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Leen Ammeraal · Kang Zhang

Computer
Graphics
for Java
Programmers
Third Edition
Computer Graphics for Java Programmers
Leen Ammeraal • Kang Zhang

Computer Graphics for Java

Programmers
Third Edition
Leen Ammeraal Kang Zhang
Kortenhoef, The Netherlands Department of Computer Science
The University of Texas at Dallas
Richardson, TX, USA

ISBN 978-3-319-63356-5 ISBN 978-3-319-63357-2 (eBook)

DOI 10.1007/978-3-319-63357-2

Library of Congress Control Number: 2017947160

© Springer International Publishing AG 2017

1st edition: © John Wiley & Sons Ltd 1998
2nd edition: © John Wiley & Sons Ltd 2007
This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of
the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations,
recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission
or information storage and retrieval, electronic adaptation, computer software, or by similar or
dissimilar methodology now known or hereafter developed.
The use of general descriptive names, registered names, trademarks, service marks, etc. in this
publication does not imply, even in the absence of a specific statement, that such names are exempt
from the relevant protective laws and regulations and therefore free for general use.
The publisher, the authors and the editors are safe to assume that the advice and information in this
book are believed to be true and accurate at the date of publication. Neither the publisher nor the
authors or the editors give a warranty, express or implied, with respect to the material contained
herein or for any errors or omissions that may have been made. The publisher remains neutral with
regard to jurisdictional claims in published maps and institutional affiliations.

Printed on acid-free paper

This Springer imprint is published by Springer Nature

The registered company is Springer International Publishing AG
The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland
Preface

It has been 10 years since the publication of the second edition. The programming
language, Java, has now developed into its maturity, being the language of choice in
many industrial and business domains. Yet the skills of developing computer
graphics applications using Java are surprisingly lacked in the computer science
curricula. Though no longer active in classroom teaching, the first author has
developed and published several Android applications using Java, the main lan-
guage for Android developers. The second author has taught Computer Graphics at
his current university for the past 17 years using the first and second editions of
this textbook, apart from his previous years in Australia using different textbooks.
We feel strongly a need for updating the book.
This third edition continues the main theme of the first two editions, that is,
graphics programming in Java, with all the source code, except those for exercises,
available to the reader. Major updates in this new edition include the following:
1. The contents of all chapters are updated according to the authors’ years of
classroom experiences and recent feedback from our students.
2. Hidden-line elimination and hidden-face elimination are merged into a single
chapter.
3. A new chapter on color, texture, and lighting is added, as Chap. 7.
4. The companion software package, CGDemo, that demonstrates the working of
different algorithms and concepts introduced in the book, is enhanced with two
new algorithms added and a few bugs fixed.
5. A set of 37 video sessions (7–11 min each) in MOOC (Massive Open Online
Course) style, covering all the topics of the textbook, is supplemented.
6. A major exercise, split into four parts, on implementing the game of Tetris is
added at the end of four relevant chapters.
Many application examples illustrated in this book could be readily
implemented using Java 3D or OpenGL without any understanding of the internal
working of the implementation, which we consider undesirable for computer
science students. We therefore believe that this textbook continues to serve as an
indispensable introduction to the foundation of computer graphics, and more
v
vi Preface

importantly, how various classic algorithms are designed. It is essential for com-
puter science students to learn the skills on how to optimize time-critical algorithms
and how to develop elegant algorithmic solutions.
The example programs can be downloaded from the Internet at:
http://home.kpn.nl/ammeraal/
or at:
http://www.utdallas.edu/~kzhang/BookCG/
Finally, we would like to thank the UT-Dallas colleague Pushpa Kumar, who has
been using this textbook to teach undergraduate Computer Graphics class and
provided valuable feedback. We are grateful to Susan Lagerstrom-Fife of Springer
for her enthusiastic support and assistance in publishing this edition.

