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Viewpoints: Mathematical Perspective and Fractal Geometry in Art

Marc Frantz, Annalisa Crannell

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An undergraduate textbook devoted exclusively to relationships between mathematics and art, Viewpoints is ideally suited for math-for-liberal-arts courses and mathematics courses for fine arts majors. The textbook contains a wide variety of classroom-tested activities and problems, a series of essays by contemporary artists written especially for the book, and a plethora of pedagogical and learning opportunities for instructors and students. Viewpoints focuses on two mathematical areas: perspective related to drawing man-made forms and fractal geometry related to drawing natural forms. Investigating facets of the three-dimensional world in order to understand mathematical concepts behind the art, the textbook explores art topics including comic, anamorphic, and classical art, as well as photography, while presenting such mathematical ideas as proportion, ratio, self-similarity, exponents, and logarithms. Straightforward problems and rewarding solutions empower students to make accurate, sophisticated drawings. Personal essays and short biographies by contemporary artists are interspersed between chapters and are accompanied by images of their work. These fine artists--who include mathematicians and scientists--examine how mathematics influences their art. Accessible to students of all levels, Viewpoints encourages experimentation and collaboration, and captures the essence of artistic and mathematical creation and discovery. Classroom-tested activities and problem solving Accessible problems that move beyond regular art school curriculum Multiple solutions of varying difficulty and applicability Appropriate for students of all mathematics and art levels Original and exclusive essays by contemporary artists Forthcoming: Instructors manual (available only to teachers)

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VIEWPOINTS This page intentionally left blank VIEWPOINTS Mathematical Perspective and Fractal Geometry in Art Marc Frantz Annalisa Crannell Princeton University Press Princeton and Oxford Copyright c 2011 by Princeton University Press Published by Princeton University Press, 41 William Street, Princeton, New Jersey 08540 In the United Kingdom: Princeton University Press, 6 Oxford Street, Woodstock, Oxfordshire, OX20 1TW press.princeton.edu Cover photo: Winter Road along the Trees, by Wil Van Dorp All Rights Reserved Library of Congress Cataloging-in-Publication Data Frantz, Marc, 1951– Viewpoints: mathematical perspective and fractal geometry in art / Marc Frantz, Annalisa Crannell. p. cm. Includes bibliographical references and index. ISBN 978-0-691-12592-3 (hardback: alk. paper) 1. Perspective—Textbooks. 2. Fractals—Textbooks. 3. Art—Mathematics—Textbooks. I. Crannell, Annalisa. II. Title. QA515 .F73 2011 742.01′51-dc22 2010053315 British Library Cataloging-in-Publication Data is Available This book has been composed in LATEX The publisher would like to acknowledge the authors of this volume for providing the camera-ready copy from which this book was printed. Printed on acid-free paper. ∞ Printed in the United States of America 1 3 5 7 9 10 8 6 4 2 Contents Preface vii Acknowledgments ix 1 Introduction to Perspective and Space Coordinates Artist Vignette: Sherry Stone 2 Perspective by the Numbers 1 9 13 Artist Vignette: Peter Galante 25 3 Vanishing Points and Viewpoints 29 Artist Vignette: Jim Rose 4 Rectangles in One-Point Perspective What's My Line?: A Perspective Game 5 Two-Point Perspective Artist Vignette: Robert Bosch 6 Three-Point Perspective and Beyond Artist Vignette: Dick Termes 7 Anamorphic Art Viewpoints at the Movies: The Hitchcock Zoom 39 43 55 59 77 85 113 117 135 Plates follow page 138 8 Introduction to Fractal Geometry 139 vi Contents Artist Vignette: Teri Wagner 9 Fractal Dimension 157 161 Artist Vignette: Kerry Mitchell 193 Answers to Selected Exercises 1; 97 Appendix: Information for Instructors 215 Annotated References 223 Index 229 Preface Viewpoints is an undergraduate text in mathematics and art suitable for math-for-liberal-arts courses, mathematics courses for ﬁne art majors, and introductory art classes. Instructors in such courses at more than 25 institutions have already used an earlier online version of the text, called Lessons in Mathematics and Art. The material in these texts evolved from courses in mathematics and art which we developed in collaboration and taught at our respective institutions. In addition, this material has been tested at, and inﬂuenced by, a series of weeklong Viewpoints faculty development workshops. Pedagogical Approach The approach of Viewpoints is highly activity based, much like the approach of teaching in art school. As many of our workshop graduates will attest, the true value of the material can only be fully appreciated by engaging in these activities, and not by merely reading the book. We have endeavored to include problems and activities of genuine interest and value to art students—problems that go signiﬁcantly beyond what students normally learn in art school. In our experience the authenticity of the problems makes them genuinely interesting, not only to art majors but to students from a broad range of disciplines. We have included a detailed appendix for instructors which includes advice on the window-taping activity of Chapter 1, a sample timetable for a ﬁrst-year seminar course based on Viewpoints, and a list of additional writing assignments. We have endeavored to make sure that the problems are real math problems. Happily, there is a wealth of problems at the boundary of mathematics and art having a number of excellent pedagogical properties: (1) the problems are natural and easily understood; (2) the problems have multiple solutions of varying diﬃculty and applicability; (3) the problems admit multiple proofs, both geometric and algebraic; (4) once arrived at, the solutions are easy to remember and rewarding to use; and (5) the search for solutions captures the essence of mathematical research and discovery. (We demonstrate in depth how this ﬁvefold approach comes to bear on a single problem viii Preface in the solution to Exercise 8 of Chapter 4.) We have sought to make the problems in this book embody each of these characteristics, at a level that is accessible to every undergraduate student. Artist Vignettes A special feature of the text is a series of personal essays which we call Artist Vignettes. During the course of our project we have been fortunate to meet a number of professional artists who have generously contributed to the book. Each Vignette contains a short biography, an artist's statement, and some images of the artist's work. Color images of the artists' work will also appear in the color plate section. We feel that the input of practicing artists informed about mathematics will be an exciting and attractive addition to the text. Origin of the Text The initial course development and the ﬁrst Viewpoints workshops were supported by the Indiana University Mathematics Throughout the Curriculum project, the Indiana University Strategic Directions Initiative, Franklin & Marshall College, and the National Science Foundation (NSF-DUE 9555408). The 2001 workshop was also supported by the Professional Enhancement Program of the Mathematical Association of America. We revised and expanded the text as part of a collaboration between our two institutions, supported by an NSF Educational Materials Development grant (NSF-DUE 0439891 and 0439713). Marc Frantz Indiana University Annalisa Crannell Franklin & Marshall College Acknowledgments The ﬁrst glimmerings of this book appeared ﬁfteen years ago, when the two authors happened to have the good luck to be in the same place (IUPUI) at the same time (fall of 1995), just as Indiana University proposed a Mathematics Throughout the Curriculum project (MTC) to the National Science Foundation. The fearless leader of that grant was Dan Maki. Since then, he's been supportive of our small part of his larger project at every stage: as we started teaching math-and-art courses at our two home institutions, as we started writing up our materials, as we put together our Viewpoints summer workshops for college instructors, and as we hosted our follow-up reunions. We couldn't imagine a better fearless leader than Dan, and to him goes our heartfelt gratitude. As memory serves, it was Bart Ng, Dan's coleader, who ﬁrst mentioned the idea of our collaboration. We're grateful to Bart for his support and for the spark that led to such a long and rewarding adventure. And of course, we're very grateful to the National Science Foundation for the grant to MTC (NSF-DUE 9555408), and for the grants to our own collaborative project (NSF-DUE 0439891 and NSF-DUE 0439713) to write and disseminate this book. Tina Straley, Brian Winkel, Pippa Drew, and Dorothy Wallace invited us to Dartmouth in the summer of 1998 for a wonderful workshop on math and the humanities. Pippa and Dorothy have done a lot of nice work with symmetry groups and art, and we learned a lot from them—including the joys of running a workshop! It's no coincidence that the ﬁrst Viewpoints workshop came two summers after that. One of our ﬁrst and most determined "Viewpointers" (as we called our workshop participants) was Sister Barbara Reynolds of Cardinal Stritch University. It was Sr. Barbara who brought Peter Galante and Teri Wagner to our attention; she came to our workshop two years in a row and sent even more of her colleagues in later years. In her position as Editor of the MAA Notes series, she played a pivotal role in encouraging our manuscript. The world should have more x Acknowledgments people like Sr. Barbara; Marc and Annalisa are glad that our own world includes at least one version of her. A second Barbara entered our life in 2001, when the emerging MAA-PREP program funded the Viewpoints workshop and assigned us an evaluator. Barbara Edwards has served (at our request) as our project evaluator since then; she's not only super at cross-country pedagogy, but also is a wonderful person! And we must have kissed the right frog, because Vickie Kearn agreed to be our editor (and Princeton University Press our publisher). We knew we were working with the right people when Vickie told us she was working all the problems herself just for fun. Viewpointer John Putz gave our manuscript a thorough reading before coming to the last workshop, and made many helpful suggestions. We're ﬂattered and thankful for his careful attention. We're likewise grateful for super-detailed comments from Viewpointers Leah Berman Williams and Andrius Tamulis. We are very grateful to each of our Viewpointers, who spent a week (or two!) in close proximity with us, poring over spreadsheets and fence posts and giving us the kind of honest and immediate feedback that made us think and rethink our material. The next round of masking tape and shish kebab skewers goes to them: AbdelRida Saleh, Alex I. Bostandjiev, Alexandra Robinson, Alice Petillo, Amanda K. Serenevy, Amy E. Wheeler, Amy N. Myers, Andrius Tamulis, Andrzej Gutek, Ann C. Hanson, Anna M. Gavioli, Anne E. Edlin, Azar N. Khosravani, Barbara Duval, Barbara E. Reynolds, Barbara Edwards, Barbara Pinkall, Bernard Mathon, Betty Cliﬀord, Bill Branson, Burkett Fleming, Calvin Williamson, Carol Piersol, Carolyn H. Campbell, Cathy W. Carter, Cliﬀord Davis, Cynthia L. McGinnis, Darwyn C. Cook, David G. Hartz, Daylene Zielinski, Dinesh G. Sarvate, Donald McElheny, Doug Norton, Douglas E. Ensley, Edwina C. Richmond, Gordon Williams, Irina Ivanova, J. Paul Balog, J. Scott Billie, Jackie Hall, Jacqueline A. Bakal, Jan G. Minton, Janette Flaws, Janice C. Sklensky, Jay M. Kappraﬀ, Jean B. Mastrangeli, Jennifer M. Rodin, Jim R. Rose, John F. Putz, Jonathan D. Schweig, Joshua Thompson, Joy H. Hsiao, Juan Marin, Judith C. Meckley, Judy A. Kennedy, Judy A. Silver, Julian F. Fleron, Julianne M. Labbiento, Kathi Crow, Kathleen M. McGarvey, Kerry E. Fields, Kerry Mitchell, Kevin Hartshorn, Lasse Savola, Laura Eden, Leah Berman Williams, Leslie Hayes, Lily Moshe, Linda H. Tansil, Lindsay B. Hilbert, Lisa A. Mantini, Lun-Yi Tsai, Lyn Miller, M. John Kezys, M. P. Chaudhary, Marian A. VanVleet, Marianne Neufeld, Marilyn Gottlieb-Roberts, Marion D. Cohen, Mark D. Binkley, Mark D. Schlatter, Martina Z. Mincheva, Mary Anne Stewart, Mary Jane Wolfe, Mary Williams, Mary Woestman, Meltem Ceylan Alibeyoglu, Michelle Y. Penner, Mike Daven, Mindi Thalenfeld, Nancy Prudic, xi Naomita Malik, Natalie Niblack, Natalie Rivera, Ozlem Cezikturk, Patricia A. Oakley, Patricia Hauss, Patricia K. Jayne, Patricia S. Hill, Penny H. Dunham, Peter Galante, Peter N. Bartram, Peter Reeves, Premalatha Junius, Rachael H. Kenney, Rachel W. Hall, Rafael