algorithms for various displacement operations. Chap-
ter 12 provides brief coverage of the algorithms for 3-D surfaces and 3-D features.
Indeed, this book provides comprehensive coverage of the algorithmic founda-
tion of multi-scale representation of spatial data. It is written at a medium level of
technical detail to make the concepts easy to understand. An attempt is also made
to use illustrations, to make the working principles of algorithms intuitive.
REFERENCES
Abler, R., The National Science Foundation National Center for Geographc Information and
Analysis. International Journal of Geographical Information Systems, 1(4), 303–326,
1987.
Buttenfield, B. P. and McMaster R. B., Eds., Map Generalization: Making Rules for Knowl-
edge Representation, Longman Scientific and Technical, London, 1991.
Joa
~
o, E., Causes and Consequences of Map Generalization. CRC Press, Boca Raton, 1998.
Marble, D., Geographic information systems: an overview. Proceedings of Pecora 9, Sioux
Falls, SD, 1984, 18–24. Reprinted in Peuquet, D. J. and Marble, D. F. Eds., Intro-
ductory Readings in Geographic Information Systems, Taylor & Francis, London,
1990, pp. 8–17.
McMaster, R. B. and Shea, K. S., Generalization in Digital Cartography, Association of
American Geographers, Washington, DC, 1992.
Müller, J.-C., Rule based generalization: potentials and impediments, in Proceedings of 4th
International Symposium on Spatial Data Handling, International Geographic Union,
1990, pp. 317–334.
Müller, J. -C., Lagrange, J. P., and Weibel, R., Eds., GIS And Generalization: Methodology
and Practice, Taylor & Francis, London, UK, 1995.
Perkal, J. D., An Attempt at Objective Generalisation (translated by W. Jackowshi from
Proba obiektywnej generalizacji, Geodezia I Kartografia, Tom VII, Zeszyt 2, 1958,
pp. 130–142), Michigan Inter-University Community of Mathematical Geogra-
phers Discussion Paper 10, Dept. of Geography, University of Michigan, Ann
Arbor, MI, 1966.
Rhind, D., A GIS research agenda. International Journal of Geographical Information Sys-
tems, 2(1), 23–28, 1988.
Tobler, W., Numerical map generalization. Michigan Inter-University Community of Mathe-
matical Geographers Discussion Paper 8. Dept. of Geography, University of Michigan,
An Arbor, MI, USA, 1966.
Töpfer, F. and W. Pillewizer, The principles of selection, Cartographic Journal, 3(1), 10–16,
1966.
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9072_C000.fm Page viii Friday, September 8, 2006 11:20 AM

Acknowledgments
First, I would like to take this opportunity to express my sincere gratitude to
Prof. Stan Openshaw for allowing me to enter into this area with great freedom.
Indeed, it was with Stan at the NorthEast Regional Research Laboratory
(NE.RRL) at The University of Newcastle upon Tyne that I was for the first
time attracted to this topic. That was really an enjoyable period (1990 to 1991)
because the ESRC (Environmental and Social Research Council) of the United
Kingdom supported three research associates for each of its eight regional
research laboratories (RRL) and gave the RRLs great freedom to select research
topics.
I would also like to express my thanks to fellow colleagues in the community
for their constructive discussions on map generalization research, especially Prof.
David Rhind, Prof. Jean-Claude Muller, Prof. Robert Weibel, Dr. Bo Su, Prof. Liqiud
Meng, Prof. Robert Bob McMaster, Prof. Barbara Buttenfield, Prof. Deren Li, Prof.
Jiayao Wang, Prof. Yue Liu, Prof. Jun Chen, Prof. Tinghua Ai, Prof. Haowen Yan,
Dr. Zesheng Wang, Dr. Dan Lee, Dr. Elsa João, and Dr. Tiina Sarjakoski. My sincere
appreciation goes to T. Ai, T. Cheng, M. Deng, H. Qi, andY. Hu for their constructive
comments on the early versions of this book and to the publisher for making this
volume available to you.
I am in debt to the National Natural Science Foundation of China and the
Research Grant Council (RGC) of Hong Kong Special Administration Region for
their support through a number of research grants.
I gratefully acknowledge the kind permissions granted by a number of organi-
zations and individuals that have allowed me to make use of their copyrighted
materials.
Last but not least, special thanks go to my wife, Lingyun Liu for her continuous
support and understanding and to our sons, Andrew and Edward for their help in
the writing of this book.
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9072_C000.fm Page x Friday, September 8, 2006 11:20 AM

