Table Of ContentTransformations and Projections in
Computer Graphics
David Salomon
Transformations
and Projections in
Computer Graphics
Professor David Salomon (emeritus)
Computer Science Department
California State University
Northridge, CA 91330-8281
USA
Email: [email protected]
Cover illustration: Adapted from figure 4.25, courtesy of Ari Salomon.
British Library Cataloguing in Publication Data
A catalogue record for this book is available from the British Library
Library of Congress Control Number: 2006923906
ISBN-10: 1-84628-392-2
ISBN-13: 978-1-84628-392-5
Printed on acid-free paper
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Dedicated to Dick Termes, whose work and talent
have contributed much to the quality of this book.
If there’s a book you really want to read, but it
hasn’t been written yet, then you must write it.
—Toni Morrison
Preface
It is probably a coincidence that the three main terms discussed in this book, namely
transformations, projections, and perspective, are ambiguous. Here is what the dictio-
nary has to say about these terms.
Transformation
(a) The act or an instance of transforming. (b) The state of being transformed.
A marked change, as in appearance or character, usually for the better.
Mathematical transformation. (a) Replacing a variable in an expression by its
value. (b) Mapping a mathematical space onto another or onto itself.
In geometry. Moving, rotating, reflecting, or otherwise systematically deforming a
geometric figure (discussed in this book).
In linguistics. (a) A rule to convert a syntactic form into another. (b) A sentence
or sentential form derived by such a rule; a transform.
Ingenetics. (a)Thechangeundergonebyacelluponinfectionbyacancer-causing
virus. (b)ThealterationofabacterialcellcausedbythetransferofDNAfromanother
bacterial cell, especially a pathogen.
Projection
The act of projecting or the condition of being projected.
(a) An object or part thereof that extends outward. (b) Spiky projections on top
of a fence. (c) A projection of land along the coast.
A prediction or an estimate of a future situation, based on current data or trends.
(a) The process of projecting a recorded image onto a viewing surface. (b) An
image so projected.
In mathematics. The image of an n-dimensional geometric figure reproduced in
n−1orfewerdimensions. Themostcommoncaseisfor n=3(discussedinthisbook).
viii Preface
In psychology. The attribution of one’s own beliefs or suppositions to others (such
as when a scientist projects his beliefs into the subjects of his research or into theories
he develops).
Perspective
(a) A view or scene. (b) A mental view or outlook.
The appearance of objects in depth as perceived by normal binocular vision.
(a) The relationship of aspects of a subject to each other and to a whole, “let’s
put this into perspective.” (b) Subjective evaluation of relative significance, “in my
perspective as an electrician, this wire is defective.” (c) The ability to perceive things
in their actual interrelations or comparative importance “in perspective, this flood is
minor.”
Thetechniqueofrepresentingthree-dimensionalobjectsanddepthrelationshipson
a two-dimensional surface (discussed in this book).
(Adjective) Of, relating to, seen, or represented in perspective.
Thisiswhy writing issuch a liberatingthing. You get to knowwhat you didn’t know
you knew.
—Richard Lederer
There is no question that computer graphics has become an important field that
pervades our lives in many areas. Many advertisements on television and in magazines
are graphical and are created on computers. The screens of computers, PDAs, cellular
telephones, and similar devices interact graphically with the user. More and more full-
length featurefilmsarebeing created entirely by computers. Graphicssoftwareenables
users to draw engineering plans, to create technical and artistic illustrations, and to
develop fonts of text. (For a short history of computer graphics, see [hocg 06].)
Computer graphics is an immense discipline, encompassing many fields, but this
bookconcentratesonthethreekeytermsmentionedabove. Followingisashortdiscus-
sion of each term.
The term “transformation” as discussed in this book refers to a geometric opera-
tion applied to all the points of an object. An object may be moved, rotated, or scaled
(shrunk or stretched). It may be reflected about a plane (as in a mirror) or deformed
in some way, as illustrated by Figures Intro.1 and 1.1. Several transformations may
be combined and may completely change the position, orientation, and shape of the
object. Many graphics operations are greatly simplified with the help of transforma-
tions. A forest can be created from a single tree by duplicating the tree several times
and moving and transforming each copy differently. An object can be animated by
moving it along a path in small steps while also rotating and scaling it slightly at each
step. Transformations, both two-dimensional and three-dimensional, are discussed in
Chapter 1.
