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There is no question that computer graphics has become an important eld that pervades our lives in many areas projection shows more of the object

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David Salomon,Transformations,and Projections in,Computer Graphics. Professor David Salomon emeritus,Computer Science Department. California State University,Northridge CA 91330 8281. Email david salomon csun edu, 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. Springer Verlag London Limited 2006, Apart from any fair dealing for the purposes of research or private study or criticism or review as permitted under. the Copyright Designs and Patents Act 1988 this publication may only be reproduced stored or transmitted in any. form or by any means with the prior permission in writing of the publishers or in the case of reprographic. reproduction in accordance with the terms of licences issued by the Copyright Licensing Agency Enquiries. concerning reproduction outside those terms should be sent to the publishers. The use of registered names trademarks etc in this publication does not imply even in the absence of a specific. statement that such names are exempt from the relevant laws and regulations and therefore free for general use. The publisher makes no representation express or implied with regard to the accuracy of the information. contained in this book and cannot accept any legal responsibility or liability for any errors or omissions that may. Printed in the United States of America EB,9 8 7 6 5 4 3 2 1. springer com,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. 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 re ecting or otherwise systematically deforming a. geometric gure 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. In genetics a The change undergone by a cell upon infection by a cancer causing. virus b The alteration of a bacterial cell caused by the transfer of DNA from another. 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 gure reproduced in. n 1 or fewer dimensions The most common case is for n 3 discussed in this book. 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 signi cance 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 ood is. The technique of representing three dimensional objects and depth relationships on. a two dimensional surface discussed in this book, Adjective Of relating to seen or represented in perspective.

This is why writing is such a liberating thing You get to know what you didn t know. Richard Lederer, There is no question that computer graphics has become an important eld 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 feature lms are being created entirely by computers Graphics software enables. 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 elds but this. book concentrates on the three key terms mentioned above Following is a short discus. 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 re ected 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 simpli ed 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 di erently 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. 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 from n 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. 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 di cult to specify A. good example is rotations In two dimensions there are only two directions clockwise. and counterclockwise for a rotation and rotations are performed about a point In three. dimensions rotations are about an axis and the terms clockwise and counterclockwise. are ambiguous, Fortunately all the important two dimensional transformations can be speci ed by.

a 3 3 transformation matrix and this matrix is easy to extend to the three dimensional. case where all the important transformations can be speci ed by means of a 4 4 matrix. 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 combinations of transformations such as a rotation followed by a re ection 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. 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 di cult to compute dimensions of. parts of the object because each face of the object may be shrunk by a di erent factor. when drawn in the projection, Oblique projections are similar to axonometric projections and employ certain pro. 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 sheye 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 gures, The heart of mathematics consists of concrete examples and concrete problems. Paul Halmos How to Write Mathematics 1973, I have collected and developed the material in this book over many years of studying.

and teaching Some of it has been published in Salomon 99 and in various class notes. but most of it is seeing the light of day for the rst time in this book I hope the readers. will nd the presentation clear and unambiguous and will immediately bring any errors. 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. requires no previous knowledge of transformations projections or perspective The key. ideas are introduced slowly are examined whenever possible from several points of. view and are illustrated by gures examples and solved exercises The discussion. must involve some mathematics but it is nonrigorous and therefore easy to grasp The. mathematical background required is the basics of linear algebra mostly vectors vector. operations and matrices The following features enhance the usefulness of the book. The book has many gures It is my belief that a book on aspects of graphics. should have gures to illustrate the concepts under discussion Drawings paintings and. photographs are included Most color gures have been printed in place in grayscale. All of them appear in color in Appendix C, Many exercises are sprinkled throughout the text These are important and should.

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 by David.

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 da er 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 handprint 06 has a detailed discussion titled Elements of Perspective. 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 dsalomon csun edu but any email sent to email address. anyname 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 xed ideas a preface without a volume. Thomas Hardy Tess of the d Urbervilles 1891,Preface vii. Introduction 1,1 Transformations 5,1 1 Introduction 5. 1 2 Two Dimensional Transformations 8,1 3 Three Dimensional Coordinate Systems 36.

1 4 Three Dimensional Transformations 37,1 5 Transforming the Coordinate System 54. 2 Parallel Projections 57,2 1 Orthographic Projections 58. 2 2 Axonometric Projections 60,2 3 Oblique Projections 67. 3 Perspective Projection 71,3 1 One Two Three In nity 73. 3 2 History of Perspective 78,3 3 Perspective in Curved Objects 84.

3 4 The Mathematics of Perspective 87,3 5 General Perspective 96. 3 6 Transforming the Object 101,3 7 Viewer at an Arbitrary Location 105. 3 8 A Coordinate Free Approach I 114,3 9 A Coordinate Free Approach II 117. 3 10 The Viewing Volume 121,3 11 Stereoscopic Images 123. 3 12 Creating a Stereoscopic Image 128,3 13 Viewing a Stereoscopic Image 132.