Author
Dr. Zhilin Li is a full professor of geo-informatics (cartography/GIS/remote sensing)
at the Department of Land Surveying and Geo-Informatics, the Hong Kong Polytechnic
University. He holds a B.Eng. and Ph.D. Since obtaining his Ph.D. from the Uni-
versity of Glasgow (U.K.) in 1990, he has worked as a research fellow at the
University of Newcastle upon Tyne, (U.K.), the University of Southampton (U.K.),
and the Technical University of Berlin (Germany). He had also worked at Curtin
University of Technology (Australia) as a lecturer for two years. He joined the Hong
Kong Polytechnic University in early 1996.
Prof. Li has published 90 papers in international journals and is the principal
author of the popular book Digital Terrain Modeling: Principles and Methodology.
He has been presented with the Schwidefsky Medal by the International Society
for Photogrammetry and Remote Sensing (ISPRS) at its 20th Congress held in 2004
and the State Natural Science Award from the Central Government of China in 2005.
Prof. Li’s research interests include multi-scale spatial representation and map gen-
eraliazation, digital terrain and 3-D modelling, and spatial relations.
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Contents
Chapter 1 Introduction .........................................................................................1
1.1 Spatial Representation: Representation of Spatial Data .................................1
1.1.1 Forms of Spatial Representation..........................................................1
1.1.2 Dynamics of Spatial Representation....................................................4
1.2 Multi-Scale Spatial Representation .................................................................7
1.2.1 Spatial Representation as a Record in the Scale–Time System .........7
1.2.2 Transformations of Spatial Representations in Time: Updating.........7
1.2.3 Transformations of Spatial Representations
in Scale: Generalization.......................................................................8
1.3 Transformations in Multi-Scale Spatial Representation................................10
1.3.1 Geometric Transformations................................................................10
1.3.2 Relational Transformations................................................................12
1.3.3 Thematic Transformations .................................................................17
1.4 Operations for Geometric Transformations in Multi-Scale
Spatial Representation....................................................................................17
1.4.1 A Strategy for Classification of Operations for Geometric
Transformations..................................................................................17
1.4.2 Operations for Transformations of Point Features............................18
1.4.3 Operations for Transformations of Line Features.............................19
1.4.4 Operations for Transformations of Area Features.............................21
1.4.5 Operations for Transformations of 3-D Surfaces and Features ........23
1.5 Scope of This Book .......................................................................................24
References................................................................................................................25
Chapter 2 Mathematical Background .................................................................29
2.1 Geometric Elements and Parameters for Spatial Representation .................29
2.1.1 Coordinate Systems............................................................................29
2.1.2 Representation of Geometric Elements in Vector
and Raster Spaces ..............................................................................30
2.1.3 Some Commonly Used Geometric Parameters.................................31
2.1.4 Dimensionality of Spatial Features....................................................34
2.2 Mathematical Morphology.............................................................................38
2.2.1 Basic Morphological Operators.........................................................38
2.2.2 Advanced Morphological Operators..................................................41
2.3 Delaunay Triangulation and the Voronoi Diagram .......................................45
2.3.1 Delaunay Triangulation......................................................................45
2.3.2 Constrained Delaunay Triangulation .................................................46
2.3.3 Voronoi Diagram................................................................................48
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2.4 Skeletonization and Medial Axis Transformation.........................................50
2.4.1 Skeletonization by Means of MAT and Distance Transform ...........50
2.4.2 Skeletonization by Means of Voronoi Diagram
and Triangulation ...............................................................................51
2.4.3 Skeletonization by Means of Thinning .............................................52
References................................................................................................................53
Chapter 3 Theoretical Background.....................................................................57
3.1 Scale in Geographical Space .........................................................................57
3.1.1 Geo-Scale in the Scale Spectrum ......................................................57
3.1.2 Measures (Indicators) of Scale ..........................................................58
3.1.3 Transformations of Spatial Representation in Scale
in Geographical Space .......................................................................59
3.2 Relativity in Scale: The Natural Principle.....................................................62
3.2.1 The Idea of a Natural Principle.........................................................62
3.2.2 Estimation of Parameters for the Natural Principle..........................64
3.3 The Radical Laws: Principles of Selection ...................................................66
3.3.1 Number of Symbols at Different Scales:
A Theoretical Analysis.......................................................................66
3.3.2 Principle of Selection: Empirical Formula
or Radical Law...................................................................................67
3.3.3 Fractal Extension of the Principle of Selection ................................68
3.4 Strategies for Transformations of Spatial Representations in Scale.............69
3.4.1 Separation of Scale-Driven from Graphics-Driven
Transformations..................................................................................69
3.4.2 Separation of Geometric Transformation
from High-Level Constraints .............................................................70
3.4.3 Distinguishing Three Levels of Transformations
for Spatial Representation..................................................................71
3.4.4 Integration of Raster-Based Manipulation
into Vector-Based Data Structure ......................................................72
References................................................................................................................72
Chapter 4 Algorithms for Transformations of Point Features ...........................75
4.1 Algorithms for Point Features: An Overview ...............................................75
4.2 Algorithms for Aggregation of a Set
of Point Features............................................................................................76
4.2.1 K-Means Clustering Algorithm .........................................................76
4.2.2 Iterative Self-Organizing Data Analysis Technique
Algorithm (ISODATA).......................................................................78
4.2.3 Determination of a Representative Point for a Cluster
of Point Features................................................................................80
4.3 Algorithms for Selective Omission of a Set of Point Features ....................81
4.3.1 Settlement-Spacing Ratio Algorithm.................................................82
4.3.2 Circle-Growth Algorithm...................................................................83
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4.4 Algorithms for Structural Simplification of a Set of Point Features............84
4.4.1 Structural Simplification Based on Metric Information....................84
4.4.2 Structural Simplification Concerning Metric
and Thematic Information .................................................................86
4.5 Algorithms for Outlining a Set of Point Features: Regionization................87
References................................................................................................................88
Chapter 5 Algorithms for Point-Reduction of Individual
Line Features......................................................................................91
5.1 Algorithms for Line Point-Reduction: An Overview....................................91
5.2 Sequential Algorithms with Geometric Parameters as Criteria ....................94
5.2.1 Algorithm Based on Number of Points.............................................94
5.2.2 Algorithm Based on Length ..............................................................95
5.2.3 Algorithm Based on Angle ................................................................97
5.2.4 Algorithm Based on Perpendicular Distance ....................................97
5.3 Iterative Algorithms with Geometric Parameters as Criteria........................98
5.3.1 Algorithm Based on Minima and Maxima .......................................99
5.3.2 Progressive Splitting Based on Perpendicular Distance .................100
5.3.3 Split-and-Merge Based on Perpendicular Distance.........................101
5.3.4 Algorithm Based on Area ................................................................102
5.4 Algorithms with Functions of Geometric Parameters as Criteria...............104
5.4.1 Algorithm Based on Cosine Value ..................................................105
5.4.2 Algorithm Based on Distance/Chord Ratio.....................................106
5.4.3 Algorithm Based on Local Length Ratio ........................................107
5.5 Evaluation of Point-Reduction Algorithms .................................................108
5.5.1 Measures for Evaluation of Point-Reduction Algorithms...............109
5.5.2 Performance of Point-Reduction Algorithms ..................................110
5.6 Attempts to Improve Point-Reduction Algorithms .....................................111
5.6.1 Attempts to Avoid Topological Conflicts ........................................112
5.6.2 Attempts to Make Algorithms Robust.............................................112
5.6.3 Attempts to Make Algorithms Self-Adaptive..................................113
5.6.4 Attempts to Make Algorithms More Efficient ................................113
References..............................................................................................................113
Chapter 6 Algorithms for Smoothing of Individual Line Features .................117
6.1 Smoothing of a Line: An Overview ............................................................117
6.2 Smoothing by Moving Averaging in the Space Domain ............................117
6.2.1 Smoothing by Simple Moving Averaging.......................................117
6.2.2 Smoothing by Weighted Moving Averaging ...................................119
6.3 Smoothing by Curve Fitting in the Space Domain.....................................120
6.3.1 Smoothing by Best Fitting: Least-Squares......................................120
6.3.2 Smoothing by Exact Fitting: Cubic Spline .....................................122
6.3.3 Smoothing by Energy Minimization: Snakes..................................123
6.4 Smoothing by Frequency Cutting in the Frequency Domain.....................127
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6.4.1 Smoothing by Fourier Transforms...................................................127
6.4.2 Smoothing by Wavelet Transforms..................................................128
6.5 Smoothing by Component Exclusion in the Space Domain ......................132
6.5.1 Smoothing by EMD.........................................................................132
6.5.2 A Comparison between EMD and Frequency-Based
Transforms........................................................................................136
6.6 Evaluation of Line Smoothing Algorithms .................................................137
References..............................................................................................................138
Chapter 7 Algorithms for Scale-Driven Generalization
of Individual Line Features..............................................................141
7.1 Scale-Driven Generalization: An Overview ................................................141
7.2 Algorithms Based on Gaussian Spatial-Scale .............................................142
7.2.1 Gaussian Line Smoothing in Scale-Space.......................................143
7.2.2 Attempts to Improve Gaussian Smoothing .....................................145
7.3 Algorithms Based on ε-Circle Rolling........................................................146
7.3.1 Perkal Algorithm Based on ε-Circle Rolling ..................................146
7.3.2 The WHIRLPOOL Approximation of the Perkal Algorithm..........147
7.3.3 Waterlining and Medial Axis Transformation
for Perkal’s Boundary Zone.............................................................148
7.4 Algorithms Based on the Natural Principle ................................................149
7.4.1 The Basic Idea of Li–Openshaw Algorithm....................................149
7.4.2 The Li–Openshaw Algorithm in Raster Mode................................150
7.4.3 The Li–Openshaw Algorithm in Raster–Vector Mode....................152
7.4.4 Special Treatments in the Li–Openshaw Algorithm .......................154
7.4.5 The Li–Openshaw Algorithm for Nonnatural Lines:
Some Remarks .................................................................................154
7.5 Evaluation of Scale-Driven Line Generalization Algorithms .....................155
7.5.1 Benchmarks for Evaluating Scale-Driven
Line Generalization......................................................................... 156
7.5.2 Performance of Scale-Driven Line
Generalization Algorithms ...............................................................156
References..............................................................................................................158
Chapter 8 Algorithms for Transformations of a Set of Line Features ............161
8.1 A Set of Line Features: An Overview.........................................................161
8.2 Algorithms for Transformation of a Set of Contour Lines.........................162
8.2.1 Approaches to the Transformation of Contour Lines .....................162
8.2.2 Selection of a Subset from the Original Set
of Contour Lines: Selective Omission.............................................163
8.2.3 Objective Generalization of a Set of Contour
Lines as a Whole..............................................................................165
8.2.4 Transformation of a Set of Contour Lines via
the Removal of Small Catchments ..................................................168
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8.3 Algorithms for Transformation of River Networks.....................................170
8.3.1 Overview ..........................................................................................170
8.3.2 Ordering Schemes for Selective Omission of Rivers......................170
8.3.3 Four Strategies for Selective Omission
of Ordered River Features ...............................................................172
8.3.4 Other Transformations for Selected River Features........................173
8.4 Algorithms for Transformation of Transportation Networks......................173
8.4.1 An Overview ....................................................................................173
8.4.2 The Stroke Scheme for Selective Omission of Roads....................175
8.4.3 Road Junction Collapse: Ring-to-Point Collapse............................178
8.4.4 Other Transformations for Selected Transportation Lines..............179
References..............................................................................................................180
Chapter 9 Algorithms for Transformations of Individual Area Features.........183
9.1 Transformation of Individual Area Features: An Overview........................183
9.2 Algorithms for Boundary-Based Shape Simplification
of an Area Feature........................................................................................184
9.2.1 Boundary-Based Area Shape Simplification: Natural
versus Extremal................................................................................184
9.2.2 Natural Simplification of the Boundary of an Area
Feature as a Closed Curve...............................................................186
9.2.3 Formation of the Convex Hull of an Area Feature .........................186
9.2.4 Formation of the MBR of an Area Feature.....................................188
9.3 Algorithms for Region-Based Shape Simplification
of an Area Feature........................................................................................189
9.3.1 Shape Simplification by Morphological
Closing and Opening .......................................................................190
9.3.2 Formation of Convex Hull and Bounding Box
by Morphological Thickening..........................................................191
9.3.3 Shape Refinement by Morphological Operators .............................192
9.4 Algorithms for Collapse of Area Features ..................................................194
9.4.1 Area-to-Point Collapse.....................................................................194
9.4.2 Area-to-Line Collapse......................................................................197
9.4.3 Partial Collapse ................................................................................199
9.5 Algorithms for Area Elimination.................................................................201
9.5.1 Elimination via Sequential Eroding
Using Monmonier Operators ...........................................................201
9.5.2 Elimination via Erosion Followed by Restoration..........................202
9.5.3 Elimination by Mode Filter .............................................................204
9.5.4 Elimination via a Change in Pixel Size ..........................................205
9.5.5 Coarsening as Elimination of an Area Feature ...............................206
9.6 Algorithms for Splitting an Area Feature....................................................207
9.6.1 Splitting via Systematic Elimination and Eroding..........................207
9.6.2 Splitting via Morphological Opening..............................................208
9.7 Algorithms for Exaggeration .......................................................................208
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9.7.1 Whole Exaggeration by Enlargement: Buffering
and Expansion..................................................................................208
9.7.2 Partial Exaggeration: Directional Thickening .................................209
References..............................................................................................................211
Chapter 10 Algorithms for Transformations of a Set of Area Features ..........213
10.1 Transformation of A Class of Area Features: An Overview .....................213
10.2 Algorithms for Simplification of the Shape of
a Polygonal Network ..................................................................................215
10.2.1 Decomposition-Based Simplification of a Polygonal
Network .........................................................................................215
10.2.2 Whole-Based Simplification of a Polygonal Network .................217
10.3 Algorithms for Combining Area Features: Aggregation
and Amalgamation ......................................................................................218
10.3.1 Boundary-Based Combination via
Equal-Spanning Polygons .............................................................219
10.3.2 Boundary-Based Combination via Convex Hulls.........................219
10.3.3 Boundary-Based Combination via Constrained Hulls .................223
10.3.4 Region-Based Combination via Gap Bridging.............................224
10.3.5 Region-Based Morphological Combination .................................225
10.4 Algorithms for Merging and Dissolving Area Features ............................228
10.4.1 Merge via a Union Operation.......................................................228
10.4.2 Dissolve via Split and Merge........................................................229
10.5 Algorithms for Agglomeration of Area Features.......................................230
10.6 Algorithms for Structural Simplification of Area Patches.........................231
10.6.1 Vector-Based Structural Simplification.........................................231
10.6.2 Raster-Based Structural Simplification.........................................233
10.7 Algorithms for Typification of Area Features............................................234
10.7.1 Typification of Aligned Area Features..........................................234
10.7.2 Typification of Irregularly Distributed Area Features ..................236
References..............................................................................................................237
Chapter 11 Algorithms for Displacement of Features .....................................239
11.1 Displacement of Features: An Overview ...................................................239
11.2 Algorithms for Translations of Features ....................................................241
11.2.1 Uniform Translation in a Single Direction
in Raster Mode..............................................................................241
11.2.2 Translation in Normal Directions in Vector Mode.......................243
11.3 Displacement by Partial Modification
of a Curved Line.........................................................................................243
11.3.1 Partial Modification with a Vector Backbone...............................243
11.3.2 Partial Modification with Morphological Algorithms ..................243
11.3.3 Partial Modification Based on Snakes Techniques.......................246
11.4 Algorithms and Models for Relocation of Features ..................................249
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11.4.1 Relocation of Features with Displacement Fields........................249
11.4.2 Relocation of Features with Finite Elements ...............................251
11.4.3 Relocation of Features with Least-Squares Adjustment ..............252
11.4.4 Relocation of Features with a Ductile Truss
and Finite Elements.......................................................................253
References..............................................................................................................253
Chapter 12 Algorithms for Transformations of Three-Dimensional
Surfaces and Features.....................................................................255
12.1 Algorithms for Transformations of Three-Dimensional Features: An
Overview.....................................................................................................255
12.2 Algorithms for Transformations of DTM Surfaces ...................................255
12.2.1 Multi-Scale Transformation of DTM Surfaces:
An Overview .................................................................................255
12.2.2 Metric Multi-Scale Representation through Filtering
and Resampling.............................................................................258
12.2.3 Metric Multi-Scale Representation Based on the Natural
Principle.........................................................................................260
12.2.4 Visual Multi-Scale Representation through
View-Dependent LoD....................................................................262
12.3 Algorithms for Transformation of 3-D Features........................................266
12.3.1 Transformation of Individual Buildings .......................................266
12.3.2 Transformation of a Set of Buildings...........................................268
References..............................................................................................................269
Epilogue ................................................................................................................271
Index......................................................................................................................273
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9072_C000.fm Page xx Friday, September 8, 2006 11:20 AM

1
1 Introduction
Spatial representation refers to the representation of spatial data in this context.
Spatial data could be represented at different scales, leading to the issues of multi-
scale spatial representation. This chapter provides an overview of the issues of
spatial representation and multi-scale spatial representation. Emphasis is on the
introduction of the essential operations for geometric transformations in multi-scale
spatial representation.
1.1 SPATIAL REPRESENTATION: REPRESENTATION
OF SPATIAL DATA
1.1.1 FORMS OF SPATIAL REPRESENTATION
Spatial data can be represented in various forms, that is, graphics or nongraphics.
Graphic representation includes maps, drawings, and animation, either two-dimen-
sional (2-D) or 3-D. Graphic representations are achieved through the use of point,
linear, and areal symbols. Colors and patterns are also in use as visual variables.
Nongraphical representation includes audio, audiovisual, text, digital numbers, and
so on. Nongraphical representations are not very popular in the geo-information
community and thus are not discussed in this book.
A map is a typical 2-D representation. It is a spatial representation of reality,
used for recording and conveying information about spatial and semantic character-
istics of the natural world and cultural phenomena. It is a traditional means for
spatial representation. Figure 1.1 shows a spatial representation on an ancient Chi-
nese map (made in the Han Dynasty before 168 BC). Maps have been a popular
mode of spatial representation and are still popularly used in practice because of
their measurability, which results from the use of mathematical laws, intuitive view
from symbolization, and overview from generalization.
Based on their contents, maps are usually classified into three types:
• Topographic maps (also called general-purpose maps): Representation of
terrain surface and the features on the surface with balance (Figure 1.2a).
(See color insert following page 116).
• Thematic maps: Representation of a theme of natural or cultural phenomena
(Figure 1.2b).
• Special maps: Representation of a few themes of natural and/or cultural
phenomena, for example, tourist maps. In other words, they are in between
topographic and thematic maps.
9072_C001.fm Page 1 Monday, September 11, 2006 4:22 PM

2 Algorithmic Foundation of Multi-Scale Spatial Representation
A topographic map is a type of qualitative map, but a thematic map could be
either quantitative or qualitative. Other criteria can also be used for classification of
maps such as scale, usage, size of area covered, color, and so on. One interesting
classification made by Moellering (1984, 1987) distinguishes maps into:
• Real maps: Directly viewable and having a permanent tangible reality.
• Virtual maps: Lacking one or both of the qualities of real maps.
Moellering (1984, 1987) has further differentiated virtual maps into three types:
• Virtual type 1: Directly viewable but without a permanent tangible reality,
for example, a screen display.
• Virtual type 2: Having a permanent tangible reality but not directly view-
able, for example, hardcopy form on a CD-ROM.
• Virtualtype3:Neitherhardcopynorviewable,forexample,datastoredondisk.
FIGURE 1.1 Representation of spatial data on an ancient Chinese map.
FIGURE 1.2 (See color insert following page 116) A topographic map (a) and a thematic
map (b) (Courtesy of LIC of HKSAR).
HONG KONG
BUILDING FOR THE FUTURE
(a) Topographic map at 1:200,000 (b) Thematic map (house development)