Kortenhoef, The Netherlands Leen Ammeraal

Richardson, TX, USA Kang Zhang
Contents

1 Elementary Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Pixels and Device Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Logical Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.3 Anisotropic and Isotropic Mapping Modes . . . . . . . . . . . . . . . . . 12
1.4 Defining a Polygon Through Mouse Interaction . . . . . . . . . . . . . 19
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2 Applied Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.1 Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.2 Inner Product and Vector Product . . . . . . . . . . . . . . . . . . . . . . . 31
2.3 The Orientation of Three Points . . . . . . . . . . . . . . . . . . . . . . . . 34
2.4 Polygons and Their Areas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
2.5 Point-in-Polygon Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
2.6 Triangulation of Polygons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
2.7 Point-on-Line Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
2.8 Projection of a Point on a Line . . . . . . . . . . . . . . . . . . . . . . . . . 55
2.9 Distance Between a Point and a Line . . . . . . . . . . . . . . . . . . . . . 57
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
3 Geometrical Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
3.1 Matrix Multiplication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
3.2 Linear Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
3.3 Translations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
3.4 Homogeneous Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
3.5 Inverse Transformations and Matrix Inversion . . . . . . . . . . . . . . 72
3.6 Rotation About an Arbitrary Point . . . . . . . . . . . . . . . . . . . . . . . 73
3.7 Changing the Coordinate System . . . . . . . . . . . . . . . . . . . . . . . . 78
3.8 Rotations About 3D Coordinate Axes . . . . . . . . . . . . . . . . . . . . 79
3.9 Rotation About an Arbitrary Axis . . . . . . . . . . . . . . . . . . . . . . . 80
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

vii
viii Contents

4 Classic 2D Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
4.1 Bresenham Line Drawing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
4.2 Doubling the Line-Drawing Speed . . . . . . . . . . . . . . . . . . . . . . . 97
4.3 Circle Drawing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
4.4 Cohen–Sutherland Line Clipping . . . . . . . . . . . . . . . . . . . . . . . . 106
4.5 Sutherland–Hodgman Polygon Clipping . . . . . . . . . . . . . . . . . . 112
4.6 Bézier Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
4.7 B-Spline Curve Fitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
5 Perspective and 3D Data Structure . . . . . . . . . . . . . . . . . . . . . . . . . 137
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
5.2 Viewing Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
5.3 Perspective Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
5.4 A Cube in Perspective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
5.5 Specification and Representation of 3D Objects . . . . . . . . . . . . . 149
5.6 Some Useful Classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
5.7 A Program for Wireframe Models . . . . . . . . . . . . . . . . . . . . . . . 172
5.8 Automatic Generation of Object Specification . . . . . . . . . . . . . . 177
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
6 Hidden-Line and Hidden-Face Removal . . . . . . . . . . . . . . . . . . . . . 191
6.1 Hidden-Line Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
6.2 Backface Culling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
6.3 Painter’s Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200
6.4 Z-Buffer Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220
7 Color, Texture, and Shading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225
7.1 Color Theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225
7.2 Additive and Subtractive Colors . . . . . . . . . . . . . . . . . . . . . . . . 227
7.3 RGB Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230
7.4 HSV and HSL Color Models . . . . . . . . . . . . . . . . . . . . . . . . . . . 234
7.5 Transparency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237
7.6 Texture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239
7.7 Surface Shading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251
8 Fractals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253
8.1 Koch Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253
8.2 String Grammars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257
8.3 Mandelbrot Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268
8.4 Julia Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279
Contents ix

Appendix A: Interpolation of 1/z . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281

Appendix B: Class Obj3D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285
Appendix C: Hidden-Line Tests and Implementation . . . . . . . . . . . . . . 293
Appendix D: Several 3D Objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313
Appendix E: Hints and Solutions to Exercises . . . . . . . . . . . . . . . . . . . . 349
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 385
Chapter 1
Elementary Concepts

This book is primarily about computer graphics programming and related mathe-
matics. Rather than discussing general graphics subjects for end users or how to use
graphics software, we will cover more fundamental subjects, required for graphics
programming. In this chapter, we will first understand and appreciate the nature of
discreteness of displayed graphics on computer screens. We will then see that x- and
y-coordinates need not necessarily be pixel numbers, also known as device coordi-
nates. In many applications, logical coordinates are more convenient, provided we
can convert them to device coordinates before displaying on the screen. With input
from a mouse, we would also need the inverse conversion, i.e. converting device
coordinates to logical coordinates, as we will see at the end of this chapter.