Espericueta, Raymond A. Beaulieu, Robert A. Bosch, Robert Lewand, Robert Wolfe, Ruth F. Favro, Sakura S. Therrien, Sandra Camomile, Sandy Yeager, Sarah A. Berten, Shangyou S. Zhang, Sharon E. Persinger, Shehraiz Husain, Sheli Petersen, Shirley L. Yap, Sid Malik, Stanislav P. Bratovanov, Stanley Eigen, Stephen F. May, Stephen I. Gendler, Steve Cope, Susan L. Banks, Susan Shifrin, Tamara Lakins, Tanea Richardson, Teresa D. Magnus, Teri G. Wagner, Thomas George, Toby Rivkin, Tom Rizzotti, Trisha M. Moller, Valerie Hollis, Vickie Kearn, William D. Ergle, William Seeley, and Wing Mui. We think you're all 8 heads tall! Marc is especially grateful to Neil Gussman, who has been a fantastic supporter of Viewpoints all along. Not only did you contribute your time when we needed help, Neil, you were there as a great husband and dad when the needs of our Viewpointers diverted Annalisa's time and energy (and even the family blankets) away from home. More than that, seeing the way you have bravely met your own challenges has impressed me and reminded me of the strength and optimism we all need to face what life has in store for us. Annalisa particularly wants to thank Paula Frantz, who has become a friend over these last dozen years. Paula, you've been courageous and creative and caring. From you, I have learned to respond to change by taking a deep breath, relaxing, and resting in faith. There aren't enough thank yous to express how much I appreciate your taking such good care of the both of us during these roller-coaster times. This page intentionally left blank VIEWPOINTS This page intentionally left blank CHAPTER 1 Introduction to Perspective and Space Coordinates ur first perspective activity involves using masking or drafting tape1 to make a perspective picture of a building on a window (Figure 1.1). It's tricky! One person (the Art Director) must stand rooted to the spot, with one eye closed. Using the one open eye, the Art Director directs one or more people (the Artists), telling them where to place masking tape in order to outline architectural features as seen from the Director's unique viewpoint. In Figure 1.1, this process resulted in a simple but fairly respectable perspective drawing of the University Library at Indiana University– Purdue University Indianapolis. O 1 Actually, half-inch drafting tape from an office supply store is better. It's less sticky and easier to find in a narrow width. Nevertheless, we'll use the more common term "masking tape." Figure 1.1. Making a masking tape drawing on a window. If no windows with views of architecture are available, then a portable "window" made of Plexiglas will do just as well. In Figure 1.2, workshop participants at the Indianapolis Museum of Art are making masking tape pictures of interior architectural details in a hallway. 2 Chapter 1 Figure 1.2. Plexiglas will do the job indoors. Figure 1.3. Using a display case. Finally, if a sheet of Plexiglas is not available, the window of a display case will also work. In this case, the Art Director directs the Artists in making a picture of the interior of the case (Figure 1.3). If the masking tape picture from Figure 1.1 is put in digital form (either by photographing and scanning, or by photographing with a digital camera) it can be drawn on in a computer program, and some interesting patterns emerge. (Figure 1.4). Lines in the real world that are parallel to each other, but not parallel to the picture plane have images that are not parallel. V1 The images of these lines converge to a vanishing point. Lines in the real world that are parallel to each other, and also parallel to the V2 picture plane have parallel images. Figure 1.4. Two observations of the library drawing. 2 A line (extended infinitely in both directions) is parallel to a plane if the line does not intersect the plane. Observation 1. Lines in the real world that are parallel to each other, and also parallel2 to the picture plane (the window) have parallel (masking tape) images. Observation 2. Lines in the real world that are parallel to each other, but not parallel to the picture plane, have images that converge to a common point called a vanishing point. Two such vanishing points, V1 and V2 , are indicated in Figure 1.4. The correct use of vanishing points and other geometric devices can greatly enhance not only one's ability to draw realistically, but also one's ability to appreciate and enjoy art. To properly understand such things, we need a geometric interpretation of our perspective experiment (Figure 1.5). As you can see from Figure 1.5, we're going to be using some mathematical objects called points, planes, and lines. To begin describing these objects, let's start with points. 3 Introduction to Perspective and Space Coordinates path of light ray (a line) eye of viewer (a point) Figure 1.5. Mathematical description of the window-taping experiment. image object window (picture plane) It's assumed that you're familiar with the idea of locating points in a plane using the standard xy-coordinate system. To locate points in 3-dimensional space (3-space), we need to introduce a third coordinate called a z-coordinate. The standard arrangement of the xyz-coordinate axes looks like Figure 1.6; the positive x-axis points toward you. For a point P (x, y, z) in 3-space, we can think of the x, y, and zcoordinates as "out," "over," and "up," respectively. For instance, in Figure 1.6, the point P (4, 5, 6) can be located by starting at the origin (0, 0, 0) and going out toward you 4 units along the x-axis (you'd go back if the x-coordinate were negative), then over 5 units to the right (you'd go to the left if the y-coordinate were negative), and ﬁnally 6 units up (you'd go down if the z-coordinate were negative). z x y z 6 the origin (0,0,0) P(4,5,6) 6 units up 4 units out 5 4 y 5 units over x We took a look at the standard xyz-system in Figure 1.6 simply because it is the standard system, and you may see it again in another course. However, it will be convenient for our purposes to use the Figure 1.6. The standard xyz-coordinate system. 4 Chapter 1 y (1,3,7) (4,2,3) z slightly diﬀerent xyz-coordinate system in Figure 1.7—it's the one we'll be using from now on. In Figure 1.7 we have included sketches of three special planes called the coordinate planes. In this case, we have to think of the x, y, and z-coordinates as "out," "up," and "over," respectively, as indicated in the ﬁgure. y x Margin Exercise 1.1. What are the missing vertex coordinates of this block whose faces are parallel to the coordinate planes? x y z 5 The xz-plane (contains the x- and z-axes). Also called "the plane y=0" because all y-coordinates on this plane are 0. P(4,5,6) 6 units over 5 units up 4 units out 6 z 4 x The yz-plane (contains the y- and z-axes). Also called "the plane x=0" because all x-coordinates The xy-plane on this plane are 0. (contains the x- and y-axes). Also called "the plane z=0" because all z-coordinates on this plane are 0. Figure 1.7. The coordinate system we will use. A ﬁrst look at how this coordinate system will be used to study perspective is presented in Figure 1.8. A light ray from a point P (x, y, z) on an object travels in a straight line to the viewer's eye located at E(0, 0, −d), piercing the picture plane z = 0 at the point P ′ (x′ , y ′ , 0) and (in our imagination) leaves behind an appropriately colored dot. The set of all such colored dots forms the perspective image of the object and hopefully fools the eye into seeing the real thing. y P(x,y,z) Margin Exercise 1.2. Suppose we are given two points A(3, 3, 2) and B(4, 2, 7) in the coordinate system of Figure 1.8. Which is higher? P'(x',y',0) viewer's eye: located on negative z-axis at E(0,0,–d) z x Which is closer to the viewer? Which is further to the viewer's left? picture plane z= 0 Figure 1.8. Perspective as a problem in coordinates. In the next chapter we will see how to use this coordinate method to make pictures in perspective, much like special eﬀects artists do 5 Introduction to Perspective and Space Coordinates in the movies. We close this chapter by taking a look at how even the most basic mathematics can help us make better drawings. 1 A Brief Look at Human Proportions 2 Most untrained artists will draw the human ﬁgure with the head too large and the hands and feet too small (Figure 1.9). To prevent these common mistakes, artists have made measurements and observations, and come up with some approximate rules, some of which may surprise you: 3 4 • The adult human body, including the head, is approximately 7 to 7 21 heads tall. • Your open hand is as big as your whole face. • Your foot is as long as your forearm (from elbow to wrist). That last one really is pretty surprising—we have big feet! To see that these principles result in good proportions, take a look at the two versions of the painting by Diego Velazquez in Figure 1.10. Figure 1.9. Detail of a family portrait by Lauren Auster-Gussman at 8 years old. Note the height of the father in heads marked on the right, and the small hands and feet of the mother. Figure 1.10. Diego Velazquez, Pablo de Vallodolid, c. 1635 oil on canvas 82.5 × 48.5 in. In the digitally altered version on the right, we see that the ﬁgure is about 7 heads tall, the left hand (superimposed) is as big as the face, and the man's right foot, when superimposed on his right forearm, just about covers it from elbow to wrist. Artists who understand human proportions also know how to bend the rules to achieve the eﬀects they want. Comic artists are a good example of this (Figure 1.11). 6 Chapter 1 1 2 Figure 1.11. In this sketch by popular comic artist Alex Ross, the DC Comics superhero Atom-Smasher is more than eight heads tall. Superimposed circles of the same diameter show that Starwoman's foot is roughly as long as her forearms. (From Rough Justice: The DC Comics Sketches of Alex Ross, Pantheon, New York, 2010. ATOMSMASHER and STARWOMAN are TM and c DC Comics. All Rights Reserved.) 3 4 5 6 7 8 In their book How to Draw Comics the Marvel Way (Simon & Schuster, New York, 1978) Marvel Comics editor Stan Lee and artist John Buscema reveal that Marvel artists generally draw superheroes eight and three-quarters heads tall, for heroic proportions. Popular comic artist Alex Ross, who has drawn for both Marvel and DC Comics, uses these proportions for the DC Comics superhero AtomSmasher in Figure 1.11, taken from Ross's book Rough Justice: The DC Comics Sketches of Alex Ross (Pantheon, New York, 2010). Having rules like this helps comic artists to visually distinguish superheroes from ordinary characters. It also helps the artists to draw the same character again and again in a consistent way. Thus we see that although artists are not bound by any one set of mathematical rules, understanding the rules can be very helpful. That's a theme we will see repeatedly throughout this book. Introduction to Perspective and Space Coordinates Exercises for Chapter 1 1. Divide your height in inches by the height of your head in inches (you'll have to measure). According to the artists' rule, the answer should be about 7 to 7.5. (a) What is your actual answer? (b) For a child, should the answer be greater or smaller than 7–7.5? 2. In each of Parts (a), (b), and (c), we consider a rectangular box with its faces parallel to the coordinate planes in Figure 1.7. Some of the coordinates of the eight corners (A, B, C, D, E, F , G, H) of the box are given; your job is to ﬁll in the rest. (a) A = (1, 1, 1), B = (1, 1, 5), C = (4, 1, 1), D = (4, 1, 5), E = (4, 7, 1), F = (4, 7, 5), G = (1, 7, 1), , H=( , ). (b) A = (1, 2, 3), B = (2, 3, 4), C =( , , D=( , , , E=( , F =( , , , G=( , , H=( , ), ), ), ), ), ). (c) A = (1, 1, 1), B = (3, 4, 5), C =( , , D=( , , E=( , , , F =( , G=( , , , H=( , ), ), ), ), ), ). (d) Which of the boxes in Parts (a), (b), and (c) is a cube? How big is it? 7 8 Chapter 1 P Q R S T U Figure 1.12. 