Introduction 3
With the advancement of computing technology, 3-D representation has
become more and more popular. Spatial data can be represented in perspective
views. Figure 1.3a is an example generated from a digital terrain model, or digital
elevation model (see Li et al., 2005). Rendering techniques can be applied to
produce more realistic representations in image form. Terrain texture and features
can also be added to perspective views (Figure 1.3b) or rendered representations
to generate more vivid representations. These are not true 3-D and are sometimes
called 21/2-D representations.
A true 3-D representation is viewed stereoscopically. Each stereo view consists
of two graphical representations (or images), one viewed by each eye. Stereo viewing
is a common technique for increasing visual realism or enhancing user interaction
with 3-D scenes. The 3-D effect can be created by introducing a visual parallax
along the eye-base direction, that is, the x-direction in the conventional coordinate
system. The principle of such a representation lies outside the scope of this book
and can be found elsewhere (e.g., Li et al., 2005). Figure 1.4 shows a pair of contour
maps with a stereo view. A special viewing device is required to force the left eye
to view the left image (or spatial representation) and the right eye to view the right
one. Optical or optical–mechanical devices are required for this purpose. Figure 1.5
shows two examples of such optical–mechanical devices.
FIGURE 1.3 Three-dimensional representation of terrain surface and terrain features.
FIGURE 1.4 Visual parallax introduced into the contour map to create a stereo view.
(a) Terrain surface in perspective view (b) Map and terrain features (Lee et al., 2001)

1.1.2 DYNAMICS OF SPATIAL REPRESENTATION
Spatial data can be represented either in static or in dynamic mode. Traditional maps
are typical examples of static spatial representation. Indeed, those discussed in
Section 1.1.1 are all static representations. In this context, dynamic representation
means that some kind of interaction or movement is involved. This classification is
slightly different from that made by other researchers (e.g., Kraak, 2001), and the
inclusion of interaction might be arguable.
In interactive representation, the information for representation is controlled by the
operator through some action. A typical example is the click (or double-click) operation.
By clicking a symbol, more information about the feature will be displayed either from
the same source or through a hyperlink. An interesting development is the use of the
mouse to control the interaction between the legend and symbol. One could display a
type of feature by clicking the feature symbol in the legend (as control-panel) or alter-
natively show the legend by clicking the feature inside the map. Another interesting
development is the mouse-over operation. When the mouse is moved over a symbol,
more information about the feature is displayed. Figure 1.6 (See color insert following
page 116) shows two examples of such developments (van den Worm, 2001).
FIGURE 1.5 Two examples of optical–mechanical devices for stereo viewing.
FIGURE 1.6 (See color insert following page 116) New developments in interactive repre-
sentations (van den Worm, 2001).
Population (pers.)
145.000
70.000
30.000
5000
Age 14–64
Age 0–14
Age > 64
Age group
(a) Mouse-over symbol (b) Legend-controlled display
Deventer
Almelo
Hengelo
Enschede
Zwolle
A35
A28
A1
A1
Road infrastructure in Overijssel
Highways
Prov. road
Main towns
All towns
All details
Population numbers and age in 1990,
Overijssel

Introduction 5
Movement can be achieved in various ways. Drag and pan are the two simple
operations to move a representation around a screen. The former moves a represen-
tation on a screen by holding the click and then moving the mouse, while the latter
makes use of the scroll bars. These two operations do not change the viewpoint of
the representation. The fly-through and walk-through operations create representations
viewed from different positions. The former mimics the representation viewed by a
bird when flying over an area, and the latter mimics the representation viewed by a
person while walking along a route. These two operations are considered a result of
animation. The fundamental technique used in such an animation is the page flipping,
leading to movies. In the animation process, a number of frames (i.e., representations)
are first prepared and these frames are then played in sequence. Figure 1.7 shows
four frames of a fly-through animation of terrain features. To control the animation
process, three variables, called dynamic variables, are available (DiBiase et al., 1992):
• Duration: The time units for a scene, e.g., second (30 frames per second).
If the duration is too long, the action will be jerky.
• Rate of change: The pace of animation or difference between two successive
scenes. If the rate is low, slow motion can be produced. Fast motion is
produced if the rate is high.
• Order: The sequence of the frames, which could be arranged according
to time, position, or attributes.
Motion can also be created by other types of animation. Blinking is an operation
at symbol level and can be achieved by animating space (location) or attributes. This
is a local operation. For a whole representation, motion can also be animated over
FIGURE 1.7 Four frames of a fly-through animation for a piece of terrain surface (Reprinted
from Li et al., 2005).
1
30

time, scale, and attributes. Figure 1.8 (See color insert following page 116) shows
two frames (i.e., 1960 and 1990) from the motion animated over time for the relative
population change in the United States over time (since 1790), which were produced
from IDL (interactive data language). Animation of attributes can be achieved by
switching layers on and off and by reclassification of data. Animation over scale
can be achieved by a zooming (zoom in and out) operation. Figure 1.9 (See color
FIGURE 1.8 (See color insert following page 116) Two frames selected from an animation
of U.S. population change since 1790.
FIGURE 1.9 (See color insert following page 116) Zooming into Beijing streets as an animation
(Courtesy of National Geometics Center of China).
1960 1990
Loss >0% >1% >10% >20% >50% >100% >200%
(a) (b)
(d) (c)

Introduction 7
insert following page 116) shows the continuous zooming into the Beijing streets.
This is a multi-scale issue and thus is the main topic of this book. More precisely,
this book introduces a set of algorithms for various operations required for multi-
scale representation of spatial data.
1.2 MULTI-SCALE SPATIAL REPRESENTATION
Section 1.1.2 discussed that zooming is a process of dynamic multi-scale spatial
representation and is one of the most exciting functions in a geographical information
system (Abler, 1987). In the context of mapping, maps are produced at various
scales, resulting in multi-scale representations in static mode. That is, multi-scale
spatial representation can be in different formats and different modes. This section
will discuss some general issues in multi-scale spatial representation.
1.2.1 SPATIAL REPRESENTATION AS A RECORD IN THE SCALE–TIME
SYSTEM
A spatial representation is a record of spatial phenomena at a particular time and at
a particular scale in the scale–time system (Li, 1993), as shown in Figure 1.10a.
Figure 1.10b shows a number of spatial representations at different times but at a fixed
scale. This figure indicates the transformations of spatial representation in time. This
is about the updating of spatial representation. Figure 1.10c shows a number of spatial
representations at different scales but at the same time. This is about multi-scale
representation and is referred to as transformations in scale in this context.
1.2.2 TRANSFORMATIONS OF SPATIAL REPRESENTATIONS IN TIME:
UPDATING
The environment changes over time. Natural processes (e.g., soil erosion or land
subsidence) usually change slowly. Dramatic changes are caused by natural disasters
and human activities. Examples are buildings being destroyed by earthquake and
land lots being subdivided. The changes of coastal lines in Hong Kong over time
(due to reclamation), as shown in Figure 1.11, (See color insert following page 116)
is another example of human activity.
FIGURE 1.10 Spatial representations as a record in the time–scale system.
Scale
Time
Scale
Time
Scale
Time
(a) A spatial representation as
a record in scale-time system
(b) Transformations of spatial
representation in time
(c) Transformations of spatial
representation in scale

The usefulness of an outdated spatial representation (e.g., a map) is very limited
for most applications except for those studies related to history. Indeed, currency
(or updatedness) has been considered as a quality measure for spatial data and spatial
representation in the literature (e.g., Burrough and McDonnell, 1998) and spatial
data transfer standards (SDTS) (see http://mcmcweb.er.usgs.gov/sdts/). That is, spa-
tial representations need to be updated frequently.
Updating has indeed become a headache for spatial data producers such as
national mapping agencies and has become a hot research topic in recent years.
Many issues are involved in the updating process, for example, how frequently to
update, how to keep track of the historical versions, how to efficiently acquire the
required data, how to automate the updating process, and how to disseminate updated
data to end users. Most of these issues lie outside of the scope of this text except
one, that is, the automation of the process. This topic will be further explored in
Section 1.2.3.
1.2.3 TRANSFORMATIONS OF SPATIAL REPRESENTATIONS IN SCALE:
GENERALIZATION
As discussed in Section 1.1.1, a spatial representation may take different forms, and a
map is a typical type of spatial representation. Maps are associated with scale. Maps at
different scales depict different levels of detail about the natural and cultural phenomena
on the Earth. Figure 1.12 (See color insert following page 116) shows maps of Kowloon
Peninsula of Hong Kong at two different scales. It can be seen clearly that the levels of
abstraction are quite different in these two maps. Indeed, different symbols may be used
to represent the same types of features but at different scale. This can be demonstrated
by using the representation of a town as an example. It may be represented:
• By streets and building on maps at large scale.
• By main streets and big building blocks on maps at a smaller scale.
• By the outline of the town on maps at an even smaller scale.
• By a point symbol on maps at a small scale.
• By nothing as it disappears on maps at a very small scale.
This can be observed in Figure 1.9. Figure 1.9d is a large-scale representation,
showing streets of Beijing in detail. Figure 1.9c is a map of the Beijing urban area,
FIGURE 1.11 (See color insert following page 116) Changes of coastal lines in Hong Kong
over time (Courtesy of LIC of HKSAR).
up to 1887 1888–1924 1925–1945
Kowloon–Canton railway
operated in 1900.

Introduction 9
showing major streets and city blocks. Figure 1.9b is map of Greater Beijing. In
this representation, the urban area is simply outlined. Figure 1.9c is a map of China.
In this representation, Beijing has almost become a point.
A national mapping agency may have maps at scales from 1:1,000 to 1:1,000,000
and even smaller. One critical issue faced by national mapping agencies is the
frequent updating of maps at so many scales. The ideal approach is to update maps
at the largest scale frequently and to derive maps at smaller scales on demand only.
The process of deriving maps at a smaller scale from those at a larger scale is called
map generalization.
All spatial representations are associated with scale. Therefore, generalization
is a common issue for different types of spatial representation and is the process of
multi-scale representation of spatial data. It has also been regarded by Li (1996) as
the transformation of spatial representations in scale.
Generalization is a process of abstraction. The terrain features represented on
maps at smaller scales are the abstractive representations of those at larger scales.
Perhaps the highest level of generalization was achieved by Sir Isaac Newton, who
represented any planet by a point in the derivation of his famous “Law of Gravita-
tion.” This means that generalization is an issue related not only to the representation
of spatial data but also to the modeling of spatial processes.
FIGURE 1.12 (See color insert following page 116) The Kowloon Peninsula represented on
maps at two different scales (Courtesy of LIC of HKSAR).
(a) Topographic map 1:20,000 (HM20C)
(b) Topographic map 1:100,000 (HM100CL)

In order to derive a spatial representation at a smaller scale from those at a
larger scale, various types of transformations at different levels should be carried out.
This also applies to the real-time zooming operation in spatial information systems.
Therefore, in the later chapters, algorithms for such transformations will be presented.
1.3 TRANSFORMATIONS IN MULTI-SCALE SPATIAL
REPRESENTATION
Before algorithms for different types of transformations can be presented, it is neces-
sary to provide an overview of the operations required for the various transformations.
A spatial representation, for example, a map, contains the following types of
information about features:
• (Geo)metric information related to position, size, and shape.
• Relational information about spatial relationships between neighboring
features.
• Thematic information related to the types and importance of features.
Therefore, during the derivation of a spatial representation at a smaller scale from
those at a larger scale, three types of transformations have been performed: geometric,
relational, and thematic.
1.3.1 GEOMETRIC TRANSFORMATIONS
Geometric transformations are clearly demonstrated in Figures 1.9 and 1.12 and
briefly discussed in Section 1.2.3. These transformations are achieved by some
operations. The issues related to geometric transformations are:
• What operations are essential for multi-scale representation?
• What operations are currently available?
• What operations are required for a particular case?
The first question will be examined in detail in Section 1.4. The third one will be
discussed in Section 1.3.3. The second question is discussed in this section.
In the classic textbook by Robinson et al. (1984), very few operations are
identified, for example, classification, induction, simplification, and symbolization.
However, these operations are too general to be computerized. That is, more concrete
operations need to be identified. A summary of currently available operations is
listed in Table 1.1. From this table, it can be seen clearly that some terms are in
common use, while others are rarely used and that some terms are more specific,
while others are more general.
Related to these operations of geometric transformations, a critical question is,
“Is there a consensus on the use of these terms in the geospatial sciences (such as
cartography and geographic information science)?” The answer is, “Not necessar-
ily.” This is revealed by a study carried out by Rieger and Coulson in 1993. In their
study, a group of 23 expert cartographers from North America with various