1.1 Pixels and Device Coordinates

The most convenient way of specifying a line segment on a computer screen is by

providing the coordinates of its two endpoints. In mathematics, coordinates are real
numbers, but primitive line-drawing routines may require these to be integers. This is
the case, for example, in the Java language, to be used throughout this book. The Java
Abstract Windows Toolkit (AWT) provides the class Graphics containing the method
drawLine, which we use as follows to draw the line segment connecting A and B:
g.drawLine(xA, yA, xB, yB);

The graphics context g in front of the method is normally supplied as a parameter

of the paint method in the program, and the four arguments of drawLine are
integers, ranging from zero to some maximum value. The above call to drawLine
produces exactly the same line on the screen as this one:

g.drawLine(xB, yB, xA, yA);

© Springer International Publishing AG 2017 1

L. Ammeraal, K. Zhang, Computer Graphics for Java Programmers,
DOI 10.1007/978-3-319-63357-2_1
2 1 Elementary Concepts

We will now use statements such as the above in a complete Java program.
Fortunately, you need not type these programs yourself, since they are available
from the Internet, as specified in the Preface. It will also be necessary to install the
Java Development Kit (JDK). If you are not yet familiar with Java, you should
consult other books, such as those mentioned in the Bibliography. This book
assumes you to be fluent in basic Java programming.
The following program draws the largest possible rectangle in a canvas. The
color red is used to distinguish this rectangle from the frame border:
// RedRect.java: The largest possible rectangle in red.
import java.awt.*;
import java.awt.event.*;

public class RedRect extends Frame {

public static void main(String[] args) {new RedRect();}

RedRect() {
super("RedRect");
addWindowListener(new WindowAdapter() {
public void windowClosing(WindowEvent e) {Sytem.exit(0);}
});
setSize(300, 150);
add("Center", new CvRedRect());
setVisible(true);
}
}

class CvRedRect extends Canvas {

public void paint(Graphics g) {
Dimension d = getSize();
int maxX = d.width - 1, maxY = d.height - 1;
g.drawString("d.width = " + d.width, 10, 30);
g.drawString("d.height = " + d.height, 10, 60);
g.setColor(Color.red);
g.drawRect(0, 0, maxX, maxY);
}
}

The call to drawRect almost at the end of this program has the same effect as
these four lines:
g.drawLine(0, 0, maxX, 0); // Top edge
g.drawLine(maxX, 0, maxX, maxY); // Right edge
g.drawLine(maxX, maxY, 0, maxY); // Bottom edge
g.drawLine(0, maxY, 0, 0); // Left edge
1.1 Pixels and Device Coordinates 3

The program contains two classes:

RedRect: The class for the frame, also used to close the application.
CvRedRect: The class for the canvas, in which we display graphics output.
However, after compiling the program by entering the command

javac RedRect.java

we notice that three class files have been generated: RedRect.class, CvRedRect.
class and RedRect$1.class. The third one is referred to as an anonymous class since
it has no name in the program. It is produced by the following program segment:

addWindowListener(new WindowAdapter() {
public void windowClosing(WindowEvent e) {System.exit(0);}
});

which enables the user of the program to terminate it in the normal way. The
argument of the method addWindowListener must be an object of a class that
implements the interface WindowListener. This implies that this class must define
seven methods, one of which is windowClosing. The base class WindowAdapter
defines these seven methods as do-nothing functions. In the above program seg-
ment, the argument of addWindowListener denotes an object of an anonymous
subclass of WindowAdapter. In this subclass we override the method
windowClosing.
The RedRect constructor shows that the frame size is set to 400
200. If we do
not modify this size (by dragging a corner or an edge of the window), the canvas
size is somewhat smaller than the frame. After compilation, we run the program by
typing the command
java RedRect

which, with the given frame size, produces the largest possible red rectangle, shown
in Fig. 1.1 just inside the frame.
The blank area in a frame, which we use for graphics output, is referred to as a
canvas, which is a subclass, such as CvRedRect in program RedRect.java, of the
AWT class Canvas. If, instead, we displayed the output directly in the frame, we