3. This exercise involves drawing sequences of straight line segments without lifting your pencil. (a) Without lifting your pencil, connect the dots in Figure 1.12 in the following order: QP RSRT U . That is, go from Q to P , from P to R, from R to S, from S back to R, etc. Notice that some vertices (dots) get visited more than once, and some edges (such as RS) get drawn more than once. What letter did you draw? (b) Referring to Figure 1.12, write down a sequence of vertices that draws the letter H. If your straight line path takes you through a vertex, then list it. For example, don't write P T , write P RT instead. (c) On the left of Figure 1.13 is a simple drawing of a house, and on the right are the vertices of the drawing. List the vertices in an order that duplicates the drawing. Can you do it so that only one edge is drawn twice? (d) Refer to the box in Problem 2(a). List the vertices of the box (with occasional repetitions) in an order so that if we connect the dots in the same order, we trace every edge of the box at least once. Your path should stay on the edges and not cut diagonally from one corner to another. T U V P Q W R Figure 1.13. S Artist Vignette: Sherry Stone SHERRY STONE is a lecturer in Foundation Studies at Herron School of Art and Design, IUPUI, with a special interest in teaching first-year art students. Degreed in printmaking, she has become a painter and printmaker who has exhibited in the Midwest and on both coasts. She writes on the topic of the education of artists—and anything else that strikes a whim—and if she hadn't decided to study art, she says she would have become either a writer or a very bad ballet dancer. f you were to ask my freshman art students what they liked to draw when they were younger, many of them would answer Manga comics. They aren't very diﬀerent from many other generations of young artists who started oﬀ by copying comics. The ﬁrst comic I tried to copy was "Nancy." The drawings were simple and I was really fascinated with her hair; it looked like a helmet with spikes sticking out of it! When I was older, I liked to copy Wonder Woman, who was a much better role model—if a comic book character can be a role model—and I enjoyed her connection to mythology. My father was a draftsman: the old-fashioned kind, one of those guys who learned to draw with rulers and mechanical instruments like compasses and protractors, not CADs and computers. My ﬁrst drawing utensils were his turquoise 2H pencils. That's "h" for "hard," which means they could make the sharp, light, accurate lines that draftsmen needed for architectural drawings. I learned not to like them very well. The marks they made were too light no matter how hard you pressed and they had no erasers on the ends. When I was growing up, he worked for a company that constructed water towers like the ones you see from the interstate that announce I "My father was a draftsman: the oldfashioned kind, one of those guys who learned to draw with rulers and mechanical instruments like compasses and protractors, not CADs and computers." 10 "Art is a great profession for someone who has a lot of interests. It's an area where the entire realm of your experiences can come together. That is why artists really need to be well educated. It's hard to make art when you have nothing to say." Artist Vignette the presence of small towns like "Sellersburg" or "Speed" to everyone passing by. Sometimes his company built water towers shaped as unusual objects like ketchup bottles or Dixie cups. They acted as signposts for companies that were so big that they needed their own water tower. I thought those towers were very cool. That was before Claus Oldenberg began making his monumental Pop Art sculptures of everyday objects like baseball bats. Years later, after my dad left, the company built the giant baseball bat that leans against the front wall of the Louisville Slugger company. It's interesting to consider how an object is regarded as art in one context and not in another. When I was in sixth grade, my dad started moonlighting as a draftsman for the developer who was building houses in our subdivision. That was the year I almost decided to become an architect rather than an artist. I learned linear perspective and I used it to design dozens of dream homes. My interest in being an architect eventually waned: my heart was set on being an artist, and my interests were too broad to be limited to houses. I have a long history of writing poetry and stories and making drawings and paintings. I love to read. Art is a great profession for someone who has a lot of interests. It's an area where the entire realm of your experiences can come together. That is why artists really need to be well educated. It's hard to make art when you have nothing to say. My interest in architecture was, however, a valuable detour. I learned linear perspective at a time when many kids decide they can't draw. Upper elementary school children want their drawings to look realistic. They are embarrassed by drawings that look childish because they are growing up and they want their drawings to look as mature as they feel. Linear perspective was one tool I could use to make my drawings look like reality. Consequently, linear perspective has never been much of a mystery to me. Today, I teach linear perspective to wary students in ﬁrst-year drawing courses. Some really enjoy it and take to it very quickly, while others treat it like a bad math test. That saddens me because it is so useful in understanding the three-dimensional nature of objects you are drawing, even when you are not speciﬁcally using it. Art students are an interesting lot, though. Some are little Da Vincis, very analytical and seem more like scientists and philosophers. Many, though, are intuitive souls and are content to feel their way through problems and don't take well to the structure and rules of perspective. I ﬁnd that very puzzling. I once taught a drawing workshop for 8- to 10-year-olds in which the coordinator had written perspective into the course description. I had great reservations about it. I decided to teach it by playing a game of "Follow the Leader": they were to draw what I drew, line for line, and guess what we were drawing. They were very excited and followed me perfectly as we 11 Sherry Stone drew a house in two-point perspective with inclined planes, auxiliary vanishing points, and doors and windows centered on the walls. And they happily duplicated it with very little help from me! I think about that every time I am faced with an impossibly confused college art student. My artwork now has very little to do with linear perspective, but I am always aware of it, even if I am drawing from the human form. Any form that can be simpliﬁed into a conﬁguration of geometric shapes can be drawn in linear perspective. By considering the body as a series of boxes and cylinders situated on a plane, it is easier to draw the human form as though it is part of a space. In my recent work, I have utilized photography and computer programs like Photoshop to do preparatory work for my paintings. I have found the distortions of planes and lines caused by viewing the subject through a lens to be very interesting and sometimes quite a departure from the invented environments one would create with linear perspective—although I have been known to purposely distort the rules of perspective for expressive reasons. Like perspective environments, photography captures environments that appear very real, yet they both walk a line between illusion of reality and abstraction. They both are two-dimensional, striving to create an illusion of three dimensions, but if artists aren't aware of the inherent limitations of the individual processes, they can create very strange illusions. Some artists ﬁnd this aspect intriguing and freely manipulate these conventions for their own purposes. For example, the Photorealist painters of the sixties were very interested in the eﬀect of photography on painting. Richard Estes painted many images of store windows. If you were actually standing in front of one of the stores he painted, you would be able to see the merchandise inside because of our eyes' ability to focus on various planes of space and to ignore some visual information in favor of other information. Estes, however, painted the store window as the camera saw it, with many reﬂections dancing across the glass and very little of the merchandise visible. Even though painters had been using photographs as resource material since the advent of photography, most artists painted from them as though they were working from life, and often would not admit they had used a photograph. Painting an image as the camera saw it—and not only admitting it but also making the work about it—was new. One question I face is how far removed from the original subject I can progress while still maintaining the essence of the original. Through how many material, developmental, and aesthetic ﬁlters can an image pass and still be considered a documentary work? The truth is that there is no truly objective work, no matter whether it's art or journalism or law or management or anything else, because "My artwork now has very little to do with linear perspective, but I am always aware of it, even if I am drawing from the human form." Sherry Stone Vampires, 2001 acrylic on canvas 11 × 17 in. 12 Artist Vignette "The subjects of my work are young women and girls at a time in their lives when they are making decisions that will affect their destinies." everything we do is colored by our experiences and our own points of view. Inﬂuences such as education, upbringing, societal attitudes, for example, go into eﬀect even as a person anticipates beginning the project. I suppose decision making is really at the core of most aspects of my work. The subjects of my work are young women and girls at a time in their lives when they are making decisions that will aﬀect their destinies. The work, in essence, is about portraiture, but I would rather consider it to be about capturing a moment in time in these young people's lives. Art is a process-oriented activity at its best, but I am constantly questioning how far I should go in processing the image, given the immediacy of the content of my work. I do not consider myself a photographer, yet maybe the photograph is the most appropriate ﬁnal form? Once I photograph, process the image in the computer, make rough sketches, and paint the image, I have to ask myself whether I am taking the image too far from its source. In answer to that, I often leave the image as an ink-jet print, but still, unable to resist tinkering, I can be found painting on them occasionally. My answer, for now, is that my objective in photographing is to capture information, not create a ﬁnished piece of photography. The process of creating something beyond the photograph is more important for now. Sherry Stone Western Rider, 2002 ink-jet print 8 × 12 in. For more of the artist's work, see the Plates section. CHAPTER 2 Perspective by the Numbers n this chapter we'll do our ﬁrst perspective drawings, using nothing but mathematics! In Figure 2.1 is our basic perspective setup. A viewer's eye is located at the point E(0, 0, −d). Out in the real world is an object, represented by a vase. As light rays from points on the object (such as the point P (x, y, z)) travel in straight lines to the viewer's eye, they pierce the picture plane, and we imagine them leaving behind appropriately colored dots, such as the point P ′ (x′ , y ′ , 0). (How do we know the z-coordinate must be 0?) The collection of the points P ′ comprise the perspective image (the perspective drawing) of the object. I y P(x,y,z) P'(x',y',0) eye of viewer at E(0,0,–d) z x image picture plane z=0 object Our job is to ﬁgure out the coordinates of P ′ , given a point P on a real object. Since we already know that the z-coordinate of any such point P ′ is 0, this boils down to ﬁnding x′ and y ′ . To do this, we use Figure 2.2. Figure 2.1. The basic perspective setup. 14 Chapter 2 y-axis picture plane P'(x',y',0) P(x,y,z) E(0,0,–d) z d y z-axis y' x' x d z z-axis E(0,0,–d) P'(x',y',0) P(x,y,z) picture plane x-axis Figure 2.2. Computing x′ and y ′ . In Figure 2.2 the viewer is represented by a person, rather than just an eye, so that we can tell the diﬀerence between the top view and the side view. (Technically, perspective drawing is intended for viewing by just one eye, but for the purpose of symmetry, the top view shows the viewer's head centered on the z-axis.) In both views, P (x, y, z) is a point on some object, and P ′ (x′ , y ′ , 0) is the perspective image of P . In the top view we see a large right triangle. Even though P and P ′ may not lie in the xz-plane, we think of the triangle as lying in the xz-plane. One side of the big triangle is x units long, and another side is z + d units long. Inside this triangle is a smaller right triangle, similar to the larger one. In the smaller triangle, the corresponding sides are x′ units long and d units long, respectively. Since the triangles are similar, the ratios of these corresponding sides must be equal: x x′ = . d z+d Multiplying both sides of this equation by d gives us the formula for computing x′ , namely, dx x′ = . z+d In the side view we have a similar setup. We think of the large right triangle as being in the yz-plane, even though P and P ′ may not be. You should convince yourself that an argument analogous to the one just given leads to the following formula for computing y ′ : y′ = dy . z+d As we noted, the z-coordinate of P ′ is always 0, so we only need the coordinates x′ and y ′ to locate a point in the picture plane. Thus, from the equations for x′ and y ′ we have the following useful theorem. 