Introduction 11
TABLE
1.1
Operations
for
Geometric
Transformations
in
Multi-Scale
Representation
(Su,
B.,
Morphological
Transformations
for
Generalization
of
Spatial
Data
in
Raster
Format.
Ph.D.
Thesis,
Curtin
University
of
Technology,
Perth,
Australia,
1997)
Steward
(1974)
Robinson
et
al
(1984)
Delicia
&
Black
(1987)
McMaster
&
Monmonior
(1989)
Keates
(1989)
Shea
&
McMaster
(1989)
Beard
&
Mackaness
(1991)
McMaster
&
Shea
(1992)
Agglomeration
÷
Aggregation
÷
÷
÷
÷
Amalgamation
÷
÷
÷
Classification
÷
÷
÷
÷
÷
Coarsen
÷
Collapse
÷
÷
÷
÷
÷
Combination
÷
÷
Displacement
÷
÷
÷
÷
÷
Enhancement
÷
÷
÷
Exaggeration
÷
÷
÷
÷
Induction
÷
÷
Merge
÷
÷
÷
Omission
÷
÷
÷
Refinement
÷
÷
÷
÷
Selection
÷
÷
÷
Simplification
÷
÷
÷
÷
÷
÷
÷
Smoothing
÷
÷
÷
Symbolization
÷
÷
Typification
÷
÷

backgrounds was interviewed regarding the use of the operations that frequently
appear in literature: simplification, classification, displacement, selection, elimi-
nation, exaggeration, symbolization, smoothing, induction, and typification. Most
of these terms were not defined in the same way by the experts, and they did not
even understand a few of the terms. Indeed, some of the terms were defined in so
many different ways that even the experts felt confused. For example, simplification
has traditionally been used to mean the reduction of complexity (Keates, 1989)
with the retention of the main structure. However, in recent years, it has been used
to mean the reduction of points (McMaster and Shea, 1992). Therefore, there is a
need for systematic classification and standardization. This will be addressed in
Section 1.4.
For each operation, one or more algorithms may be developed. All algorithms
for these operations together form a mathematical foundation for multi-scale spatial
representation, leading to the title of this book. These algorithms can be stored as
subroutines, such as the conformal and affine transformation models, and can be
called whenever there is a need.
1.3.2 RELATIONAL TRANSFORMATIONS
After a geometric transformation is applied to a feature or a set of features, the
relationship between neighboring features may have undergone a transformation
(Dettori and Puppo, 1996). For example, if one generalizes buildings into street
blocks, then the disjoint relation between individual buildings is changed, as the
streets separating them disappear. There are three types of spatial relations:
• Topological relations: The connectivity and adjacency of spatial features.
• Order relations: The order between spatial features such as the directions,
orientation, and comparison.
• Metric relations: The relations between metric properties of spatial fea-
tures such as the distance and relative positions.
Spatial relations between features on a spatial representation may have changed
after a geometric transformation. Inversely, such relations may be of help in the
detection and resolution of spatial conflicts caused by geometric transformations.
Topological relations are the most fundamental (Freeman, 1975). Figure 1.13
shows an example of change in topological relations between area features. In the
first instance, area features A and C were separated by feature B, but they became
immediate neighbors after a transformation.
Eight basic types of topological relations between area features have been
identified (Egenhofer and Franzosa, 1992). They are shown in Figure 1.14. Com-
paring Figure 1.13 with Figure 1.14, it can be seen that the change in topological
relation in the case of Figure 1.13 is from “disjoint” to “meet.”
Sometimes a change in topological relation may result in a spatial conflict. Figure
1.15 shows such an example. In this case, a small building falls into the water after
geometric transformations. This can be detected by checking the topological relations
between the water as an area feature and the small building as another area feature.

Introduction 13
FIGURE 1.13 Change in topological relation after a transformation in scale.
FIGURE 1.14 Eight basic topological relations between area features A and B.
FIGURE 1.15 Spatial conflicts caused by a change in topological relation.
A
B
C
A
C
(a) Areas A and C separated by B (b) Areas A and C being connected
A A A
B A
A B
B
Disjoint Meet Overlap Equal
A
B
Cover
B
A
Covered by
A
B
Contain
B A
Contained by
B
(a) The original map composed of
two layers: water and settlement
(b) Topological relations altered, i.e.,
a building falling into water
Water
Water

Comparing this with the cases shown in Figure 1.14, it can be seen that the topo-
logical relations between the water and the small building have changed from
“disjoint” to “contain.” This is geographically not acceptable in normal cases, and
the problem needs to be solved. However, detailed discussion of the detection and
resolution of spatial conflict lies outside the scope of this book.
To detect the change in topological relations, a mathematical model is required
to describe the topological relations as shown in Figure 1.14. The classic model for
formal description of topological relations is the four-intersection model by Egen-
hofer and Franzosa (1992). An extension to this model is the nine-intersection model
(Egenhofer and Herring 1991). However, it has been pointed out (Chen et al., 2001)
that the extension from four to nine intersections is invalid because there is linear
dependency between the three topological components. It has also been pointed out
that the four-intersection model cannot be used for all types of spatial features
because the definitions of topological components are dimension dependent (Li et al.,
2000). For example, in 1-D space, the two end points define the boundary of a line.
However, this definition is not valid in 2-D space. If one simply adopts the definition
in 1-D space to 2-D space, a topological paradox will be caused. To solve this
problem, a Voronoi-based spatial algebra for topological relations has been devel-
oped by Li et al. (2002), which makes use of the features themselves and their
Voronoi regions (see Section 2.3.3 for a more detailed discussion) only if necessary.
One interesting development is the Voronoi-based k-order adjacency model
(Chen et al., 2004) for the more detailed differentiation of the disjoint relation. This
model makes the use of Voronoi neighbors. In this model, the neighbors with one-
order Voronoi-adjacency to a given feature are those features whose Voronoi regions
are connected to these of the given feature. For example, the one-order Voronoi-
adjacency neighbors of the given feature P in Figure 1.16a are the highlighted ones.
The two-order Voronoi-adjacency neighbors of P are those features whose Voronoi
regions are connected to the Voronoi regions of the one-order Voronoi-adjacency
neighbors, as highlighted in Figure 1.16b. In another sense, this model might be
regarded as a model for qualitative distance relations because it makes use of Voronoi
regions of spatial features as a distance measure.
FIGURE 1.16 Voronoi-based k-order adjacency model to further differentiate the disjoint
relation.
(a) 1-order (b) 2-order

Introduction 15
The model described above uses the term order. This is different from order in
an ordinary sense, which means the arrangement of two or more features (or events)
in accordance with stated criteria. For example, spatial features can be ordered
according to size, distance, and direction. The order relations of spatial features can
be described qualitatively or quantitatively. They can also be described in a relative
sense or an absolute sense. Examples of relative descriptors are front and back, left
and right, and above and below. The followings descriptors are absolute: N, NE, E,
SE, S, SW, W, and NW.
Directional relations can be easily represented by a quantitative measure. The
bearing in units of degree is a common practice. A direction is traditionally defined
as a kind of distance-independent relation between two points in space that specifies
the angular position of either with respect to the other. However, in a spatial repre-
sentation, there are point, line, and area features. Attempts have been made to
describe the directional relations among all these three types of spatial features. The
cone-based four- and eight-direction models are in common use (e.g., Peuquet and
Zhan, 1987). This is illustrated in Figure 1.17a, in which feature B is located on the
north side of A if the four-direction model is used.
Sometimes the shape of an area is very complicated, and thus it is difficult to specify
the directional relation in such a way. An alternative is the MBR (minimum bounding
rectangle) matrix model (Goyal, 2000), as shown in Figure 1.17b (see Section 9.2.4 for
a more detailed discussion). In this model an area is represented by its MBR. The whole
space is divided into nine tiles. The relations are represented by a matrix. If feature B
is located within the MBR of A, the directional relation is called “the same.” In the
case of Figure 1.17b, feature B is partially located in the N and NE directions. Weighting
can also be used according to the proportion of the area in each tile.
A third model is the projective model used for reasoning the directional relations
between a given area feature and other two area features (Billen and Clementini,
2004). It divides the space into five regions: before, after, left side, right side, and
between. Figure 1.17c shows such a model, where feature C is on the left side of
A and B. Another interesting development is the reduction of area features into line
features by generalization for the establishment of directional relations between two
area features (Yan et al., 2006).
FIGURE 1.17 Directional relations between spatial features.
NE
SE
N
E
S
SW
W
NW
A
B B
A EA
WA
SA
NA NEA
SEA
NWA
SWA
(a) Cone-based model (b) MBR matrix model
B
A
Before
(A,B)
Leftside
(A,B)
Rightside (A,B)
After
(A,B)
Between
(A,B)
(c) Projective relation model
C

Figure 1.18 shows the change in directional relations between areas A and B
after a geometric transformation. With reference to the cone-based model, area
feature B is to the east of A. After the geometric transformation in scale, B is
completely on the northeast side of A.
Metric relations describe the relations between metric properties of spatial
features such as distance and relative positions. The descriptors can be either quan-
titative or qualitative. The Voronoi-based k-order adjacency model is a kind of
semiquantitative distance mode. Figure 1.19a shows another type of qualitative
measure. However, it is based on the concept of distance.
The Euclidean distance is a quantitative metric between two points. Efforts
have been made to extend the concept to describe the metric between lines and
areas. Minimum distance, maximum distance, and centroid distance are widely
used in geo-information science. It has also been suggested to make use of
Hausdorff distance (Figure 1.19b) and generalized Hausdorff distance (Deng
et al., 2005).
FIGURE 1.18 Change in directional relations after transformation in multi-scale representation.
FIGURE 1.19 Distance relations between two spatial features.
(a) Area B in the east of A (b) Area B in the northeast of A
E
S
W
N
A
B
Far away
Near
Distance in-between
(a) Qualitative distance
(b) Hausdorff distance
ρ (A, B) = max {εA, εB}
A
B
εA
εB

Introduction 17
It is clear that after a scale reduction the map space is reduced. The distance
between two features is reduced, therefore, the representation needs to be modified.
For example, some features are too close to be separated and thus need to be
combined; some need to be displaced and others deleted. In other words, a set of
operations for geometric transformation needs to be applied to make the result
suitable for representation at a smaller scale. Such operations will be identified in
Section 1.4.
1.3.3 THEMATIC TRANSFORMATIONS
After a geometric operation is applied to a feature or a set of features, the thematic
meanings of features may have undergone a thematic transformation. For example,
at a large scale, individual buildings are represented. Residential, commercial, and
administrative buildings are identifiable. However, at a smaller scale, buildings of
different types are grouped into city blocks. In this way, features with new thematic
meanings are created and old features disappear. Similarly, different types of farm-
lands are represented on land-use maps at large scales, such as irrigated land,
irrigable land, and dry land. However, they may be aggregated into a new type,
called farmland, at a smaller scale.
A reverse process can be applied to thematic information, that is, to make use
of thematic information for formalization of rules to control geometric transforma-
tions. For example, based on the biogeographical principle, there should be a piece
of shrub land between lowlands and grassland (Pun-Cheng et al., 2003). This can
be used as a rule for the transformations in spatial representation. Detailed discussion
of thematic transformation lies outside the scope of this book, but more information
can be found in Muller (1990), Buttenfield and McMaster (1991), Muller et al.
(1995), Li and Choi (2002), and Gao et al. (2004).
1.4 OPERATIONS FOR GEOMETRIC
TRANSFORMATIONS IN MULTI-SCALE SPATIAL
REPRESENTATION
To derive small-scale spatial representations from large-scale ones, various types of
geometric transformations need to be performed. A list of operations for geometric
transformations is given in Table 1.1. However, as pointed out previously, no con-
sensus has been made for some of these operations. Therefore, a systematic classi-
fication and redefinition of essential operations needs to be carried out so that
algorithms for these operations can be described in later chapters.
1.4.1 A STRATEGY FOR CLASSIFICATION OF OPERATIONS
FOR GEOMETRIC TRANSFORMATIONS
Classification means to place together in categories those operations that resemble
each other. A systematic classification should be complete (i.e., exhaustive),