Fig. 1.1 Largest possible

rectangle and canvas
dimensions
4 1 Elementary Concepts

would have a problem with the coordinate system: its origin would be in the top-left
corner of the frame; in other words, the x-coordinates increase from left to right and
y-coordinates from top to bottom. Although there is a method getInsets to obtain the
widths of all four borders of a frame so that we could compute the dimensions of the
client rectangle ourselves, we prefer to use a canvas.
The tiny screen elements that we can assign a color are called pixels (short for
picture elements), and the integer x- and y-values used for them are referred to as
device coordinates. Although there are 200 pixels on a horizontal line in the entire
frame, only 192 of these lie on the canvas, the remaining 8 being used for the left
and right borders. On a vertical line, there are 100 pixels for the whole frame, but
only 73 for the canvas. Apparently, the remaining 27 pixels are used for the title bar
and the top and bottom borders. Since these numbers may differ on different Java
implementations and the user can change the window size, it is desirable that our
program can determine the canvas dimensions. We do this by using the getSize
method of the class Component, which is a superclass of Canvas. The following
program lines in the paint method show how to obtain the canvas dimensions and
how to interpret them:
Dimension d = getSize();
int maxX = d.width - 1, maxY = d.height - 1;

The getSize method of Component (a superclass of Canvas) supplies us with the

numbers of pixels on horizontal and vertical lines of the canvas. Since we start
counting at zero, the highest pixel numbers, maxX and maxY, on these lines are one
less than these numbers of pixels. Remember that this is similar with arrays in Java
and C. For example, if we write:

int[] a = new int[8];

the highest possible index value is 7, not 8. In such cases, the “index” is always one
less than “size”. Figure 1.2 illustrates this for a very small canvas, which is only
8 pixels wide and 4 high, showing a much enlarged screen grid structure. It also
shows that the line connecting the points (0, 0) and (7, 3) is approximated by a set of
eight pixels.

Fig. 1.2 Pixels as

coordinates in a 8
4
canvas (with maxX ¼ 7
and maxY ¼ 3)
1.1 Pixels and Device Coordinates 5

The big dots approximating the line denote pixels that are set to the foreground
color. By default, this foreground color is black, while the background color is
white. These eight pixels are made black as a result of this call:

g.drawLine(0, 0, 7, 3);

In the program RedRect.java, we used the following call to the drawRect method
(instead of four calls to drawLine):
g.drawRect(0, 0, maxX, maxY);

In general, the call:

g.drawRect(x, y, w, h);

draws a rectangle with (x, y) as its top-left and (x + w, y + h) as its bottom-right

corners. In other words, the third and fourth arguments of the drawRect method
specify the width and height, rather than the bottom-right corner, of the rectangle to
be drawn. Note that this rectangle is w + 1 pixels wide and h + 1 pixels high. The
smallest possible square, consisting of 2
2 pixels, is drawn by this call:

g.drawRect(x, y, 1, 1);

To put only one pixel on the screen, we cannot use drawRect, because nothing at
all appears if we try to set the third and fourth arguments of this method to zero.
Curiously enough, Java does not provide a special method for this purpose, so we
have to use this method:
g.drawLine(x, y, x, y);

Note that the method:

g.drawLine(xA, y, xB, y);

draws a horizontal line consisting of jxB – xAj + 1 pixels.