Currently, virtually all our graphics output devices are two dimensional, but many
graphics projects and objects are three-dimensional. Converting a three-dimensional
graphics object or scene into two dimensions is a mathematical operation called projec-
tion. In general, a projection transforms an object fromn dimensions to n−1 or fewer
Preface ix
dimensions, but in computer graphics n is always 3. Because of the loss of dimensions,
an object loses some of its details when projected. It is therefore important to study
the various types of projections and always use the right one. Chapters 2 through 4
describe the three main classes of projections: parallel, perspective, and nonlinear.
(cid:2) Exercise Pre.1: Discuss the impossible fork of Figure Pre.1.
Figure Pre.1: An Impossible Fork.
Perspective (or more accurately, linear perspective) is the general name of several
techniques that create the illusion of depth in a two-dimensional drawing. The rules of
perspective determine where and how to place objects in a painting or drawing so that
they appear to have depth and seem to be at the correct distance from the observer. A
picture in perspective creates in the viewer’s brain the same sensation as the original
three-dimensional scene. The main tool employed by linear perspective is vanishing
points. Perspective, including its history, its use in art, its applications to computer
graphics, and its mathematical representation, is the topic of Chapter 3.
Following is a short description of the chapters and appendices of the book.
Chapter 1 introduces geometric transformations. Both two-dimensional and three-
dimensional transformations are included, and it is shown that the latter are more
plentiful and more complex than the former and are also more difficult to specify. A
good example is rotations. In two dimensions there are only two directions, clockwise
andcounterclockwise,forarotationandrotationsareperformedaboutapoint. Inthree
dimensions, rotations are about an axis and the terms clockwise and counterclockwise
are ambiguous.
Fortunately,alltheimportanttwo-dimensionaltransformationscanbespecifiedby
a3×3transformationmatrix,andthismatrixiseasytoextendtothethree-dimensional
case,wherealltheimportanttransformationscanbespecifiedbymeansofa4×4matrix.
Thus, the use of a transformation matrix is elegant and leads to a deep understanding
of transformations.
Other topics discussed in this chapter are (1) the use of homogeneous coordinates,
(2)combinationsoftransformations, suchasarotationfollowedbyareflection, and(3)
transforming the coordinate system instead of the object.
The remainder of the book is devoted to projections, and Chapter 2 introduces
parallel projections. These are used mostly in engineering drafting but can also be
found in Eastern art. There are three classes of parallel projections: orthographic,
axonometric, and oblique (although it is shown at the end of this chapter that the last
two types are similar). An orthographic projection displays one side or one face of the
x Preface
object. The downside of this type is that three projections are needed in order to see
the entire object. On the other hand, it is easy to compute dimensions of object details
from measurements made on the projection.
Axonometric projections normally show three sides of the object. Thus, a single
projection shows more of the object, but it is more difficult to compute dimensions of
parts of the object because each face of the object may be shrunk by a different factor
when drawn in the projection.
Obliqueprojectionsaresimilartoaxonometricprojectionsandemploycertainpro-
jection angles in order to simplify the process of measuring and computing dimensions.
Perspective projections are the topic of Chapter 3. The chapter starts with an
intuitive explanation of the important concept of vanishing points. It follows with a
short history of perspective, its origins, and its applications to art. The short but im-
portant Section 3.3 is devoted to perspective projection in curved objects, a topic that
is neglected by most texts on perspective. The bulk of the chapter develops the math-
ematics of perspective in a systematic way, approaching this topic from several points
of view and illustrating it with examples. The chapter ends with a long presentation of
stereoscopic images, an important application of perspective.
Chapter 4 treats the important (and alas neglected) topic of nonlinear projections.
The most important nonlinear projections are the fisheye projection (Section 4.2), the
panoramic projection (Section 4.4), and the many sphere projections (Section 4.14). In
addition, this chapter includes material and examples on circle inversion (Section 4.3),
six-point perspective (Section 4.8), panoramic cameras (Section 4.10), telescopic and
microscopic projections (Sections 4.11 and 4.12), and anamorphosis (Section 4.13).
Appendix A, on vector products, and Appendix B, on quaternions, provide infor-
mation on these mathematical topics that may be unfamiliar to some readers. Finally,
Appendix C consists of color figures.
The heart of mathematics consists of concrete examples and concrete problems.
—Paul Halmos, How to Write Mathematics (1973).