nonoverlapping, and objective. It is very difficult to have an exhaustive classification
because no consensus has been made with the criteria for the assessment of multi-
scale representation. Therefore, as one can imagine, the operations to be discussed
in this section can only be regarded as essential operations.
Various criteria could be used for such a classification. For example, McMaster
and Monmonior (1989) first used data mode as a criterion to classify these operations
into two categories: raster-mode and vector-mode operations. Then they further
identified a number of operations in raster and vector. However, the author believes
that the data mode is not an issue anymore, and raster-based and vector-based
algorithms can be easily integrated. Therefore, no special emphasis is needed on the
difference in data mode. Instead, in this text emphasis is on the implementation of
algorithms, and thus classification is determined based on the dimension of geometric
elements, as shown in Figure 1.20.
1.4.2 OPERATIONS FOR TRANSFORMATIONS OF POINT FEATURES
For an individual point feature, the possible operations for geometric transformations
are illustrated in Table 1.2 and their definitions are as follows:
• Displacement: To move a point away from a feature or features because
the distance between the point and the other feature(s) is too small to be
separated.
• Elimination: To eliminate an individual point feature, as it will be too
small to represent.
• Magnification: To make the size of a point feature become large enough
to be represented, although it appears too small to be represented.
In some literature magnification is also called exaggeration or enlargement.
For a set of point features, the operations are illustrated in Table 1.3 and the
definitions are as follows:
FIGURE 1.20 Classification of operations for geometric transformations in multi-scale rep-
resentation based on the dimensions of geometric elements.
3 − D features
3 – D surfaces
3 − D
A set of areas
Individual areas
2 − D
A set of lines (line networks)
Individual lines
1 − D
A set of points (point cluster)
Individual points
0 − D
Geometric transformations

Introduction 19
• Aggregation: Grouping a number of points into a single point features.
• Regionalization: Outlining the boundary of the region covered by points
so as to make this region an area feature.
• Selective omission: Selection of more important point features to be
retained and omission of less important ones, if space is not adequate.
• Simplification: Reducing the complexity of the structure of a point cluster
by removing some point, with the original structure retained.
• Typification: Keeping the typical pattern of the point feature while removing
some points.
1.4.3 OPERATIONS FOR TRANSFORMATIONS OF LINE FEATURES
Representation of lines in multi-scale has been a heavily researched topic for a long
time because (a) it is a staring point for new studies of the multi-scale representation
TABLE 1.2
Operations for Geometric Transformations of Individual Point Features
TABLE 1.3
Operations for Geometric Transformations of a Set of Point Features
Operators Large-scale Photo-reduced Small-scale
Displacement
Elimination
Magnification
Operation Large-scale Photo-reduced Small-scale
Aggregation
Regionalization
Selective
omission
(Structural)
Simplification
Typification

of other features and (b) over 80% of features on a map are line features. A huge
body of literature is available on the manipulation of line features.
For an individual line feature, the possible operations for geometric transforma-
tion are illustrated in Table 1.4, and the definitions are as follows:
• Displacement: Moving the line in a given direction.
• Elimination: Eliminating an individual line feature because it will be too
narrow to represent.
• (Scale-driven) generalization: Producing a new line in which the main
structure is retained but small details are removed. This operation is
dependent on the scales of input and output representations.
• Partial modification: Modifying the shape of a segment within a line.
• Point reduction: Reducing the number of points for representation by
removing the less important points from a line so that only the so-called
critical points are retained.
• Smoothing: Making the line appear smoother.
• Typification: Keeping the typical pattern of line bends while removing some.
There are two types of smoothing: filtering and curve fitting. Filtering means to
filter out the high-frequency component (or small details) of a line so that the line
appears smoother. Curve fitting is another type of smoothing, which tries to fit a
TABLE 1.4
Operations for Geometric Transformations of Individual Line Features
Displacement
Elimination
(Scale-driven)
generalisation
Partial modification
Point reduction
Curve fitting
Smoothing
Filtering
Typification

Introduction 21
curve through a set of points. In the author’s viewpoint, smoothing is not necessarily
an operation for multi-scale representation of lines. However, smoothing does create
some effects required for multi-scale representation, although this operation is not
directly related to scale.
The term point reduction is not in wide use. In some literature, line simplification
is used to refer to point reduction, as some kind of simplification might be created in
some cases. In computing literature, curve approximation and corner detection are the
two operations used to retain critical points and remove less important points. In the
author’s viewpoint, point reduction should not be part of the operations for multi-scale
representation because traditional generalization has nothing to do with point reduction
(Li, 1993). Point-reduction algorithms try to make best approximations of the original
line with a minimum number of points. It must be emphasized here that no scale change
is involved in such an operation. The output is for the representation of the line at the
same scale. These algorithms are good for weeding operations. Reduction of the number
of points on a line was an important issue in the early development of digital represen-
tation because of limited computing power. Indeed, at that time, data volume was a big
concern. As a consequence, many algorithms have been developed for such a purpose.
In practice, point reduction should also be applied to lines to reduce the number of points
as preprocessing of scale-driven generalization because the points on a line will appear
to be too dense after a scale reduction, as can be seen from Table 1.4.
Scale-driven generalization is a type of smoothing. In scale-driven generaliza-
tion, the smoothing effect is computed based on the scale of the input data and the
scale of the output data. In the end of such a process, the main trend of the line is
retained and small variations removed.
For a set of line features, possible operations for geometric transformations are
illustrated in Table 1.5, and the definitions are as follows:
• Selective omission: Selecting the more important lines to be retained.
• Collapse:Making the dimension changed. Two types are identified: ring-
to-point and double-to-single-line.
• Enhancement: Making the characteristics still clear.
• Merging: Combine two or more close lines together.
• Displacement: Moving one away from the other or both lines away from
each other.
In Table 1.5 the example of selective omission is a river network. There is also a
selective omission problem for contour lines and a transportation network.
1.4.4 OPERATIONS FOR TRANSFORMATIONS OF AREA FEATURES
The operations for individual area features are listed in Table 1.6 and are defined as
follows:
• Collapse: Making the feature represented by a symbol with lower dimension.
• Displacement: Moving the area to a slightly different position, normally
to solve the conflict problem.

TABLE 1.5
Operations for Geometric Transformations of a Set of Line Features
TABLE 1.6
Operations for Geometric Transformations of Individual Area Features
Operation Large-scale
Photo-
reduced
Small-scale
Selective omission
Ring-to-
point
Collapse
Double-
to-single
Enhancement
Merging
Displacement
Operation Large-scale
Photo-
reduced
Small-
scale
Area-to-
point
Area-to-line
Collapse
Partial
Displacement
Directional
thickening
Enlargement
Exaggeration
Widening
Elimination
(Shape) Simplification
Split

Introduction 23
• Exaggeration: Making an area with small size still represented at a smaller
scale on maps on which it should be too small to be represented.
• Elimination: Eliminating small and unimportant areas.
• Simplification: Making the shape simpler.
• Split: Splitting an area’s features into two because the connection between
them is too narrow.
There are three types of collapse: area-to-point collapse (e.g., representing a
city with a point feature on small-scale maps), area-to-line collapse (e.g., represent-
ing a river with a single line), and partial collapse (representing the thin part of an
area feature with a line while the other part is still a region).
There are also three types of exaggeration: directional thickening (making the
area feature exaggerated in a given direction), enlargement, or magnification (making
the whole feature enlarged in all directions), and widening (making the bottleneck
of an area feature wider to make it observable at a smaller scale).
For a set of area features, in addition to simplification, displacement, selective
omission, collapse, and exaggeration, the following form a subset of operations:
• Agglomeration: Making area features bounded by thin area features into
adjacent area features by collapsing the thin area boundaries into lines.
• Aggregation: Combining area features (e.g., buildings) separated by open
space.
• Amalgamation: Combining area features (e.g., buildings) separated by
another feature (e.g., a road).
• Dissolving: Splitting a small area into pieces (according to adjacent areas)
and merging these pieces into their corresponding adjacent areas.
• Merging: Combining two adjacent areas into one.
• Relocation: Moving more than one feature around normally to solve a
conflict problem.
• Structural simplification: Retaining the structure of area patches by select-
ing important ones and omitting less important ones.
• Typification: Retaining the typical pattern, for example, a group of area
features (e.g., buildings) aligned in rows and columns.
1.4.5 OPERATIONS FOR TRANSFORMATIONS OF 3-D SURFACES
AND FEATURES
For 3-D surfaces, two types of multi-scale representation are differentiated (Li et al.,
2005): metric and visual. In metric multi-scale representation, the features on the
same representation have the same scale. Filtering and pyramid structuring are the
methods commonly used (de Floriani, 1989). Scale-driven generalization has also
been discussed by Li and Li (1999) and Li et al. (2005). However, in visual multi-
scale representation the features on the same representation may have different
scales. In other words, the scale of the representation may vary from place to place
on the same representation. Level of detail (LoD) is the concept used to refer to
such visual multi-scale representations (Luebke et al., 2003).

For 3-D features, the operations identified by researchers are still similar to those
for area features listed in Table 1.7. However, some terms may have slightly different
meanings. For example, exaggeration has been used to refer to enlarging the size
of the doors of a building instead of the building itself (Bai and Chen, 2001). Such
exaggeration will be defined as partial exaggeration in this context. In addition,
bunching and injoining have been in use (Bai and Chen, 2001), but they are similar
to the typification and aggregation operations for 2-D representations.
In essence, there is not much difference between the operations used for 2-D
and 3-D representations. Therefore, no further discussion of the operations for the
geometric transformation of 3-D features will be conducted here.
1.5 SCOPE OF THIS BOOK
In Section 1.4 essential operations for geometric transformations of spatial represen-
tation are systematically classified. One or more algorithms are required for each of
these operations. This book is an attempt to provide a comprehensive coverage of these
algorithms. Only the low-level algorithms (or operators) for such transformations will
be presented. High-level algorithms, that is, those based on neural networks and
compound algorithms, are not included. High-level rules for controlling the algorithms
and the spatial relations between the spatial features are not also discussed.
TABLE 1.7
Operations for Geometric Transformations of a Set of Area Features
Aggregation
Agglomeration
Amalgamation
Dissolving
Merging
Relocation
(Structural)
Simplification
Typification

Introduction 25
In order to make the description of such algorithms more convenient, mathe-
matical tools that are widely used for algorithm development are included in Chapter
2, and some principles and strategies are presented in Chapter 3. From Chapter 4
on, algorithms for multi-scale spatial representations will be presented. The presen-
tations are organized according to the classifications described in Section 1.4.
Chapter 4 is dedicated to the multi-scale representation of point features. The
elimination of individual point features is an easy operation and there is no need of
any algorithm. The displacement of a point feature is similar to displacement of a
line or an area feature and will be discussed in Chapter 11, which is dedicated to
the topic of displacement. The magnification of a point feature means the enlarge-
ment of a small area feature and will be discussed in Chapter 9. Therefore, in that
chapter, only algorithms for a set of point features are presented. Typification will
be discussed in Chapter 10, which is dedicated to a set of area features because
point features under typification are small area features.
Lines have been well studied because they are the most frequently occuring
features on a topographic map and they can also be used to represent area features
(i.e., by boundaries). Various types of treatments have been made to lines. In this text,
three chapters are dedicated to individual lines. Chapter 5 presents some algorithms
for point reduction, Chapter 6 for line smoothing, and Chapter 7 for scale-driven
generalization. Displacement and partial modification are discussed in Chapter 11.
Typification of line bends is omitted here because of its subjectivity. Readers who are
interested in this operation are referred to the articles by Plazanet et al. (1995) and
Burghardt (2005). Chapter 8 presents algorithms for multi-scale representation of three
types of line networks: contours, hydrological networks (i.e., rivers), and transportation
networks (i.e., roads).
Chapter 9 presents algorithms for the multi-scale representation of individual
area features and Chapter 10 for the multi-scale representation of area features at
the class level.
Chapter 11 is dedicated to displacement. Algorithms for various types of displace-
ment are presented that are common for point, line, and area features. The last chapter,
Chapter 12, presents some more recent developments for 3-D surfaces and 3-D features.
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isation, Cartographica, 30(2), 69–80, 1993.
Robinson, A. H., Sale, R., Morrison, J. L., and Muehrcke, P. C., Elements of Cartography,
5th ed., Wiley, New York, 1984.
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and how to generalise, in Proceedings of Auto Carto 9, Baltimore, MD, 1989, pp. 56–67.
Steward, H. J., Cartographic generalization: some concepts and explanations, Canadian Car-
tographer, 11(10) (Suppl. 1), 1974.
Su, B., Morphological Transformations for Generalization of Spatial Data in Raster Format.
PhD thesis, Curtin University of Technology, Perth, Australia, 1997.
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Kraak, M. J. and Brown, A., Eds., Taylor & Francis, London, 2001, pp. 87–107.
Yan, H., Chu,Y., Li, Z. L., and Guo, R.,A quantitative description model for direction relations
based on direction groups, GeoInformatica, 10(2), 177–195, 2006.