15 Perspective by the Numbers Theorem 2.1: The Perspective Theorem. Given a point P (x, y, z) on an object, with z > 0, the coordinates x′ and y ′ of its perspective image P ′ (x′ , y ′ , 0) are given by x′ = dx z+d and y ′ = dy , z+d where d is the distance from the viewer's eye at E(0, 0, −d) to the picture plane z = 0. The formulas in Theorem 2.1 may be simple, but much can be done with them! For instance, movies that employ computer animation or computerized special eﬀects make use of formulas similar to those in Theorem 2.1. One spectacular example is the movie Jurassic Park (see Figure 2.4 in the exercises). In many scenes, the dinosaurs were essentially made of mathematical points in 3-space! People who enjoyed the movie were delighted, thrilled, and terriﬁed by the corresponding computer-generated image points rampaging across the picture plane (the movie screen)! 16 Chapter 2 Exercises for Chapter 2 1. The rectangle ABCD in Figure 2.3 is parallel to the yz-plane, so that all of its x-coordinates are the same positive number. Draw the top view to visualize this. Also, the y-coordinates of A and B are the same. With the viewer located as shown, what can we say about the x-, y-, and z-coordinates of the image points A′ , B ′ , C ′ , and D′ ? A B C D Figure 2.3. edge of picture plane 2. Think of the Jurassic Park image in Figure 2.4 as being painted on the picture plane, with the people and the Velociraptor existing in the same space. Let P (x, y, z) be the lower left corner of the actual doorway, and let Q(x, y, z) be the actual tip of one of the raptor's claws. The points P ′ and Q′ are the respective images of these points. Which is bigger: (a) the x-coordinate of P , or the x-coordinate of Q? (b) the y-coordinate of P , or the y-coordinate of Q? (c) the z-coordinate of P , or the z-coordinate of Q? (d) the x′ -coordinate of P ′ , or the x′ -coordinate of Q′ ? (e) the y ′ -coordinate of P ′ , or the y ′ -coordinate of Q′ ? Figure 2.4. Scene from the movie Jurassic Park (copyright Universal Studios). Two paleontologists, played by Laura Dern and Sam Neill, attempt to protect a child from a fierce Velociraptor. For the most part, the dinosaurs in the film were computergenerated mathematical perspective images. For the image to appear consistent, it must come from a virtual 3-D creature existing in the same space as the human characters. P′ Q′ Perspective by the Numbers 3. This exercise deals with a point P whose x- and y-coordinates do not change (they are equal to 2 and 3, respectively), but whose z-coordinate gets bigger and bigger. That is, the point moves farther and farther away from the picture plane and the viewer. Assume that the viewing distance d is 5 units. (a) Referring to Theorem 2.1, suppose P = (2, 3, 5). What are the values of x′ and y ′ ? (b) Now suppose P = (2, 3, 95). What are x′ and y ′ ? (c) Suppose P = (2, 3, 995). What are x′ and y ′ ? (d) Draw one TOP VIEW and one SIDE VIEW like those in Figure 2.2, and include all the points P and P ′ from Parts (a)–(c), along with light rays to the viewer's eye (the drawings need not be to scale). Can you see what's happening? (e) Consider a point P (x, y, z). If x and y do not change, but z gets bigger and bigger, what happens to the picture plane image P ′ of P ? (f) Our everyday experience tells us that objects appear smaller as they get farther away. Explain how this is consistent with your answers to Parts (a)–(e). 4. Our ﬁrst perspective drawing will be of a box with its faces parallel to the coordinate planes, and a viewing distance of d = 15 units. (a) If two corners of the box have coordinates (−10, −6, 12) and (−4, −2, 24), then: How wide is the box in the xdirection? How high is the box in the y-direction? How deep is the box in the z-direction? List the coordinates of the other 6 corners. (b) Using Theorem 2.1, ﬁnd the x′ and y ′ coordinates of the images of all 8 corner points. Plot them in the xy-plane and connect the dots with straight lines to obtain the perspective image. Used dashed lines to indicate hidden edges. Your drawing should look something like Figure 2.5. (Note that all of the image points have negative coordinates.) 17 18 Chapter 2 y x Figure 2.5. Your box should look something like this. 5. In this more involved problem of drawing a house, we'll use a computer to make the work easier. In Figure 2.6 is the perspective setup of a viewer, a house, and the picture plane. Since the house is small in relation to the viewer, you can think of it as a large dollhouse. You might also want to look at Figure 2.9 to help visualize the shape of the house. G 18 z picture plane x E(0,0,–15) Figure 2.6. TOP VIEW y C(–12, 3, 99) F picture plane E(0,0, –15) z B(–12,–6,93) A(–12, –18,81) D(–12, –18,117) SIDE VIEW (a) What are the coordinates of the point F ? What are the coordinates of the point G? If the measurements are in inches, how high is the dollhouse? (b) The rest of this problem assumes some familiarity with Microsoft Excel, or some other spreadsheet program. In the ﬁrst 3 columns of the spreadsheets in Figure 2.7(a) and (b) are the xyz-coordinates of all 17 vertices of the Perspective by the Numbers 19 house. In the 4th column is the viewing distance (d = 15 units). The 5th and 6th columns are for the x′ -and y ′ coordinates of the images of the points. To compute the ﬁrst value of x′ (Figure 2.7(a)), select cell E2. The relevant formula from Theorem 2.1 is x′ = dx/(z + d), but you should type =D2*A2/(C2+D2) as indicated, because d is in cell D2, x is in cell A2, etc. Then hit the return key. Similarly, to compute y ′ (Figure 2.7(b)) you should type =D2*B2/(C2+D2). Note in Figure 2.7(b) that the ﬁrst value of x′ should be −1.875. The other values of x′ can be computed by selecting and copying cell E2 and pasting into the other cells in column E. Similarly, the other values of y ′ can be computed by selecting and copying cell F2 and pasting into the other cells in column F. (a) Figure 2.7. Applying Theorem 2.1 in a spreadsheet. (b) After this is done, the entries in columns E and F can be selected as a group, and a scatterplot can be made to display the images of the vertices. Alternatively, the points can be plotted by hand on graph paper. The result should look something like Figure 2.8. Finally, you can print out your result and use a pencil to connect the vertices correctly. This will take some thought! Then you can color and shade your drawing if you like. The result should look something like Figure 2.9. 20 Chapter 2 Figure 2.8. Scatterplot (vertices of the house). Figure 2.9. Connect the dots and color to your taste! 6. The house in Figure 2.9 is a bit featureless, so your job in this problem is to add to the x, y, and z columns of your Excel spreadsheet the correct 3-space coordinates for the vertices of the following items: (a) two or more windows on the near wall; (b) a door on the right-hand wall, centered under the dormer; (c) a small, rectangular yard for the house; (d) a chimney somewhere on the roof, at least partially visible to the viewer. The bottom vertices of the chimney should lie on the roof, not above or below it. To complete the exercise, compute the new x′ and y ′ values, and draw and "paint" the picture of the house as you did in Problem 5, with the new details included. 21 Perspective by the Numbers Practice Quiz for Chapter 2 This quiz refers to a rectangular box in space with its faces parallel to the coordinate planes of the xyz-coordinate system we use for perspective. 1. The box has eight corner points, three of which are listed below. If the viewer is located at E(0, 0, −6) (viewing distance of 6), write down the space coordinates of the other corners of the box, and the corresponding picture plane coordinates of the perspective images of those points. Space coordinates of corners Picture plane coordinates (−4, −2, 6) x′ = , y′ = , z=0 (−4, −2, 3) x′ = , y′ = , z=0 (−6, −4, 3) x′ = , y′ = , z=0 ( , , ) x′ = , y′ = , z=0 ( , , ) x′ = , y′ = , z=0 ( , , ) x′ = , y′ = , z=0 ( , , ) x′ = , y′ = , z=0 ( , , ) x′ = , y′ = , z=0 22 Chapter 2 Practice Quiz for Chapter 2 (continued) 2. The box casts a shadow onto the horizontal plane y = −5 (all y-coordinates are −5 in this plane). Assume that the shadow is cast by parallel vertical light rays, like the sun overhead at noon. Write down the space coordinates of the 4 corners of the shadow, and then write down the picture plane coordinates of their perspective images (same viewer location as in Problem 1). Space coordinates of shadow corners Picture plane coordinates ( , , ) x′ = , y′ = , z=0 ( , , ) x′ = , y′ = , z=0 ( , , ) x′ = , y′ = , z=0 ( , , ) x′ = , y′ = , z=0 23 Perspective by the Numbers Practice Quiz for Chapter 2 (continued) 3. Draw the perspective image of the box and its shadow in pencil on the grid below, and shade in the shadow. The shadow should be partially hidden by the box—make sure you draw it that way! (You can also shade or color the box if you like.) y -6 -5 -4 -3 -2 -1 x -1 -2 -3 -4 -5 -6 This page intentionally left blank Artist Vignette: Peter Galante PETER GALANTE is an Associate Professor of Art and University Creative Director at Cardinal Stritch University in Milwaukee, Wisconsin. Primarily responsible for the Graphic Design program, he also teaches undergraduate and graduate Digital Imaging. In his role as Creative Director, Peter supervises the Advanced Design Group, an in-house design practicum where students, as part of their academic course work, design and produce most of the university's marketing and communication materials. Balancing the responsibility of two full-time roles makes the time he spends on his current passion of filmmaking all the more precious. (Photograph by Peter Galante) n my mind I am a printmaker ﬁrst and a photographer second— even though the majority of my work is photographic. This is because I approach image making, in the camera or on a zinc etching plate, with the same interest in geometry and formal composition that began in Renaissance printmaking. My earliest memories are of television, of a large mahogany veneer set with its horizontally ovoid picture. The images I recall are not of the children's programs of the ﬁfties, but the ﬁlm noir genre movies of the forties that dominated WPIX's weekend schedule. Dark, empty streets with a hardened, solitary detective struggling to correct some injustice. The image is very clear to me, but by the time the ﬁlmed original was reproduced by our television, it was reduced to a fuzzy, high-contrast abstraction. The highlights became a green-white glow, and the shadows became murky and indiscriminate. This eﬀect— a consequence of early television's technical limitations—heightened the sense of drama as it allowed a willful suspension of reality. Throughout this decade when, as a child, I was forming my impression of the world, camera images began to dominate the world's access to visual media. Thereafter, the camera's inherent abstrac- I ". . . I approach image making, in the camera or on a zinc etching plate, with the same interest in geometry and formal composition that began in Renaissance printmaking." 