Ihavecollectedanddevelopedthematerialinthisbookovermanyyearsofstudying
and teaching. Some of it has been published in [Salomon 99] and in various class notes,
butmostofitisseeingthelightofdayforthefirsttimeinthisbook. Ihopethereaders
willfindthepresentationclearandunambiguousandwillimmediatelybringanyerrors,
omissions, and misprints to my attention.
I cannot tell my learned reader (whose eyebrows, I suspect, have by now
traveled all the way to the back of his bald head), I cannot tell him how
the knowledge came to me.
—Vladimir Nabokov, Lolita (1955).
Readership of the Book
This book is aimed mostly at mathematically mature readers (i.e., those who can deal
comfortably with mathematical abstractions), who are familiar with computers and
Preface xi
computer graphics and are looking for a mathematically easy presentation of the trans-
formations and projections used in computer graphics. The material presented here
requiresnopreviousknowledgeoftransformations,projections,orperspective. Thekey
ideas are introduced slowly, are examined, whenever possible, from several points of
view, and are illustrated by figures, examples, and (solved) exercises. The discussion
must involve some mathematics, but it is nonrigorous and therefore easy to grasp. The
mathematicalbackgroundrequiredisthebasicsoflinearalgebra,mostlyvectors,vector
operations, and matrices. The following features enhance the usefulness of the book:
The book has many figures. It is my belief that a book on aspects of graphics
shouldhavefigurestoillustratetheconceptsunderdiscussion. Drawings,paintings,and
photographs are included. Most color figures have been printed in place in grayscale.
All of them appear in color in Appendix C.
Manyexercisesaresprinkledthroughoutthetext. Theseareimportantandshould
be worked out. The answers are also provided, but should be consulted only to verify
the reader’s own answer, or as a last resort.
Learn from other people’s mistakes. Life isn’t long enough to make them all yourself.
—Harry S. Truman
Books and Internet resources for transformations and projections.
Godel, Escher, Bach: An Eternal Golden Braid, by Douglas Hofstadter. Basic
Books, 20th Anniversary edition, 1999. This classical volume discusses symmetries in
art, literature, and science.
Transformation Geometry: An Introduction to Symmetry, by George E. Martin.
Springer-Verlag, 1982. An excellent mathematical reference.
Symmetry Discovered: Concepts and Applications in Nature and Science, by Joe
Rosen. Dover Press, 1975. An accessible introduction to the ideas of symmetry.
The New Ambidextrous Universe, by Martin Gardner. W. H. Freeman and Com-
pany, 1990. A beautifully written exploration of symmetry.
Symmetry, by Hermann Weyl. Princeton University Press, 1952. A classic illus-
trated introduction to symmetry.
The Renaissance and Rediscovery of Linear Perspective, by Samuel Y. Edgerton.
Harper and Row, 1976 (especially chapters 9 and 18).
Secret Knowledge: Rediscovering the Lost Techniques of the Old Masters,byDavid
Hockney. Viking, 2001.
The Science of Art, Optical Themes in Western Art From Brunelleschi to Seurat,
by Martin Kemp. Yale University Press, 1990.
The Life of Brunelleschi, by Antonio Tuccio Manetti, edited by Howard Saalman.
Penn State University, 1970.
Geometry: An Investigative Approach, and Laboratory Investigations in Geometry,
by Phares G. O’daffer and Stanley R. Clemens. Addison-Wesley, 1976.
Reference [Wolfram 06] has information, examples of, and code to create many
panoramic projections and map projections.
Reference[handprint06]hasadetaileddiscussiontitled“ElementsofPerspective.”
xii Preface
Currently, the book’s Web site is part of the author’s Web site, which is located
at http://www.ecs.csun.edu/~dsalomon/. Domain name DavidSalomon.name has
been reserved and will always point to any future location of the Web site. The
author’s email address is [email protected], but any email sent to email address
(cid:3)anyname(cid:4)@DavidSalomon.name will reach the author.
This book is dedicated to Dick Termes whose work and talent have contributed
much to the quality of the book. The many images by Dick that are included in the
book serve to illustrate important concepts. In addition, I would like to thank Ari
Salomon for Figure 4.25 and Professor Shinji Araya, Fukuoka Institute of Technology,
for Figure 4.32.
Lakeside, California David Salomon
The university as a step to anything but ordination seemed,
to this man of fixed ideas, a preface without a volume.
—Thomas Hardy, Tess of the d’Urbervilles (1891)