29
Mathematical
Background
In the algorithms to be presented in the later chapters, mathematical tools at various
levels are involved. To facilitate those discussions, this chapter provides some basic
mathematical background.
2.1 GEOMETRIC ELEMENTS AND PARAMETERS
FOR SPATIAL REPRESENTATION
2.1.1 COORDINATE SYSTEMS
To make a spatial representation possess a certain level of metric quality, a
coordinate system needs to be employed. The Cartesian coordinate system is the
basic system in Euclidean space and is most familiar to us. It could be three-
dimensional (3-D) or 2-D. The latter is a result of the orthogonal projection of
the former. A geographical coordinate system is also a fundamental system for
spatial representation, consisting of longitude and latitude. A geographical coor-
dinate system can be defined on a sphere (or spheroid) or on a 2-D plane. The
latter is a result of a projection of the former. Such a projection is called a map
projection. Polar coordinate systems are also possible but are not widely used in
spatial representation. Figure 2.1 shows such systems in a 2-D plane. The Cartesian
coordinate system is normally used for spatial representation at large and medium
scales, and the geographical coordinate system is used for spatial representation
at small and very small scales.
In recent years raster systems have been in wide use. Through the history of
spatial information science, there has been an almost constant debate about the
nature and importance of raster (e.g., Peuquet, 1984; Maffini, 1987; Maguire et.
al., 1991). The importance of working with multi-scale representation in raster was
realized in the early 1980s (e.g., Monmonior, 1983). Mathematically elegant algo-
rithms have been developed since the 1990s (Li, 1994; Li and Su, 1996; Su et al.,
1997a, 1997b, 1998).
A raster space is a discrete space. It is a result of partitioning a connected space
into small pieces, but these pieces together cover the whole space contiguously. This
kind of partition is called space tessellation. Various approaches are possible for space
tessellation (Lee and Li, 1998), such as grids, hexagons, triangular irregular networks,
andVoronoi diagrams, as shown in Figure 2.2. (The latter two will be discussed in Section
2.3). Raster, as a grid-based tessellation, is only one among many possible alternatives.
2
9072_C002.fm Page 29 Friday, September 8, 2006 11:52 AM

Exploring the Variety of Random
Documents with Different Content

“Softly, softly, my dear boy. In ten or twelve years from now
I hope, D.V., to have a good balance for you at the bank,
and an income of five or six hundred pounds for you. I have
almost settled all the personal debts, and am now working
to reduce the mortgages.”
“Yes. But how about the present? Can I not realise any
money?”
“I have made some arrangements with my own bank, and
can let you have a lump sum of say five hundred pounds on
your note of hand, only if I do this it will mean drawing your
income until you are of age.”
“I am quite willing to sign any document you like if you can
do this without inconveniencing yourself, Mr Romer,” said
Ned, eagerly.
Mr Jabez Raymond gazed upon his ward for a few moments
silently, while he appeared to think. Then he spoke.
“It is not exactly professional; but as you wish to try your
fortune in Africa, while from all reports, Africa is the coming
land for fortune-making, I think it can be done. I wish you
first to read over carefully these documents, in which all I
have told you is written, and also examine carefully this
parchment, which I have drawn up for you to sign—examine
them and consult together about them. When you have
done this, if you decide to have the money down now,
instead of drawing it quarterly for the next four years, you
can let me know on my return at lunch-time.”
Mr Jabez rose as he said these words, and leaving the boys
to study the pile of parchment, he went out to attend to his
other business.

They had been accustomed to difficult problems at Dr
Heardman’s Academy, but those were simple to solve
compared to the understanding of these legally obscured
documents. After a long hour of bewilderment, Ned laid
them down respectfully in a heap, and turned dolefully upon
his trusty but equally befogged comrades.
“Well, boys, what do you make out of these mysteries?”
“Nothing, except a dry throat and an aching head,” replied
Clarence Raybold.
“Let’s go for a walk, and get some gooseberries. I’m not
going to attempt the impossible,” cried Ned, decidedly.
This proposal was grateful to the others, so together they
went out to the sun, and enjoyed themselves till lunch-time.
After lunch the lawyer read over the document which
required the signature of Ned. It sounded all right, although
terribly garnished with obscure phrases. There were blank
spaces to be afterwards filled up, such as the amounts
received, with the terms of repayment, dates, and
conditions, which Mr Jabez explained as he read in his most
fatherly tones.
It sounded all right, therefore Ned put his name boldly at
the bottom in the presence of Miss Priscilla Raymond and
his two friends, who afterwards signed theirs as witnesses.
This document dried, and locked with other papers into Mr
Raymond’s safe, the lawyer drew a cheque in his own name
and went to the bank to get it cashed.

Chapter Four.
Stephanus Groblaar.
Five hundred pounds seemed a big fortune to the three
young adventurers, who had hitherto been more than
passing wealthy on an odd half-sovereign. It was a vast
sum to think about, and its possibilities seemed limitless.
They felt likewise, as they talked over matters, that
appearances were unjustly against Mr Jabez Raymond, and
how his face and manners belied his real nature. If he had
the face of a fox, and that peculiarly slinking manner
generally ascribed to false natures and treacherous dogs,
his present actions all went to prove that he was entirely
the opposite to what these outward signs betokened. Ned
remembered how the ancient physiognomist had misread
the great Socrates; and how the good philosopher
confirmed his opinion, by telling his disciples that he might
have been the degraded being the physiognomist said he
was, but for his power of self-restraint. Perhaps Mr Jabez
Raymond had the gift of Socrates, and had mastered his
original tendencies. If, therefore, he looked and grinned like
a wily fox, while he listened to their plans, and heard them
joy over their store of cash, he certainly showed that he had
full faith in their discretion by placing this large sum so
freely at their disposal. He also exhibited the active side of
his appreciation and sympathy by aiding them in every way
that he could.
Indeed, older heads than theirs might have been a little
surprised at the extraordinary zeal he showed in advancing
them on their journey. He devoted himself so entirely to the

lads during their short stay at his house, that more
experienced people might have grown suspicious.
But to the young fellows, this exclusive attention, which
prevented them from talking with any one outside the
lawyer’s household—this eager zeal that made him
accompany them to London and attend to their comfort
while there, were so many signs that he was their best
friend and well-wisher.
Their first unpleasant impressions faded quickly away, and
they even forgot to shiver before that long and crafty grin.
Why should any man be distrusted because at times he may
remind one of a beast of some kind, when his acts are those
of a benefactor? It is by their actions people prove
themselves, not by their looks. Mr Jabez Raymond took one
of his rare holidays from business and accompanied them to
London.
He took them to a staid and respectable inn in Holborn, and
went with them to the best theatres, music-halls, and
picture-galleries during the week. On Sunday he took them
to hear two of the most celebrated preachers.
During the day he accompanied them to the shops where
outfits were to be had, and insisted on getting for them the
best that could be purchased, paying for everything himself
without a murmur. They had considered him to be a hard
man at first sight, but now they were forced to alter their
opinions when they witnessed his generosity.
He introduced them to the Dutch agent of a big South
African firm at the Cape, who was called Johannes Groblaar,
and who not only gave them much sage advice, but told
them that his nephew was going out in the next steamship,
and would accompany them if they liked. This friendly offer

they gladly accepted, as Stephanus Groblaar was a native
of Pretoria, and knew the country thoroughly. Thus they
found everything made easy for them at the start by this
benevolent and generous guardian, and after a pleasant
sojourn of eight days in London, they bade farewell to Mr
Raymond at the East India Docks, and prepared to enjoy
three weeks’ sea-voyage as saloon passengers.
Everything they could think about, even their passage-
money, had been paid from the purse of Mr Raymond, in
spite of their protests, and they were carrying their store of
funds untouched. By this time their hearts were completely
won, and they vowed that old Raymond was a jolly fine
fellow.
This jolly fine fellow stood on the deck until the bell rang for
the tender to return. He rubbed his lean hands together
when not engaged spreading one hand over his wide mouth
to cover a yawn. Then, with a hearty handshake, he
returned to the train, while the steamship proceeded on her
voyage.
To amuse himself on the journey back, he took out his
pocket-book and added up the expenses he had been put to
for the young gentlemen.
The grin had left his jaws at the last wave of his
handkerchief. He now looked grim, yet on the whole not
dissatisfied.
“Priscilla will grumble at the large outlay,” he muttered; “but
it was necessary to keep the young cub in a proper mood,
and leave a good impression. Now he can have no
suspicions, and I have four years to turn myself about, even
if he ever comes back to claim his own, which is extremely
doubtful.”

Ned Romer was going away full of faith and loving-kindness
towards this knave who had been robbing him
systematically for years. What he had given had not been a
quarter’s interest on the money due to Ned, therefore no
wonder that he felt it needful to expend this sum—
particularly with that document in his possession.
The trusting father had left him entire control of the estate,
with the possession of all papers and deeds; thus he had
not many fears about his peculations being discovered. The
parchment which Ned had signed was really a deed of sale
of all that he had inherited from his father. Being a minor, it
was as yet useless in a legal sense, but as the dates were
not yet added, Mr Raymond was prepared to advance these
dates by four years, if what he expected happened. If Ned
added his bones to the number of those who had left them
in the wilds of that fatal country, this could easily be done
without any dispute. Indeed, Mr Jabez would hardly require
any deed to step into the property which he had already
marked out as his own. Yet possible heirs might turn up
unexpectedly, and it would quash their claims. If Ned
returned, more wide awake than he had left, the wily lawyer
had all these years to prepare for him.
“I don’t think this cub will come back, and it is not likely
that he will make name enough for his death to be much
noticed.”
Mr Jabez Raymond belonged to the singular sect of
Bedlitonians, and amongst them he was a shining light as a
local preacher. The thought of what he had done, and what
he planned, did not disturb his conscience in the least. He
possessed the not uncommon quality of being able to
separate business entirely from religion; therefore the
following Sunday evening he preached a very edifying
sermon to his brethren, and went home to sister Priscilla as

full of rectitude and self-righteousness as any local preacher
could be. Humanity is crammed with such anomalies.
Stephanus Groblaar, the new companion and shipmate of
our heroes, was a pleasant and affable young Boer of about
twenty-two. As a specimen of his countrymen, he impressed
them most favourably.
He was straw-haired and grey-eyed, with skin suntanned to
a warm amber tint. Tall, burly, yet well formed, he was a
picture of rude strength and solid resolution. In repose, his
heavy features gave him rather a morose appearance,
however.
But he could be very frank and engaging when he liked, and
as he set himself to win the friendship of the young men, it
was not long before he did so.
They passed a very pleasant time going to the Cape, and
the river and coast scenery made them decide to keep
diaries.
These diaries began all right with a description of the river
Thames and coast as far as Southampton, but long before
they reached Madeira the diaries were laid aside, and never
again taken up. Jotting down ordinary events did not appeal
to our three heroes. They resolved to give their pens a rest
until they had killed their first real wild beast.
Stephanus Groblaar spoke to them sensibly and sagely.
While they listened to his prudent advice, they felt they
could not do better than act upon it.
“You will find that five or six hundred pounds will not go far
in Africa. It will be best to bank it, and try to make your
way without breaking upon your capital, or rather work for
money to add to it.”