26 Artist Vignette tion was understood to be faithful duplication of "reality" and was therefore perceived as "the" truth. Therein, mechanical rather than interpretive means gave us the look of things and, more important, our outlook on them. It must be some law of nature that existentialists are to be reared in urban environments. Existentialism is a philosophical theory that emphasizes the existence of the individual person as a free and responsible agent determining his or her own development through acts of the will. It seems somewhat ironic that this approach—an approach that tends to be atheistic, disparages scientiﬁc knowledge, and denies the existence of objective values—would provide the foundation of my work, as I am a deeply spiritual person trained in both the arts and natural sciences. I suppose it is the obvious disconnection from the earth in the cities' concrete canyons that emphasizes existential alienation. In places like New York, where I grew up, distinctions of nature are blurred: it is always night in the subway, and it is always day in the streets, under the glare of unnatural light, even if the sun is shining. Peter Galante Waterfront Wall, Brooklyn, NY, 1986 Transmedia, 35 mm film original, 1994 "Emerson, an unlikely existentialist, said correctly, 'Every wall is a door.' Walls define our existence, and so I look for the door and the way out in my personal walls." To reduce the city to a fundamental unit, I would say that the city is made entirely of walls. I'm sure this contradicts the majority opinion that the city is made of people, but I just don't "see" them. I only see the walls and how the city's unique light plays on them. Emerson, an unlikely existentialist, said correctly, "Every wall is a door." Walls deﬁne our existence, and so I look for the door and the way out in my personal walls. For me, the walls reference a time before the camera's inﬂuence, and it is for these reasons that I use the camera itself to search for 27 Peter Galante fundamental meanings and human equivalents. I do this even though the camera is the very instrument that induced an altered perception of society. It is necessary for me to work with our culture's primary visual medium in order to understand its power over the traditions it consumed. Peter Galante Warehouse, Brooklyn, NY, 1986 Transmedia, 35 mm film original, 1994 Traditionally, art has used material means (canvas, oil, stone) to gain spiritual ends. The traditions of Renaissance printmaking provide a conceptual foundation, without which my use of the camera image would be as empty as the pervasive media images. Society nostalgically accepts the mechanical photographic likeness in the family picture album, but I do not accept the camera's view, de facto, as truth. Nor do I attempt to manipulate the image in a surrealist or Dada fashion simply to create shocking juxtapositions or anti-art sentiments. My work began simply and directly with traditional printmaking techniques to heighten visual and emotional impact, and today has become entirely digital: I am moving into the emerging genre of high-deﬁnition video. Throughout this transition I have attempted to strike a balance between the unsettling impact of technology and the stability inherent in the spiritual dimensions of tradition. For me, neither the camera nor the processes of printmaking are reproductive techniques, but, rather, are investigative tools with almost mythic signiﬁcance. For all the technology in the world, there is nothing quite like the printmaker's experience—the odors and eﬀort of mixing bone, vine, and burnt oil in handmade ink—to awaken the mythic sense of tradition. "For me, neither the camera nor the processes of printmaking are reproductive techniques, but, rather, are investigative tools with almost mythic significance." 28 Peter Galante Cabin, Belleayre, NY, 1983 Transmedia, 35 mm film original, 1994 Peter Galante Cabin, Belleayre, NY, 1983 Transmedia, 35 mm film original, 1994 Artist Vignette I have been rephotographing photographs as part of still lifes for as long as I can remember; some of my ﬁrst published examples appeared in Vogue in 1979. In my current ﬁlm projects, the concept of using animated stills is as much practical as aesthetic. The techniques that I have been utilizing in both my still and motion picture work frequently elicit comparison with the work of Ken Burns. He is a contemporary of mine (we were both born the same year); since I am a teacher and probably also out of professional pride, I would suggest that we likely had the same or similar inﬂuences. The "Ken Burns" eﬀect—as it has become widely known in the United States since his 1990 Civil War documentary—may well have its conceptual roots in the work of French ﬁlmmaker Chris Marker and his legendary apocalyptic 1962–4 ﬁlm La Jetée. I have never heard Mr. Burns speak of his inﬂuences, but for me living and working in New York in the late '70s, the Museum of Modern Art brought me into contact with László Moholy-Nagy and avant-garde ﬁlmmaking. These inﬂuences (along with others like Harry Callahan, Aaron Siskind, and Richard Avedon) have become such a part of my subconscious that I am unaware of their constant presence. This is also true of my fascination with the geometry underlying the formal composition of the picture plane. It is not possible for me to compose an image without intuitively imposing some structural framework or grid. I do not suppose that the casual viewer will become aware of my somewhat rigid organizational structure, but I cannot even imagine working without one. For me, geometry has become the glue that keeps an existential universe from ﬂying apart. For nearly the past ten years the focus of my creative work has been on meeting the needs of the university. The nature of a private Catholic university keeps recruiting and advancement requirements in the forefront. Hence, my work centers on telling the story of the Franciscan Intellectual Tradition. As a consequence little of my work has been exhibited in gallery settings, though all of it has been published and distributed widely. I have never felt limited or encumbered by these circumstances; conversely, I have felt quite empowered, as if I were following in the footsteps of Cimabue and Giotto, whose renowned fresco paintings document the life of St. Francis of Assisi. It has been an honor and a privilege to immerse myself in such a rich tradition and to translate my understanding of the tradition for today's postmodern, if not existential, world. For more of the artist's work, see the Plates section. CHAPTER 3 Vanishing Points and Viewpoints Y ou've probably heard the term "vanishing point" before, and we mentioned it in Chapter 1 when we discussed the masking tape drawing of a library (see Figure 3.1). Figure 3.1. Masking tape drawing of a library (above) and the vanishing points of the drawing (below). V1 V2 In the drawing on the bottom of Figure 3.1 are two vanishing points V1 and V2 . The three lines which converge to V1 , for example, represent lines in the real world (architectural lines of the building) that are actually parallel to one another. Clearly, however, the images of these lines are not parallel, because they intersect. The reason 30 Chapter 3 we say "vanishing point" instead of "intersection point" is that a single line, all by itself, can have a vanishing point; the explanation is illustrated in Figure 3.2. vanishing point vanishing point line vanishes line vanishes parallel lin ev isi line ble still line s till vis visi ble infinite edge of picture plane parallel lin ible eL straight lin ev isi line ble still line s till vis visi ible ble ight line L infinite stra edge of picture plane Figure 3.2. The vanishing point is where the line appears to vanish. In Figure 3.2 a viewer looks along various lines of sight (dashed lines) at a line L in the real world. The viewer looks at farther and farther points on the line, and keeps seeing the line as long as his line of sight intersects it (imagine looking through a thin soda straw). At a certain moment, however, the line of sight becomes exactly parallel to L and no longer intersects it. That's the precise moment at which the line L seems to vanish (when looking through a soda straw), simply because the viewer isn't looking at it anymore. For this reason, the intersection of this special line of sight with the picture plane is called the vanishing point of the line L. A vanishing point always lies in the picture plane: its height is determined by the side view in Figure 3.2, and its right-left location is determined by the top view. Notice that the special line of sight parallel to L would also be parallel to any other line M that was also parallel to L; that is, if any line is parallel to L, then it has the same vanishing point as L. To state these results in a theorem, recall that a line is parallel to a plane if it does not intersect the plane. Clearly the line L in Figure 3.2 is not parallel to the picture plane. Thus we have Theorem 3.1: The Vanishing Point Theorem. If two or more lines in the real world are parallel to one another, but not parallel to the picture plane, then they have the same vanishing point. The perspective images of these lines will not be parallel. If fully extended in a drawing, the image lines will intersect at the vanishing point. Notice that the photograph in Figure 3.3 has one obvious vanishing point. It would seem reasonable to call this "one-point perspective," but there is usually one more requirement before this term is 31 Vanishing Points and Viewpoints used. We often use the term when rendering rectangular box shapes such as buildings, and we will assume one face of this box is parallel to the picture plane. (In Figure 3.3, notice that the images of the vertical beams and the horizontal crossbars on the roof are likewise vertical and horizontal.) If one face of the box is parallel to the picture plane, then the line L is orthogonal (perpendicular) to the picture plane as in Figure 3.4. Thus the special line of sight parallel to L (the sight line which goes through the vanishing point) is also orthogonal to the picture plane. This means that the correct location for the viewer's eye is directly in front of the vanishing point, and this is the situation which is usually referred to as one-point perspective. Figure 3.3. Notice in this picture how the long beams of the ceiling, which are parallel in the real world, have images that almost touch at the vanishing point. This is a World War II photograph of Henry Ford's Willow Run bomber plant near Ypsilanti, Michigan. Over half a mile long, the plant was the largest factory in the world under a single roof when it was completed in 1941. vanishing point V vanishing point V line vanishes lin ev line isi ble line vanishes parallel still line s visi ble till vis lin ible ev line isi ble still ble infinite straight line L edge of picture plane parallel line s visi till vis ible infinite straight line L edge of picture plane Figure 3.4. In one-point perspective, the only lines with vanishing points are those orthogonal to the picture plane. We will say that a perspective drawing is in one-point perspective if (a) there is only one vanishing point V to which lines that are part of the drawing converge, and (b) those image lines that converge to V represent lines in the real world that are orthogonal to the picture plane. It's often possible to tell by looking that a drawing is in true onepoint perspective. In this case it may be easy to ﬁnd the exact viewing 32 Chapter 3 position for the viewer's eye. One such example is the drawing of a rectangular box in Figure 3.5. V Figure 3.5. A box in one-point perspective. parallel A parallel D picture plane a V' V d Figure 3.6. The shaded triangles are similar. First, let's see how we can tell that Figure 3.5 exhibits true onepoint perspective. Clearly there is one vanishing point V (conveniently located at the base of a tree), but we must also verify that the image lines (dashed) which converge to V represent lines in the real world (the edges of the real box) which are orthogonal to the picture plane (the plane of the page). This will be true if the front face of the box is parallel to the picture plane. But this must be the case, because the image lines of the edges of the front face appear to be parallel; if the front face of the actual box were not parallel to the picture plane, then at least one opposing pair of its edges would not be parallel to the picture plane, and by Theorem 3.1, their images in the drawing would converge to a vanishing point. In other words, since the front face of the box appears undistorted, the drawing is in true one-point perspective. It therefore follows that the correct viewing position is somewhere directly opposite the vanishing point V —but how far from the page? To determine that, we need the top view of the perspective setup for the box (see Figure 3.6). In Figure 3.6 we see the vanishing point V for the edges of the box that are orthogonal to the picture plane, and we also see the vanishing point V ′ for the diagonal of the top face of the box. Since the indicated pairs of lines are parallel, the two shaded triangles are similar. Thus the ratios of corresponding sides are equal: D d = . a A We can easily solve for the viewing distance d to get D d=a . A (3.1) 33 Vanishing Points and Viewpoints Obviously we need more information to ﬁnd d, but often this can be gleaned from the context of the artwork. Suppose, for instance, we know that the box in Figure 3.5 is a cube. This may seem strange, because the box doesn't look like a cube—it looks too elongated (more like a dumpster), but the picture will look better when we determine d. If the box is a cube, then the top is a square (even though it wasn't drawn as a square in Figure 3.6), and A and D in Figure 3.6 must be equal. In this case, (D/A) = 1, so by Equation (3.1) we have d = a. But a is the distance between V and V ′ , so the viewing distance for a cube is the same as the distance between the main vanishing point V and the vanishing point V ′ of the diagonal of the top face (see Figure 3.7). This distance can be measured directly on the drawing! In the drawing (Figure 3.7) we locate V ′ (base of the other tree) by drawing the dashed diagonal line of the top face of the box. How do we know that V ′ is on the same horizontal line as V ? Because the dashed lines are images of real lines which are level with the ground, so the sight lines of the viewer to their vanishing points must be level also. To test out our determination of d, use the large drawing in Figure 3.8 as follows: viewing distance d V' V • Close your right eye. • Hold the page vertically and place your left eye directly in front of the point V (not