3 14 Autostereoscopic Displays 142,xiv Contents,4 Nonlinear Projections 145. 4 1 False Perspective 145,4 2 Fisheye Projection 147. 4 3 Circle Inversion 162,4 4 Panoramic Projections 166. 4 5 Cylindrical Panoramic Projection 167,4 6 Spherical Panoramic Projection 174. 4 7 Cubic Panoramic Projection 180,4 8 Six Point Perspective 183.

4 9 Other Panoramic Projections 185,4 10 Panoramic Cameras 188. 4 11 Telescopic Projection 194,4 12 Microscopic Projection 196. 4 13 Anamorphosis 197,4 14 Map Projections 199,A Vector Products 221. B Quaternions 227,C Color Figures 231,Answers to Exercises 243. Bibliography 277, The contents as in part I understand them are to blame.

William Shakespeare King Lear act I scene II,Introduction. The 1960s were the golden age of computer graphics This was the time when many. of the basic methods algorithms and techniques were developed improved and im. plemented Two of the most important concepts that were identi ed and studied in. those years were transformations and projections Workers in the graphics eld im. mediately recognized the importance of transformations Once a graphical object is. created the use of transformations enables the designer to create copies of the object. and modify them in signi cant ways The necessity of projections was also realized. early Sophisticated graphics requires three dimensional objects but graphics output. devices are two dimensional A three dimensional object has to be projected on the at. output device in a way that will preserve its depth information Thus early researchers. in computer graphics developed the mathematics of parallel and perspective projec. tions and implemented these techniques Nonlinear projections deform the projected. image in various ways and are mostly used for artistic and ornamental purposes These. projections were also studied and implemented over the years by many people. Exercise Intro 1 Most nonlinear projections are valued for their artistic and orna. mental e ects but there is at least one type of nonlinear projection that has important. applications What is it, Today transformations and projections are important components of computer. graphics and computer aided design CAD Transformations save the designer work. and time while projections are necessary because three dimensional output devices are. still rare but see deeplight 06 for a new revolutionary technique for three dimensional. displays hence this book, Figure Intro 1 shows the power of even the simplest two dimensional transforma. tions It illustrates from left to right the following transformations rotation re ection. deformation shearing and scaling see also Figure 1 1 It is not di cult to imagine. the power of combining these transformations but it is more di cult to imagine and. visualize the power and exibility of three dimensional transformations. The basic two dimensional transformations are translation rotation re ection. scaling and shearing They are simple but it is their combinations that make them. powerful It comes as a surprise to realize that these transformations can be speci ed. 2 Introduction, Figure Intro 1 Elementary Two Dimensional Transformations. by means of a single 3 3 matrix where only six of the nine elements are used The. same ve basic transformations also exist in three dimensions but have more degrees. of freedom and therefore require more parameters to fully specify them The general. transformation matrix in three dimensions is 4 4 where 13 of the 16 elements control. the transformations and 3 are used to specify the orientation of the projection plane in. the case of perspective projections, Exercise Intro 2 What transformations are possible in one dimension.

In contrast with the ve basic transformations there are more than ve types of. projections As Figure Intro 2 illustrates we distinguish between linear and nonlinear. projections The former class consists of parallel and perspective projections while the. latter class includes many di erent types Each type of projection has variants Thus. parallel projections are classi ed into orthographic axonometric and oblique while. perspective projections include one two and three point projections. Projections,Linear Nonlinear,Fisheye Panorama Telescopic. Parallel Perspective,Orthographic One point Microscopic Map others. Axonometric Two point,Oblique Three point,Figure Intro 2 Classi cation of Projections. Nonlinear projections are all di erent and employ di erent approaches and ideas. Linear projections on the other hand are all based on the following simple rule of. projection,Introduction 3, Rule A three dimensional object is projected on a two dimensional plane called. the projection plane The object must be fully located on one side of the plane and. we imagine a viewer or an observer located on the other side On that side we select a. point termed the center of projection and it is the location of this point that determines. the class of linear projection parallel or perspective A three dimensional point P. on the object is projected to a two dimensional point P on the projection plane by. connecting P to the center of projection with a straight segment Point P is placed. at the intersection of this segment with the projection plane When the center of. projection is at in nity the result is a parallel projection If the center of projection is. at the observer the projection is perspective, dictionary de nition of projection the representation of a gure or solid on a plane.

as it would look from a particular direction, Emma was not sorry to be pressed She read and was surprised The. style of the letter was much above her expectation There were not. merely no grammatical errors but as a composition it would not have. disgraced a gentleman the language though plain was strong and. una ected and the sentiments it conveyed very much to the. credit of the writer It was short but expressed good sense. warm attachment liberality propriety even delicacy of feeling.

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