“What do you think we should do?” enquired Ned.
“Anything that turns up. I may get you a bullock team to
help to drive up to the Transvaal. That would show you a lot
of the country, and give you plenty of experience as well. It
is rough a bit, and will take you some time, but you don’t
mind that, I suppose?”
“Not at all; it will train us to rough it in the wilds, and we
can afford to spend a little time getting colonial experience.”
This conversation took place between Madeira and the
island of Tenerife.
A curious, and what might have been a tragic adventure
had happened to Ned Romer just outside of Funchal, while
they were seeing the sights of that lovely and precipitous
island of Madeira.
The three young men had gone inland with Stephanus
Groblaar. While standing on the edge of one of the cliffs
with a sheer drop of seven hundred feet, Stephanus had
suddenly made a stumble and lurched against Ned with his
full force.
The guide, who was near at hand, saved our principal hero
from a horrible death, by what seemed like a miracle.
Ned was just going over, when the guide caught hold of his
coat-tails, and by a sudden and powerful tug, landed him on
his back over the body of Stephanus, who had fallen on his
face.
It was a considerable shock to Ned’s nerves, and he rose a
little chalky about the gills. But his pallor was nothing to
that which overspread the face of the young Boer, making
his bronzed skin look like old ivory. He shook as if he had

the palsy, and for some moments could not utter a word.
When he did find his voice, his expressions of regret and
self-reproach were painful to listen to, considering that it
was only an accident.
He said he had been seized with a sudden giddiness which
he could not account for. The guide listened to his
explanation and apologies with a stolid expression, but took
good care during the rest of the journey to keep a firm hold
of his arm when they were near any dangerous ledge.
It was while they were lying at Tenerife that the second
attack of giddiness seized Stephanus, and once more Ned
was the object against which he fell.
A portion of the ship’s rail had been removed, and Ned was
standing by the open gap, looking over the moonlit sea.
It was a lovely night and hot. Ned had come on deck in his
pyjamas to have a cool down before turning in for the night.
The deck at this part was quite deserted, as it was past
midnight. Clarence and Fred had walked over to the engine-
room, and Ned fancied that he was quite alone at that
moment.
Suddenly he felt a violent push from behind, and next
instant he was in the sea with a splash.
When he rose to the surface and cleared the water from his
eyes, he found a rope within reach, and very quickly
clambered on board, nothing the worse for his unexpected
bath.
In a few more moments, Stephanus Groblaar with Clarence
and Fred were also pulled up safely from the shark-infested
waves. The two friends had seen the accident which caused

Ned and Stephanus to tumble overboard, and without a
pause they had sprung in also.
It was lucky for the Boer that they did this, also that they
were such expert swimmers, as it appeared he could not
swim a stroke. Indeed, he was almost drowned before they
could get hold of him. It had happened as before, through
his unfortunate giddiness; this time Ned had instinctively
made a clutch at him and pulled him over, otherwise he
might only have fallen to the deck.
When Stephanus recovered his senses, he enquired
anxiously who had saved his life, and, when told that it was
Clarence and Fred, he expressed his gratitude in a few
heartfelt words, and vowed that he would never forget this
great service.
He also said how sorry he was to have imperilled the life of
Ned, and hoped he would forgive him.
Ned treated the matter as a first-class joke, but told
Stephanus that he should avoid open spaces near the sea,
since he could not swim, and mountain ledges, since he
could not fly.
“For myself, I am as much at home in the water as on dry
land; so also are my chums. It was much more dangerous,
however, at Madeira.”
“I trust this may be my last attack of giddiness,” answered
Stephanus, huskily. “It is the extra fine living on board ship
which must have made me bilious, I think.”
“Funny, isn’t it, that you should have been seized twice
when near me?” remarked Ned, unsuspiciously.

The young Boer shot a rapid and furtive glance at Ned, but
seeing how open he looked, he smiled and held out his
hand.
“It was lucky for me both times that you were in front of
me, if not so for you. In a sense you have also saved my
life, Edward Romer.”
“Not at all; only you ought to learn to swim.”
“And fly,” added Fred, who was standing beside them.
For the rest of the voyage Stephanus had no more giddy
attacks, and his young friends quickly forgot the accidents.
The Boer, however, did not forget his obligations. He was
more profuse in his expressions towards Ned; yet if quieter
towards the others, he attached himself more to them, and
showed by many signs that he liked them better than he
appeared to like Ned.
Thus the days passed pleasantly until they dropped anchor
at Cape Town, and went ashore to begin their new life.

Chapter Five.
Amongst the Cape Boers.
The first week at Cape Town shook them up more than
years of living in England could have done. They had been
only boys when they first sighted Table Mountain, but in a
week’s time they felt and acted like men.
“It is a queer place, this Cape Town,” observed Ned, as they
walked through the streets, and looked about them.
It was queer because it was all so strange and new to these
English-bred lads. The sandstorm that greeted them on
their landing did not surprise the two colonial boys as it did
Ned Romer. They endured the infliction philosophically,
while Ned groaned, and wished for a few moments that he
had stopped in dear old England.
But this gust passed, and, being the first of his experience,
it seemed the worst. In a short time he became accustomed
to sand, shortness of water, and the lack of a host of
conveniences which had appeared as necessities to him at
one time.
Stephanus Groblaar continued his protection and friendship
to them all the time they were at Cape Town and its
surrounding districts. He took them to his uncle’s house,
and so saved them the expense of living at any of the
hotels, which was a great saving to them.
The South Africans are a hospitable people, and the town-
educated Dutch very different from their country cousins,
the Transvaal Boers.

The lads were delighted with their reception and generous
treatment. They explored Table Mountain, and passed
several happy days before they had exhausted the sights of
this ancient African capital.
The uncle of Stephanus was the owner of a large and
prosperous vineyard in Stellenbosch, and he had half a
dozen fair, plump, and lively female cousins, ranging from
seven years of age to twenty-three. Stephanus was
engaged to the second oldest, a girl of nineteen. They had
also eight brothers, all living at home and assisting in the
different departments of the wine business.
It was, therefore, a large household, and when the day’s
work was over, a merry, home-like party in the evenings.
It seemed to the lads as if they were transported back a
couple of centuries while they rested in this vine farm. The
buildings were nearly the same age as the great oak trees
that surrounded them and shaded the roadways. The tiles
and bricks with which they were built had been made in and
brought from Holland. Everything was quaint, old-fashioned,
and picturesque. The master of the house was patriarchal
with his family and servants, and the mother was a real
mistress after the good old style.
Morning and evening the old Bible was brought out, and
every one was forced to join in the religious exercise. The
master did not greatly believe in his coloured servants
having souls, yet as this had come to be a disputed
question amongst some of the advanced Boers, Van
Groblaar gave them the benefit of the doubt, and made
them also attend family worship. He was a strict and severe
master with these dark-skinned bondmen and bondwomen,
yet his patriarchal system appeared to be the right one as
far as they were concerned. On this farm they did their

work much better than they would have done under the
English system.
The girls had been educated at the best Cape schools. They
could play on the piano, and had all the other
accomplishments of young ladies.
Yet this did not make them disdain household and farm
work. They were all able to milk the cows, make butter and
cheese, and do all the other duties expected from a Dutch
housewife. They reserved their fancy accomplishments for
the evenings, and were up to their daily work long before
the sun rose.
Although it was a remarkably enjoyable life which the boys
led at Stellenbosch, they quickly wearied of it, and began to
long for something more exciting. The riding lessons which
they took with the sons, and the gun practice were all very
useful, yet humiliating also, since they could never hope to
compete with those born marksmen and centaurs. It is
almost impossible for a true Africander to miss his mark or
be unseated from his horse.
As soon, therefore, as they had learnt something about the
managing of cattle and Kaffirs, and had found their way
about the country, they began to find the society of their
puritanical burgher friends slightly irksome. The charming
scenery became monotonous, and the tinkle of a piano
almost as hard to endure as a barrel-organ is to some ears.
The desire to trek had come upon them, and whenever men
or boys get that desire, no fertile oasis, no earthly paradise,
can hold them back from the desert.
Stephanus, who was in their confidence, had a private
conversation with his uncle Groblaar, and communicated the

result one morning to them as they were moping amongst
the ripening grapes.
It was not easy for the young ladies or the stolid sons of
Van Groblaar to understand how any human being could be
melancholy as long as there was plenty to eat and drink. In
their own placid minds three of the daughters had decided
that Ned, Fred, and Clarence had the makings of very good
farmers and husbands in them, and for this felt gratified to
Cousin Stephanus for bringing them.
They were considerably startled, therefore, and not a little
distressed, when they saw how our heroes brightened up
after they heard the result of that family confab.
The old Dutchman, who took a long time to decide upon
anything, had been persuaded to send up his yearly
consignment of wines and brandy to Johannesburg without
any further delay. It would go by road as usual, and the
new comrades were to go with the waggons.
By doing this they would see the country, while the journey
would not cost them anything.
This offer was gladly accepted by the young men—for they
were now, in their own and the estimation of the young
ladies, such. They no longer wondered how time was to be
killed, but eagerly began to prepare for the long and slow
overland journey.
The Groblaar wines and brandy were greatly prized, and
fetched big prices everywhere in the market. In the
Transvaal particularly they were vastly appreciated. The age
was to be depended upon, and the quality; while the grower
considered that the contents of these matured hogsheads
would be ruined if transported by any other mode than
oxen.

Another reason they had for going by road instead of rail.
There were numerous customers to be served en route, at
places outside the line of the railway.
Three of the eldest sons were deputed to go on this trek
along with our heroes and Cousin Stephanus, and as they
looked upon this journey as their annual holiday, they
provided themselves with everything needful to enjoy
themselves.
Twenty teams were required to carry the stores, provisions,
and merchandise. The oxen were all specially selected, and
the waggons and drays reliable as well as strong; so that
when they mounted their horses and inspanned, they were
a very smart and prosperous-looking caravan.
Our heroes made their farewells joyously, for they were
heart-whole. They did not notice the sad looks that followed
after them. Yet three of Van Groblaar’s young daughters did
not display their customary appetite at dinner that day, nor
did they seem much inclined for supper either that night.
Next day, however, they all made up for their unusual fast.
Ned was a little surprised when he came to say good-bye to
the young lady who had given him most of her company
during his stay, by her saying to him, in a slightly tremulous
voice—
“You are going out to a strange land, where there are many
dangers. Take care!”
“Oh, I’ll look out for number one, you bet, Miss Santa.”
“Take care of the wild-beast traps.”
“Oh yes, I know; open gaps, and that sort of thing.”

“Yes; and”—she flushed scarlet while she whispered softly
—“and look out also for Cousin Stephanus; he does not like
you.”
She turned from him swiftly as she gave this warning, and
ran indoors, while he mounted his horse, wondering what
she could mean.
Then, as he rode slowly on, he recalled the accidents on the
outward voyage, with other signs which might have escaped
his notice but for this last whisper from the young Dutch
maiden. He was not quite so guileless as he had been a few
months before. Whatever the reasons were, he felt himself
forced to the conclusion that Stephanus Groblaar did not
care greatly for him, although he seemed attached to his
two chums. Stephanus avoided him as much as possible
while they had been on the farm, and he had caught sundry
sullen and furtive glances which looked almost like hatred at
times.
Well, forewarned is forearmed to some extent. Ned shook
the momentary uneasiness and depression from his heart,
and soon was riding along merrily with the others.
Not being a fool, however, he resolved to keep a wary eye
on this supposed evil-wisher, and look out for any more
awkward fits.
It is nasty for any one to feel that he is disliked, much more
so if he has done nothing to incur that disagreeable
sentiment. Ned Romer was guiltless of anything as far as he
knew. He was the most generous and happy of the party. As
yet he had never entertained a single animosity towards a
human being. Everything that he saw entertained him and
provided him with amusement. He had no fear, and tried to
make friends with every one.