in the center of the page!). • Move the page until your left eye is d units away from V . (You may want to use a thumb and foreﬁnger to measure the distance between the trees.) • Without changing your position, let your eye roll down and to the left to look at the box. Although it may be a bit close for comfortable viewing, it should look much more like a cube! Figure 3.7. The viewing distance for a cube. 34 Chapter 3 viewing distance d V Figure 3.8. To see a cube, look with one eye, directly opposite V , at the indicated distance d. 35 Vanishing Points and Viewpoints We have only used a little mathematics, but we have accomplished a lot. For one thing, we see the importance of the unique, correct perspective viewpoint (sometimes called the "station point"). If we view art from the wrong viewpoint, it can appear distorted—a cube can look like a dumpster. For another thing, the majority of perspective works in museums are done in one-point perspective, with clues that can help determine the viewing distance. Thus our simple trick can actually be used in viewing and enjoying many paintings in museums and galleries. In Figure 3.9 we see the trick applied to ﬁnding the viewpoint for the painting, Interior of Antwerp Cathedral. Since the ﬂoor tiles are squares, they serve the same purpose as the square top of the cube in the previous discussion. The viewing distance is as indicated, with the correct viewpoint directly in front of the main vanishing point V . V' viewing distance tile dia V gon al Although it's not possible to tell by viewing this small reproduction of Antwerp Cathedral, the eﬀect of viewing the actual painting in the Indianapolis Museum of Art gives a surprising sensation of depth, of being "in" the cathedral. The viewing distance is only about 24 inches, so most viewers never view the painting from the best spot for the sensation of depth! Of course you can't draw lines on the paintings and walls of an art museum, so some other method is needed to ﬁnd the main vanishing point and the viewing distance. A good solution is to hold up a pair of wooden shish kebab skewers, aligning them with lines in the painting to ﬁnd the location of their intersection points. First, the main vanishing point V is located. Then one skewer is held horizontally so that it appears to go through V , and the other is held aligned with one of the diagonals of the square tiles; the intersection point of the skewers is then V ′ . Figure 3.10 shows workshop participants at the Indianapolis Museum of Art using their skewers to determine the viewpoint of a perspective painting. Then, one by one, the viewers Figure 3.9. Peter Neeffs the Elder, Interior of Antwerp Cathedral, 1651. (Courtesy of the Indianapolis Museum of Art.) 36 Chapter 3 assume the correct viewpoint, looking with one eye to enjoy the full perspective eﬀect. If shish kebab skewers aren't practical, any pair of straight edges, such as the edges of credit cards, will work almost as well for discovering viewpoints of perspective works. Figure 3.10. Viewing art with shish kebab skewers at the Indianapolis Museum of Art. Certainly there are other important ways to view a painting. It's good to get very close to examine brushwork, glazes, and ﬁne details. It's also good to get far away to see how the artist arranged colors, balanced lights and darks, etc. Our viewpoint-ﬁnding techniques add one more way to appreciate, understand, and enjoy many wonderful works of art. 37 Vanishing Points and Viewpoints Exercises for Chapter 3 1. (Drawing your own cube.) In Figure 3.11 a start has been made on the drawing of a cube in one-point perspective. The front face is a square, V is the vanishing point, and the dashed lines are guide lines for drawing receding edges of the cube. Suppose you want to choose the viewing distance first. Let's say the viewing distance should be 7 inches. Finish the drawing of the cube. (HINT: For help in thinking about it, look at Figure 3.7. The idea is to draw the same lines, but in a diﬀerent order!) V Figure 3.11. How do you finish the cube if the viewing distance is 7 inches? 2. Suppose the box in Figure 3.12 is not a cube. Let's say its front face is a square, but its top face is in reality twice as long as it is wide from left to right. In this case, the viewing distance is not equal to the distance between the two trees. What is the viewing distance? (Figure 3.6 can help you think about this problem.) What if the top is three times as long as it is wide from left to right? 38 Chapter 3 V' V Figure 3.12. What if the box is not a cube? 3. Now do the real thing: go to a gallery or museum and practice your viewing techniques! 4. Top views, side views, and similar triangles are very useful for ﬁnding viewpoints, setting up drawings, and generally understanding what we see in pictures. For example, we often see photographs of the moon shot with telephoto lenses to make the moon seem dramatically large. However, when we see the moon in ordinary photographs, it appears quite small. To see why, suppose you want to make a drawing of the full moon rising over the ocean, with a viewing distance of two feet. What should the diameter of your moon image be? (You'll need to do a little astronomical research.) 5. Take a drawing pad and go outside to sketch a street, alleyway, or walkway, using one or more vanishing points to help make your drawing look realistic. Artist Vignette: Jim Rose JIM ROSE is a professional illustrator, graphic designer, and videographer using both digital and hands-on techniques; he also teaches at Clarion University of Pennsylvania and is a student of the creative process. He holds a Master of Fine Arts in Illustration from Syracuse University. In Philadelphia he graduated from the Pennsylvania Academy of the Fine Arts and studied with Evangelos Frudakis for anatomy. At the Cape Art School in Provincetown, Massachusetts, he studied color with Henry Hensche. His biggest influence was his mentor, Porter Groff. (Photograph by Jim Rose) grew up in inner-city Philadelphia, you know the place; it's where they ﬁlmed Rocky. As a matter of fact, my grandmother lived three blocks from the pet store that Adrian worked in. My father was a welder, paratrooper, and a kind and gentle man. My mother was a homemaker and gave me a wonderful childhood. Mom always told me that I could do anything I set my mind to do; I actually believed her. We didn't know any artists and there wasn't much talk about ﬁne arts in our home, which was on a small street nestled among giant, red-brick factories. There were a number of trees in the library park, which most people used while walking their dogs. It was years before I realized that air didn't smell like mints (there was a mint factory on the next block). I loved to draw pictures, so Mom suggested that I go to Sears and Roebuck, which was a bus ride away, to apply for an artist's job. I did this with great expectations. The advertising director said, "We have one artist that does all the illustration for Sears on the East Coast. Go to the second ﬂoor and ask for Porter Groﬀ." I "We didn't know any artists and there wasn't much talk about fine arts in our home, which was on a small street nestled among giant, redbrick factories." 40 Artist Vignette "[Porter Groff] was my Mentor in Art and many other things. He was quiet, elegant, kind, artistic, and humble. He taught me some of these traits, but not all." Jim Rose Agent Orange, 2005 Digital I entered a small cubicle where a quiet man, dressed formally with a starched white shirt and tie, sat working on ink wash drawings of washing machines, fashions, and toys. He did ﬁne renderings of just about everything that Sears sold except guns. He refused to draw anything to do with the military. I introduced myself, telling him why I was there. He looked over his glasses and welcomed me in with a smile. He looked at each sample of my work intently. I was sure I had the job. He looked at me and said, "This is what I do," handing me a beautiful rendering of a living room suite done in pen and ink. I stood there disappointed and feeling a bit foolish, knowing that my work was not up to par. I said, "I'll be back." He sensed my embarrassment and oﬀered to give me lessons. He was the ﬁrst artist I ever met. He was also the ﬁrst conscientious objector, farmer, woodworker, Quaker, and intellectual I ever met. After many years of visiting Porter once a week in his studio in Cheltenham, Pennsylvania, where there were many trees, I became a layout man for the Philadelphia Bulletin and Sears. He was my Mentor in Art and many other things. He was quiet, elegant, kind, artistic, and humble. He taught me some of these traits, but not all. One evening I asked Porter, "How do I know if I'm a conscientious objector?" He said, "You're a conscientious objector if you have a gun and the man across from you has a gun and you let him shoot you." From the time that I met Porter Groﬀ, drawing, painting, and design have always been part of my life. As a matter of fact, drawing has saved my life more than once. One instance was when I was in Vietnam as an illustrator for the army, trying to ﬁnd out if I was a conscientious objector. I found that I was not, but that's another story. I entered one of the many cul-de-sacs found in Saigon, Vietnam, where I lived. The afternoon was warm and humid, and the dust powdered around my boots. At the community fountain/bathtub that sat in the middle of the fortlike houses glazed with barbed wire, I turned right toward my door. I drew pictures for the residents, whose families lacked men of military service age. I drew the young children's dreams: horses, cowboys, circus animals, anything that they wished. On occasion I was invited to dinner and enjoyed watching The Wild Wild West with the children, who thought all Americans knew karate from watching the show dubbed in Vietnamese. It was easy to convince them that I knew karate because they weighed about 25 pounds and I weighed 180. We played, we ate ﬁsh and rice, and drawing was my link to them. One evening I retired to my enclave, listening to the soup man 41 Jim Rose banging his sticks and ﬁreﬁghts oﬀ in the distance—far enough to make them a fantasy to me. I dozed oﬀ slowly, counting the small lizards scampering on the ceiling eating the bugs. I woke up abruptly. Barrrrrrrrrrrup! Someone had climbed the barbed wire enclosure, riddling the entire room next to mine with bullets. An American soldier and his girlfriend died instantly. He did not draw pictures and he was not kind and gentle. I moved the next day. My job in the army was illustrating numbers. I worked directly under General Creighton Abrams, commander in chief of all U.S. forces in Vietnam. Oﬃcers would supply me with numbers of MIA and DOA and I would do the charts and graphs to be presented to various allies. These numbers rose constantly, so unfortunately I had job security. Each number represented a human. When I was discharged I returned to Philadelphia with no job. I freelanced and decided to move out to the country. It was time for me to live with the trees and not breathe air that smelled like mints. I stayed in the suburbs for a few years doing photography and freelancing in graphic design. While I enjoyed doing advertising layout, I needed more. I wanted to learn what the "Fine" meant in Fine Arts, so I returned to school. The Pennsylvania Academy of the Fine Arts is where I learned how to paint, draw, and think like an artist. I was still doing advertising design and illustration. Porter warned me of my presumption that I was an artist. He said, "You should not call yourself an artist; other people should." I spent three summers studying color at the Cape Art School under Henry Hensche because my mentor studied there. After graduating from the academy I moved north, settling in Starrucca, Pennsylvania, where I freelanced in illustration and graphic design. I also made many new friends from all over the country. I did community theatre, got divorced, played Santa Claus for the Baptist church, earned a Master of Fine Arts degree from Syracuse University, walked through the woods, got remarried, had a son, and became the mayor. I loved Starrucca, where I lived among the trees as a trout stream lullabied me to sleep every night. It was a magical place. It was there that I met my wife Linda. We were married behind our house at the base of a small mountain with all of our favorite people. A year later our son James was born. Needing more security, I found a job at Western Illinois University (where I taught graphic design and illustration, and helped create their new computer lab), requiring me to move to Macomb, Illinois. While in the Midwest I was very productive and we had our second son, John. I also created a series of paintings called Genetic Signatures having to do with meditation, family, and calligraphy. I had two one-man shows and Jim Rose Is It Really Over, 2000 Digital (Photographs of Rose's children, father, and son) "I did community theatre, got divorced, played Santa Claus for the Baptist church, earned a Master of Fine Arts degree from Syracuse University, walked through the woods, got remarried, had a son, and became the mayor." 