In mathematics, lines are continuous without thickness, but are discrete and at
least one pixel thick in computer graphics output. This difference in the interpre-
tation of the notion of lines may not cause any problems if the pixels are very small
in comparison with what we are drawing. However, we should be aware that there
may be such problems in special cases, as Fig. 1.3a illustrates. Suppose that we have
to draw a filled square ABCD of, say, 4
4 pixels, consisting of the bottom-right
triangle ABC and the upper-left triangle ACD, which we want to paint in dark gray
and light gray, respectively, without drawing any lines. Strangely enough, it is not
clear how this can be done: if we make the diagonal AC light gray, triangle ABC
contains fewer pixels than triangle ACD; if we make AC dark gray, it is the other
way round.
6 1 Elementary Concepts

Fig. 1.3 Small filled

regions

A much easier but still non-trivial problem, illustrated in Fig. 1.3b, is filling a
checker-board with, say, dark and light gray squares instead of black and white
ones. Unlike squares in mathematics, those on the computer screen deserve special
attention with regard to the edges belonging or not belonging to the filled regions.
We have seen that the call:
g.drawRect(x, y, w, h);

draws a rectangle with corners (x, y) and (x + w, y + h). The method fillRect, on the
other hand, fills a slightly smaller rectangle. The call:
g.fillRect(x, y, w, h);

assigns the current foreground color to a rectangle consisting of w
h pixels. This

rectangle has (x, y) as its top-left and (x + w  1, y + h  1) as its bottom-right
corner. To obtain a generalization of Fig. 1.3b, the following method, checker,
draws an n
n checker board, with (x, y) as its top-left corner and with dark gray
and light gray squares, each consisting of w
w pixels. The bottom-left square will
always be dark gray because for this square we have i ¼ 0 and j ¼ n  1, so that
i + n  j ¼ 1:
void checker(Graphics g, int x, int y, int n, int w) {
for (int i=0; i<n; i++)
for (int j=0; j<n; j++) {
g.setColor((i + n - j) % 2 == 0 ?
Color.lightGray : Color.darkGray);
g.fillRect(x + i * w, y + j * w, w, w);
}
}

If we wanted to draw only the edges of each square, also in dark gray and light
gray, we would have to replace the above call to fillRect with
g.drawRect(x + i * w, y + j * w, w - 1, w - 1);

in which the last two arguments are w – 1 instead of w.

1.2 Logical Coordinates 7

1.2 Logical Coordinates

The Direction of the y-axis

As Fig. 1.2 shows, the origin of the device-coordinate systems lies at the top-left
corner of the canvas, so that the positive y-axis points downward. This is reasonable
for text output, that starts at the top and increases y as we go to the next line of text.
However, this direction of the y-axis is different from typical mathematical practice
and therefore often inconvenient in graphics applications. For example, in a dis-
cussion about a line with a positive slope, we expect to go upward when moving
along this line from left to right. Fortunately, we can arrange for the positive
y direction to be reversed by performing this simple transformation:

y 0 ¼ maxY  y

Continuous Versus Discrete Coordinates

Instead of the discrete (integer) coordinates at the lower, device oriented level, we
often wish to use continuous (floating-point) coordinates at the higher, problem-
oriented level. Other usual terms are device and logical coordinates, respectively.
Writing conversion routines to compute device coordinates from the corresponding
logical ones and vice versa is a bit tricky. We must be aware that there are two
solutions to this problem: rounding and truncating, even in the simple case in which
increasing a logical coordinate by one results in increasing the device coordinate
also by one. We wish to write the following methods:
iX(x), iY( y): converting the logical coordinates x and y to device coordinates;
fx(x), fy( y): converting the device coordinates X and Y to logical coordinates.
One may notice that we have used lower-case letters to represent logical coordi-
nates and capital letters to represent device coordinates. This will be the convention
used throughout this book. With regard to x-coordinates, the rounding solution
could be:

int iX(float x){return Math.round(x);}

float fx(int x){return (float)x;}

For example, with this solution we have

iXð2:8Þ ¼ 3 and fxð3Þ ¼ 3:0

The truncating solution could be:

int iX(float x){return (int)x;} // Not used in

float fx(int x){return (float)x + 0.5F;} // this book.
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