Besides, he felt specially obliged, in many ways, to
Stephanus Groblaar, and therefore would have sacrificed a
good deal to be his friend.
But a new instinct had been roused in his nature by those
parting words of Santa. The first seeds of suspicion were
sown in that generous soil. This seed would grow until it
destroyed the unwise trust of boyhood, and make of him a
vigilant and discriminating man in the future. Truly he had
left adolescence behind him when his horse walked under
the shady oak avenues of Stellenbosch.
Nothing occurred, however, to mar their harmony as they
moved slowly upward through the populated portions of
Cape Colony.
Day after day went along with varying incidents and
amusements. When they were able they spent the night at
some friendly settler’s homestead, and were most
hospitably welcomed and entertained. These were, without
exception, Dutch farmers, and old friends of the Groblaars,
so that they saw little enough of the British members of the
community.
They had mastered enough of the Cape Dutch and “Kitchen
Kaffir” idioms to understand what was said, as well as
express themselves to be understood by those they were so
constantly thrown amongst by this time. As every one was
alike free and kind, if a bit rough and homely, they took the
most favourable impression possible of this industrious if
slow-going and bigoted race.
It was not nice to hear Englishmen so constantly spoken
about with such contempt as a nation of cowards and
oppressors; yet as the Boers gave their opinions good-
naturedly, and exhibited such an utter want of knowledge in

their statements, the lads could not help laughing also as
they listened.
The farther up they travelled the more crassly ignorant and
prejudiced they found their hosts to be; yet, although they
universally insulted and tried to bespatter the Union Jack,
they universally made their English guests as heartily
welcome as were their Dutch friends. The rites of hospitality
were most generously observed. It was not that these
Dutch Africanders were all uncouth and ignorant men and
women. The majority of them were as well and even more
highly educated than are these classes in England. A large
proportion of them had likewise travelled and seen England
and the Continent. It seemed the fashion to be prejudiced
against England. They had taken their preconceived notions
along with them wherever they went, accepting only such
evidences and historical facts as suited their own side of the
disputed question. “The English are a nation of liars, and
don’t know much about anything useful. They are no use
anywhere, and they are almost done for.”
This was the universal opinion of the Dutch natives of
Africa, and no argument could move them one iota. They all
spoke banteringly and with good-tempered irony, as one
might speak of something settled and past curing or
dispute. They despised the English as a nation, abhorred
Cecil Rhodes, and laughed at Gladstone as a friendly old
imbecile. But they did not object to individuals.
The boys listened and laughed with their bigoted but
generous friends, and took all this talk in the same good
part.

Chapter Six.
The Secret Message.
There were many incidents on this overland journey, both
humorous and adventurous, which might have formed
subjects for future talk.
But the after events dwarfed these minor adventures so
completely that they were hardly ever mentioned.
Small game was plentiful on some of the open parts, and
afforded them good enough sport after a tame fashion.
Here the Dutchmen displayed their wonderful skill as
marksmen, and won unqualified admiration and respect.
When they saw the unfailing and deadly precision of that
shooting, and how little lead was wasted, the lads no longer
felt any surprise at the surrender of Dr Jameson at
Krugersdorp. Surrounded as he had been by such
sharpshooters, he had not a chance of holding out, almost
shelterless as he was. The Dutchmen were all mightily
proud of the achievements of their friends in the Transvaal,
and not at all delicate in their boasting. They were never
tired of hearing and speaking about “Bronkhurst Spruit,”
“Laing’s Nek,” and “Majuba Hill,” as well as this latest defeat
at Krugersdorp. As for Johannesburg and its craven citizens,
long before the lads saw this golden city of the veldt, its
degradation had been forced deep into their hearts by this
contemptuous banter.
Stephanus Groblaar altered his manner in a most marked
degree as they progressed up the country. On the voyage
out and at Cape Town he had seemed one of the most
advanced and liberal-minded of young Boers. He even

appeared to take the part of the Uitlanders then, and thus
had won their respect and confidence.
But now he became the loudest and most insulting of the
despisers and denouncers of everything British. He lost the
small amount of humour that he seemed to have
possessed, and which his franker cousins still retained, and
grew savage instead of bantering in his expressions.
He was returning home to Pretoria, after two years of social
intercourse with Englishmen, as full of race hatred as any of
his untravelled countrymen.
Clarence Raybold saw this new phase with silent surprise,
and listened to his exasperating observations with tightly
closed mouth and lowering eyes.
At last one night matters were brought to a crisis. They had
crossed the Vaal river, and were outspanning on the open
veldt.
Eight of their heavy-laden teams were all that remained
with them. The contents of the other twelve drays had been
disposed of on the way up, and the teams sent down the
country again with chance loads. The eldest of Santa’s
brothers alone remained with the young men and
Stephanus to look after the Transvaal business. He was a
stolid, good-natured fellow, who did his utmost to keep
peace in the camp, and turn his cousin’s ill-timed remarks
into jokes.
But Stephanus seemed bent on a quarrel that night,
although with whom it was not easy to say.
Clarence seemed to feel the insults the most keenly. Ned
Romer, however, sat quietly, and watched the young Boer
while he listened and waited. For the first time a strong

desire to measure his strength with this Dutchman came
upon him—the kind of desire that young Zulus have when
they want to wash their virgin spears.
A full moon shone over their heads and lighted up the level
landscape with pale but vivid distinctness.
“Well,” at last observed Clarence, with a lisping drawl; he
always spoke slow and lazy-like when primed up for fighting
—“well, not being in Johannesburg during the time you
speak about, Stephanus Groblaar, I cannot contradict you
as to the colour of their flag; yet if I had been, I think I’d
have done my best, young as I am, to show that there was
an equal mixture of red and blue as well as white about it.”
“Hold on till you get to Pretoria. There we make Uitlanders
walk with Kaffirs in the middle of the street.”
“Is this the rule in Pretoria?” asked Ned, gently.
“Yes, for the like of you; and we’ll make them do the same
in Johannesburg before we have done with them,” cried
Stephanus, turning on Ned with an ugly scowl.
“Nonsense. I always like the side path, and I shall use that
wherever I am,” answered Ned, laughing.
“Will you? Why, curs like you could not use this veldt as you
like unless with our permission, far less the sides of our
streets.”
“Ah, indeed, Mr Groblaar,” said Ned, rising to his feet slowly.
“Is there any particular portion of this place that you as a
free burgher might prohibit tonight?”
“Yes; I defy you to pass me now.”

They were all standing now with the exception of the cousin
Groblaar, who lay on his back snoring.
“Wait a moment, Ned,” said Clarence, softly. “I think
Stephanus only meant to stop me from walking past him.”
“No,” growled the Boer; “I did not mean you. I don’t want to
interfere with you, nor with Fred either, for you are both
colonial born and bred. It is this cur of a John Bull that I’d
teach to keep his place.”
“Good,” answered Ned. “Then this cur of a John Bull accepts
your gentlemanly challenge, and will show you that he
knows his place, and that place is, whatever spot of the
earth he finds it expedient for the advance of civilisation to
tread upon.”
He walked steadily up to the Boer with his arms held limply
down; then, before the other could put up his fists, Ned
suddenly gripped him and sent him sprawling some feet
away, while he stood where Stephanus had been.
“This is Imperial ground, you Dutch Boer, upon which the
Lion of Britain permits your people to play for the present.”
It was a grand speech, which Ned felt proud to give voice
to, and which his chums cheered. Another clear voice
behind them cried, “Bravo, young cub!” but none looked
round to hear who spoke. Stephanus did not give them time
for that.
With a hoarse roar he picked himself up, and made the rush
like a wounded buffalo. He was a powerful young man come
to his full strength, whereas Ned Romer was only ripening.
But he was heavily built, and slow in his movements in spite
of his rage. He had not had the training nor discipline which

Ned could boast of; and lastly, he had been drinking “Cape
smoke” that day, which rendered him stupid and careless.
Possibly also the overweening conceit and insolence of his
race made him contemptuous of this slender lad.
Ned, on the other hand, was in splendid condition, as lithe
and agile as a young panther, and as quick in the glance as
he was active and cool. The past three months of horse
exercise and open-air life had made his muscles like steel.
As Stephanus rushed upon him with swollen features and
blood-charged eyes, Ned waited quietly; then, with a
sudden spring aside, he shot out one fist, and landed the
Dutchman a thumper on the bridge of his nose, which
caused him to see a perfect flare of fireworks, while it made
him stagger in his tracks.
For an instant he paused, and put up both hands to his
bruised organ; then as he turned once more and removed
his hands, a dark stream burst from his nostrils, and
deluged his chin and shirt-front.
“First blood, and well drawn,” cried the clear voice again.
“Go it, my hearty; you have shown him the red, let him
have the blue next stroke.”
Fred and Clarence glanced round, to see a tall, broad-
chested stranger in a light suit and soft felt hat standing
behind them, with his horse beside him and its bridle over
his arm.
As he spoke Ned got in his second blow, and as the stranger
had advised, smote his adversary higher up and right
between the eyes. It was a loud-sounding smash, which
completely blinded Stephanus, and made it apparent to all
the onlookers that he had received his blue badge.

“These will be pretty peepers tomorrow morning,” said the
stranger; then, making a hasty step forward, he raised his
heavy riding-whip, as he exclaimed, “Ha! you would show
the white next, you treacherous dog, would you? Drop that
knife instantly.”
As he spoke he brought the stock of his whip smartly upon
the wrist of Stephanus, causing him to utter a loud yell,
while his glittering sheath-knife dropped gleaming to the
ground. Holding his damaged wrist with one hand, the
Transvaaler staggered blindly back, and abandoned the field
to the calm and victorious Ned.
“He has had enough of your fists, young man, for the
present, I expect, only be on your guard with him for the
future. Boers don’t forget blows, neither do they care much
about fighting in the open. He will try a bead on you next
from behind a kopje.”
He was an immense figure of a man who had come out of
the veldt so unexpectedly, considerably over six feet in
height and broad in proportion. His skin was ruddy, with
bold features, light, keen eyes, and he wore a small, fair
moustache. As the boys looked at him, they each thought
they had seen him somewhere before, but where they could
not at the time remember. There was about him an air of
kingly authority which fascinated them.
“Have you any coffee left?” he asked gently.
Clarence went instantly to the half-empty billy at the fire,
and brought a pannikin filled. The stranger took it with a
nod, and slowly sipped the contents, looking at them
scrutinisingly as he drank.
Cousin Groblaar still lay sleeping heavily within the shadow
of one of the waggons. Stephanus had moved away to some

considerable distance to brood over his defeat and bathe his
eyes and nose at a water-hole. The Kaffirs were also sound
asleep on their side of the fire, therefore they had this
contested part of the veldt to themselves.
“You managed that onslaught in very good style, my lad,
and have made for yourself a pretty dangerous enemy, or I
am much mistaken in my reading of faces.”
“An avowed enemy is better than a secret one, sir, and I
have good reasons to suspect Stephanus Groblaar of being
one before this night,” replied Ned.
“Ah, Groblaar is his name! Any friend of Groblaar, the vine-
grower, of Stellenbosch?”
“His nephew, sir. Yonder lies his son asleep.”
“Let him sleep,” said the stranger, hastily. “Then the young
man you punished must be the son of Burgher Groblaar, of
Pretoria?”
“I believe so, sir. At least, his home is in that city,”
answered Ned.
“Hum! thanks for this information. Then take my advice,
part company with this Stephanus Groblaar as soon as
possible, and also—don’t air those Imperial ideas too freely
when you are going to Johannesburg. They are not
fashionable there at present.”
“I will never hear my nation insulted without resenting it,
sir,” replied Ned, boldly.
“Better swallow insult than run the risk of imprisonment.”
“No, sir; I cannot endorse that sentiment.”

“It is the sentiment generally held by the Transvaal
Uitlanders.”
“I do not care. It shall never be mine.”
“Nor mine!” “Nor mine!” cried Fred and Clarence in chorus.
“Good lads,” said the stranger, in feeling tones, holding out
his large hand to our heroes, who grasped it by turns. “I
like you for your pluck and freshness. Tell me your names,
so that I may remember them if I can serve you at any
time.”
The lads at once produced their cards and presented them.
The stranger smiled humorously as he took the paste-
boards.
“Ah, you are fresh from England, I see. All the better. You
will see some sad and humbling sights in Johannesburg. But
keep up your pluck, and don’t forget that you are sons of a
mighty nation of free men.”
“Depend upon it we shall never do that, so long as the great
Cecil Rhodes stays in Africa, at any rate.” The stranger
started, and a dusky tint seemed to overspread his face.
Then he smiled and looked at the cards.
“Edward Romer! I knew a Paul Romer, of Devonshire.”
“That was my father, sir.”
“Indeed! Then I must do something for you. Clarence
Raybold. Ah, I know your father, if he lives at
Johannesburg.”
“He does, sir,” answered Clarence.

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