42 "I work in many different media: paint, Photoshop, pen and ink, mathematics. Oops—I would have never thought I would ever consider mathematics as a medium, but I do." Artist Vignette then moved on to more digital projects. After six years I left Illinois kicking and screaming because of disagreements with the dean and the faculty. A creative person must be a politician in the academic environment. I had not yet developed that skill. The following year I started at Clarion University of Pennsylvania, closer to our families and making more money, so I thank my colleagues at Western Illinois for pushing me forward. At Clarion I met a man who lives on my avenue and grew up ﬁfteen minutes away from where I was raised in Philadelphia. Dr. Steve Gendler is a mathematician, professor, day trader, and somewhat eccentric Jewish philosopher. Steve introduced me to the connection between art and mathematics. He coerced me to attend the workshops given by Annalisa Crannell and Marc Frantz, where I realized that I was doing math all along. Dr. Gendler and I taught a cluster course called Art in Perspective, combining the mysteries of art and mathematics to create a clear understanding of how things work in the world. Steve and I presented reports on the course we taught at the Viewpoints conference at Franklin & Marshall College and the 2002 International Bridges Conference at Towson University. I recently presented and exhibited my work called NAMAN Dream Altars, Vietnam: A Search for use of the Golden Mean and its Effect on Design and Content at the 2005 International Bridges Conference in Banﬀ, Alberta, Canada. Recently, I have had a one-man show of watercolors at Michelle's Café in Clarion and exhibited my Vietnam Memory work in Clarion's faculty show and at the Bridges Conference in Banﬀ, Canada. I have just received a grant to work on the design and production of an illustrated book of poems regarding my NAMAN Vietnam Memory Project. I am designing Web pages and working on a series of watercolors that hopefully will come from my inner self, inspired by my readings of Zen Buddhist philosophies. I work in many diﬀerent media: paint, Photoshop, pen and ink, mathematics. Oops—I would have never thought I would ever consider mathematics as a medium, but I do. The use of the golden section, fractals, perspective, and tessellations has expanded my vision. I feel closer to the great artists such as da Vinci, M.C. Escher, Raphael, and many more. In today's art world many artists have combined digital and hands-on media to create their art. The realization that my paradigm was changing happened when I was in the woods painting a landscape—a break from the computer screen. I found myself thinking, "That color is 53% cyan, 10% magenta, 25% yellow and 12% black." A new day has begun and I'm going to give it 110%. For more of the artist's work, see the Plates section. CHAPTER 4 Rectangles in One-Point Perspective n our perspective work with the house in Chapter 2, we saw how the perspective transformation equations of Theorem 2.1 could be used to create a perspective drawing, by mathematically imitating what we did physically when we used masking tape to draw buildings on the windows in Chapter 1. However, neither of these techniques is useful to artists using traditional painting or drawing media, so we need to come up with some more practical techniques for perspective drawing. In this chapter we'll concentrate on techniques for correctly subdividing and duplicating rectangles in one-point perspective. This will enable you to draw a variety of things in one-point perspective, and many of the techniques also work for perspective with more than one vanishing point. Although the techniques are based on mathematics, most people ﬁnd them easy and fun to use in practice, and sensible from a "common sense" point of view. (Nevertheless, even a small alteration can change an easy problem into a more challenging one.) Before presenting the techniques, we list the rules they are based on. I Rule 1. The perspective image P ′ Q′ of a line segment P Q is also a line segment (unless P Q is seen end-on by the viewer, in which case the image is a point). Rule 2. A line segment P Q that is parallel to the picture plane (i.e., lies in a plane parallel to the picture plane) has a perspective image P ′ Q′ that is parallel to P Q. 44 Chapter 4 y Q Q' Figure 4.1. A line segment P Q and its image P ′ Q′ . z E x P' P picture plane z=0 plane z = k The reasons for Rules 1 and 2 can be seen in Figure 4.1. First, observe that the triangle △EP Q lies in a plane, and the intersection of that plane with the picture plane is the line containing the segment P ′ Q′ . Second, if P Q lies in a plane z = k parallel to the picture plane, then the lines containing P Q and P ′ Q′ cannot intersect, and furthermore, these two lines are coplanar, since they lie in the plane containing △EP Q. Since nonintersecting coplanar lines are parallel, this shows that P Q and P ′ Q′ are parallel. Rule 3. If two line segments P Q and RS in the real world are parallel to each other and also parallel to the picture plane, then their perspective images P ′ Q′ and R′ S ′ are parallel to each other. By Rule 2, P Q is parallel to its image P ′ Q′ . Since P Q is also parallel to RS, it follows that P ′ Q′ is parallel to RS. But Rule 2 also says that RS is parallel to its image R′ S ′ , so it follows that the two images P ′ Q′ and R′ S ′ are parallel to each other. An illustration of Rule 3 can be seen in Figure 3.1 of Chapter 3. The vertical edges of the library building are parallel to one another, and also parallel to the picture plane (the window). Thus their images (the masking tape lines on the window) are all vertical, and parallel to one another. Rule 4. If two or more lines in the real world are parallel to each other, but not parallel to the picture plane, then they have the same vanishing point. The perspective images of these lines will not be parallel. If fully extended in a drawing, the image lines will intersect at the vanishing point. Rule 4 is just a restatement of Theorem 3.1 in Chapter 3. Rule 4 is also illustrated by Figure 3.1 of Chapter 3. Edges of the library 45 Rectangles in One-Point Perspective building that are parallel to one another in the real world, but not parallel to the picture plane, have images that converge to a common vanishing point. Rule 5. A shape that lies entirely in a plane parallel to the picture plane has a perspective image that is an undistorted miniature of the original. surface parallel to picture plane picture plane CAT FOOD can't happen (diagonals not parallel) picture plane y' CAT FOOD y' CAT FOOD CAT FOOD x' perspective image (undistorted) x' ? viewer viewer (a) (b) Figure 4.2. An illustration of Rule 5. Rule 5 is illustrated in Figure 4.2(a). The image is a "miniature" in the sense that the distance between any two points on the cat food box is always greater than the distance between their images. This is apparent from the "squeezing" process of projecting everything to a point, and you can prove it using the transformation equations and the distance formula. The fact that the image must be "undistorted" is illustrated in Figure 4.2(b). At ﬁrst, everything seems OK, because the top and sides of the box are parallel to their images, which is consistent with Rule 2. However, the too-skinny image of the cat food box results in a situation in which the diagonal of the box is not parallel to its image! Since this contradicts Rule 2, the distortion cannot take place. Tricks with Rectangles Knowing how to subdivide or duplicate rectangles in perspective is very useful, not only because rectangular shapes occur frequently in architecture, but also because other shapes can be drawn with the help of (properly drawn) rectangles. For instance, a circle inscribed in a square will be tangent to the square at the midpoints of the sides. 46 Chapter 4 Thus if you can draw the square and the midpoints in perspective, you have a guide for sketching the circle in perspective. Center of a rectangle. The perspective image of the center of a rectangle is the intersection of the images of the diagonals (because the actual center is the intersection of the actual diagonals). Figure 4.3. Putting string around a package by finding centers of rectangles. (a) (b) (c) (d) In Figure 4.3 we use this fact to draw string around a box-shaped package that has been drawn in one-point perspective in Figure 4.3(a). In Figure 4.3(b) we locate the perspective centers of the three visible faces as the intersections of the corresponding diagonals. In Figure 4.3(c) we use Rule 3 to draw the images of the sections of string that are parallel to the picture plane. By Rule 3, we simply draw the horizontal sections as horizontal lines, and the vertical sections as vertical lines. However, there are two horizontal sections of string that are not parallel to the picture plane, so we have to draw them by another method. This is done in Figure 4.3(d), where we use the fact that these two sections of string must go through the centers of the corresponding faces of the box. You'll soon discover that there is more than one way to solve most perspective drawing problems. For instance, by Rule 5, the near face of the box, which is parallel to the picture plane, has an undistorted image. Thus the center of that face could have been located by measuring, rather than drawing the diagonals. (However, measuring doesn't work for the other two visible faces.) As another example, Rectangles in One-Point Perspective 47 the last two sections of string we drew could have been drawn by locating and using the vanishing point of the box. Both methods lead to the same solution, as illustated by Figure 4.4, where we have extended the lines of the two strings to show that they converge to the vanishing point. If we know that the actual top face of the package in Figure 4.4 is twice as long (into the distance) as it is wide, can you locate the correct viewpoint? Figure 4.4. The strings are automatically consistent with the vanishing point. Duplicating a rectangle. The next technique involves duplicating a rectangle in perspective, a trick that's often used to draw fences, bricks, sidewalks, tiles, etc. To see how to do it, let's ﬁrst imagine the steps while looking at the rectangle straight on, as in Figure 4.5(a), where we have sketched a duplicate of the original rectangle in black dashed lines. In Figure 4.5(b) we locate the center of the ﬁrst rectangle. In Figure 4.5(c) we draw the horizontal center line. Since the midpoint (black dot in Figures 4.5(c) and (d)) of the right side of the original rectangle is the center of the larger rectangle formed by the pair, we can locate the upper right corner of the duplicate rectangle by drawing a line from the lower left corner of the original rectangle, through the midpoint, and continuing it until it intersects the top line of the two rectangles, as in Figure 4.5(d). Now we can draw the right side of the duplicate rectangle as a solid line in Figure 4.5(d). 48 Chapter 4 Figure 4.5. Duplicating a rectangle. (a) (b) (c) (d) The analogous steps in perspective are carried out in Figure 4.6. Can you explain how they work? Figure 4.6. Drawing fenceposts by duplicating rectangles. (a) (b) (c) (d) It's fun to keep going and make a really long fence, as in Figure 4.7. Making the fenceposts thinner and paler as they go into the distance helps, too; this is called atmospheric perspective. 49 Rectangles in One-Point Perspective Figure 4.7. Extending the fence. We see examples of the principles of perspective every day. For example, Figure 4.8 shows the walkway in front of the Franklin Dining Hall at Franklin & Marshall College. Notice in Figure 4.8 how certain lines converge to a vanishing point, and how the vertical column edges conform nicely to the construction in Figure 4.6. In a sense, a photograph is a very accurate perspective drawing made by a machine (a camera). That's why, using the rules of perspective, an artist or architect can start with a blank piece of paper and make a convincing drawing of a building before it even exists! Figure 4.8. Photographs automatically conform to the rules of perspective. 50 Chapter 4 Exercises for Chapter 4 1. Draw an 8 × 8 chessboard in perspective. 2. Draw the string around the package in Figure 4.9. Observe that the package has two vanishing points, like the library building in Figure 1.4. Figure 4.9. 3. Draw the rest of the sidewalk tiles in Figure 4.10. Figure 4.10. Rectangles in One-Point Perspective 4. Make a copy of Figure 4.11 and ﬁnish drawing the perspective letters "TJC." Assume that the actual letters all have the same depth. (Compare this exercise to Exercise 5 in Chapter 5.) Make copies of the picture of a fence panel in Figure 4.12, and use them to solve the following drawing problems. 5. Draw 7 more fenceposts inside the fence panel to divide the panel (solid outline) into 8 equal sections. 6. Draw a duplicate of the fence panel (in the same plane as the original) with the top of its near fencepost at point P . 7. Draw a duplicate of the fence panel (in the same plane as the original) with the top of its far fencepost at point P . 8. Draw 2 more fenceposts inside the fence panel to divide the panel into 3 equal sections. (This is harder, but you would do the same thing to draw, say, the Italian ﬂag in perspective, or the ﬂags of many other nations.) 9. Draw 4 more fenceposts inside the fence panel to divide the panel into 5 equal sections, without any measuring. (Really hard!) 51 52 Figure 4.11. Chapter 4 Rectangles in One-Point Perspective Figure 4.12. P 53 This page intentionally left blank What's My Line? A Perspective Game∗ he purpose of this game is to draw, in excellent 1-point perspective, the house that we ﬁrst constructed in Excel in Chapter 2. The idea is to do it one line at a time, without coordinates, the way artists do! The class takes turns at the blackboard (or whiteboard), working on the same drawing. The only tools needed are a piece of chalk and a couple of yardsticks (two are occasionally needed to draw long construction lines). T Rules of the game: • The drawing must be of a house with the same proportions as the house in Chapter 2. Students will therefore need their books open to the appropriate page to recall these proportions. • Students may work individually or in teams. • When it is their turn, students may make one measurement, one line, or occasionally two construction lines (e.g., an "X" to ﬁnd a midpoint), subject to the approval of the instructor. • When direct measuring is valid (and only when it is valid), students may use the yardstick to make a measurement. • When students come to the board, they must ﬁrst tell their instructor what they intend to draw and get permission. If the instructor denies them permission, students lose their turn and must wait one full round to have another chance. • Instructors may disallow students from drawing a line if the construction is not valid, or if a diﬀerent construction for the same part of the house has already been started, or if the instructor judges the construction to be insuﬃcient or inelegant in any way. ∗ The authors invented the game, but not the title. What's My Line? was a popular game show on CBS television in the 1950s and '60s. The show would bring on a guest contestant, and celebrity panelists would attempt to guess the contestant's line of work. In our game you have to guess your own line—literally! 56 What's My Line? • The instructor will help to hold the yardsticks; students should draw lightly unless told otherwise. • The instructor is allowed to darken up important or visible lines and to erase lines that have become extraneous. While playing the game, a student (or a team) may wish to tackle a particularly diﬃcult part of the drawing to show oﬀ their skills. This is of course good. On the other hand, they often have the option of choosing an easy line, thereby passing the hard problems along to someone else (the sneaky approach). In any case, students should continue to think about the hard problems, because the game can reach a point where no easy options are available until someone "breaks through" one of the hard parts. Suggestions for instructors. Since the drawing is fairly large, a few guidelines must be drawn using two yardsticks placed end-toend, so it's important to have a couple handy. The instructor starts by drawing the front, right vertical edge of the house—the ﬁrst line of the game—along with a (lightly drawn) horizon line and a vanishing point (see Figures A and B). Putting these in good locations will help the construction greatly. Note: viewing distance is 72" >20" horizon line 6" Figure A. Setup of the game. >27" 20" 12" first line of the game Here are some suggested measurements: • Draw the horizon line more than 20′′ below the top edge of the board. • Draw the vertical line more than 27′′ from the left edge of the board. 57 A Perspective Game • Make the vertical line 18′′ tall, with 6′′ of it above the horizon and 12′′ below. • Draw the vanishing point on the horizon, 20′′ to the right of the vertical line. Instructors may ﬁnd it necessary to remind students that the house they are drawing must have the same proportions as the house in Chapter 2. It's of course much easier to draw a house with, say, a cubical lower section. But the whole point is to show students that they can accurately draw the house in Chapter 2 without a computer, using only a yardstick and a piece of chalk. It helps to see the house better if you think of it as "open" in the front: that is, the front pentagon is glass, and the rest is translucent. As students add lines to the house, darken the visible lines and leave the hidden lines fainter. If you no longer need construction lines, erase them so they don't make the picture even more confusing. In a class period of less than eighty or ninety minutes, students may not completely ﬁnish the drawing. It's diﬀerent every time. However, students should get enough done so that the structure of the house becomes apparent. When the house is done, each of the students should get a chance to view the picture (with one eye only, of course!) from the proper viewing location. Figure B. The finished house with a "glass wall" on the near end, and the hidden lines completely erased. first line of the game As an alternative, this could be a "turn it in" project in The Geometer's Sketchpad or GeoGebra. Students could be given a worksheet with the diagram of Figure A and an indicated viewing distance, with the assignment to draw construction lines in light gray and house lines in black. This page intentionally left blank CHAPTER 5 Two-Point Perspective n addition to one-point perspective, another common perspective drawing technique is two-point perspective, illustrated in Figure 5.1. Unless otherwise stated, we will use the term "twopoint perspective" to refer to a picture that is set up in such a way that the picture plane is perpendicular to the plane of the ground (outdoors) or the ﬂoor (indoors), and as a consequence of this, only two vanishing points on the horizon line are needed to render buildings or other objects whose adjacent vertical walls or sides are perpendicular to each other. This is a typical situation when dealing with architectural subjects, both indoors and outdoors. I V1 V2 Suppose the rectangular box in Figure 5.1 represents some kind of building, but we don't know anything about its size or proportions. Can we say anything about the correct location of the viewer? It turns out that we can. First, we can tell by looking that the picture plane must be perpendicular to the plane of the ground—that is, vertical—because the images of the vertical lines of the building are parallel to one another Figure 5.1. A simple example of two-point perspective. 60 Chapter 5 in the picture, and hence do not converge to a vanishing point. This can only happen if the picture plane is parallel to the vertical lines of the building, and hence perpendicular to the plane of the ground. Since our line of sight to any point on the horizon must be level, the viewer's eye is the same height as the horizon line in the picture. Thus the viewer's eye lies in a horizontal plane (the eye-level plane) H containing the horizon line (see Figure 5.2). Since the picture plane is vertical, H is perpendicular to it. It is convenient to think of H as a half-plane existing in the room where the viewer is to view the painting, as in Figure 5.2. The question is, exactly where in the plane H should the viewer's eye be located? picture plane V 1 Figure 5.2. The eye-level plane H. The correct viewpoint for the painting is somewhere in this plane, but where? ? ? the eye-level plane H ? V 2 If the picture were the result of a window-taping experiment, then the building would still be located beyond the window, and we could ﬁnd out all sorts of things about it. Even though it's not a window, we can think of the painting as the projection of a building that was once behind the canvas. At this stage we don't yet know how the building would be situated to make such a projection. Figure 5.3 shows two possible cases. We can narrow down our choices for the viewpoint E by recalling an important fact about perspective on windows. When the viewer's eye is at the correct viewpoint E, the line (of sight) from E to any vanishing point V on the window must be parallel to the actual line in the real world whose image has V as its vanishing point. Thus, regardless of how the building was oriented, the lines EV1 and EV2 in Figure 5.3 must be parallel to the corresponding building edges; since adjoining building edges are perpendicular, EV1 and EV2 are also perpendicular to each other. Figure 5.3 shows two examples of all possible cases. 61 Two-Point Perspective parallel parallel V1 of g top ildin bu to bui p of ldi ng parallel parallel V1 V2 V2 top edge of picture plane Figure 5.3. Two of many possible locations for the viewpoint E. Because the edges of the building form a right angle, the lines of sight to the vanishing points must form a right angle at the point E. E E the eye-level plane H This brings up a question: What is the set of all points E in the eye-level halfplane H such that EV1 and EV2 are perpendicular? It turns out that this set is a semicircle whose endpoints are V1 and V2 (see Figure 5.4). V1 V2 V1 E V2 H top view Theorem 5.1. The viewpoint E for a standard two-point perspective painting (drawing, photograph) with vanishing points V1 and V2 lies on a semicircle with endpoints V1 and V2 . The plane of the semicircle is perpendicular to the picture plane. Proof. Consider a possible viewpoint E in the half-plane H, as on the left of Figure 5.5. Since E is a possible viewpoint, the lines EV1 and EV2 are perpendicular. Let M be the midpoint of V1 and V2 , so that the two segments M V1 and M V2 have the same length r. Let Figure 5.4. The viewpoint E must lie on a horizontal semicircle in the halfplane H. 62 Chapter 5 s denote the length of EM . We are done if we show that s = r, for that will mean that all possible viewpoints E are r units away from M , and therefore lie on a semicircle. Now EV1 and EV2 are adjacent sides of a rectangle, so draw the entire rectangle EV1 F V2 , as indicated on the right of Figure 5.5. It's a well-known fact from geometry that the diagonals of a rectangle have equal lengths and meet at their common midpoint. Referring to the ﬁgure, this implies that s = t = r. F M V1 Figure 5.5. Looking down on the plane H and the top edge of the picture plane. V2 r V2 r s E Figure 5.6. When working in twopoint perspective, art students often tape strips of paper to their drawings so they can spread the vanishing points far apart. This makes for a larger viewing circle, one that is more likely to be occupied by the casual viewer's eye. M r r s H t V1 E H Theorem 5.1 explains a common trick used in art classes. Notice in Figure 5.4 that the farther apart the vanishing points V1 and V2 are, the bigger the "viewing circle"; that is, the farther away the potential viewpoints will be from the picture. We know from Chapter 3 that when viewpoints are unusually close to pictures, viewers perceive distortions, because they won't suspect that the correct viewpoint is so close. To prevent this from happening in two-point perspective drawings, art teachers often have their students tape strips of paper to their drawing paper (as in Figure 5.6) so that one or both of the vanishing points can be located beyond the edges of the paper. An art teacher would say, "We spread the vanishing points to avoid distortion." In view of Theorem 5.1, we could also say that "We spread the vanishing points to enlarge the viewing circle." Is a small viewing circle really so bad? To convince yourself that it is, look at Figure 5.7. It's a drawing of some boxes in two-point perspective, with both vanishing points V1 and V2 in the drawing, making them very close together. The drawing has been set up so that the viewpoint is directly in front of the midpoint C of V1 and V2 . Imagine a semicircle coming out of the page with V1 and V2 as its endpoints. It should be clear that the viewing distance in this case is just the radius of the semicircle, which is the distance between C and V1 (or V2 ). Close one eye and hold the page so that your open eye is very close to C and directly in front of it. Gaze at C for a second, then let your eye roll down and look at t