Table of Contents

Cover

Title

Publisher’s Note

Dedication

Acknowledgments

About the Author

Introduction

Why I Wrote This Book
Who Should Read This Book
What You Will Learn
Pretest
How to Contact the Author

Chapter 1: Understanding the Relationship between Math and Art

Understanding How Math and Art Differ
Understanding How Math and Art Are Similar
Using Math to Create Digital Art
Using Math in Traditional Drawing
Using Math to Help You Draw
Examining Existing Drawing Methods
Stop Separating Math from Art

Chapter 2: Extracting Graphical Structures—A Scientific Point of View

Introducing CG Concepts and Principles
Unifying 2D and 3D Cases
Understanding the Properties of Basic Shapes
Understanding Graphical Structures
Extracting GSs with Triangulation

Chapter 3: Drawing with the ABC Method

Meeting the Criteria for a New Drawing Method
Introducing the ABC Method

Chapter 4: Drawing Simple Objects

Preparing to Draw
Extracting Basic Information
Demonstrations of 3D Objects
Demonstrations of 2D Images
Exercises

Chapter 5: Drawing Complicated Objects

Breaking Down Complexity
Demonstrations
Exercises

Chapter 6: Drawing Complicated Scenes

Introducing Techniques for Merging Objects
Demonstrations
Exercises

Chapter 7: Using Advanced Techniques

Working with Curved Triangles and Polygons
Drawing a Group of Ovals
Using Shadow Maps
Using Rhythms for Partitioning
Exploring Trade-Offs between Accuracy and Speed
Demonstrations
Summary
Exercises

Chapter 8: Using Special Techniques for Drawing Human Bodies

General Tips
Tips for Drawing Faces
Tips for Quick Life Sketches
Demonstrations
Summary
Exercises
Post-Test: Drawing Van Gogh

Index

End User License Agreement

List of Illustrations

Introduction

Figure I-1: My 13-year old daughter’s pencil drawing after two years of practice with the ABC method
Figure I-2: My 7-year-old son’s pencil drawing after one year of practice with the ABC method
Figure I-3: Drawing pad, papers, and pencils
Figure I-4: Image for the pretest

Chapter 1: Understanding the Relationship between Math and Art

Figure 1-1: An example of a math problem solved in two ways
Figure 1-2: An example of a basic drawing exercise
Figure 1-3: Golden ratio in the Great Pyramid
Figure 1-4: Fibonacci structures found in Mona Lisa
Figure 1-5: The golden ratio used for composition
Figure 1-6: Stars used in flag design
Figure 1-7: A geometric pattern
Figure 1-8: 3D structures in an observational drawing
Figure 1-9: Perspective view in the Last Supper
Figure 1-10: Graphic proof of the Pythagorean theorem
Figure 1-11: Bézier curve construction (left) and vertex dependency (right)
Figure 1-12: The Mandelbrot set
Figure 1-13: A painting using IMPaSto
Figure 1-14: An ancient drawing device
Figure 1-15: Picture plane front view (left) and side view (right)
Figure 1-16: Yin and yang
Figure 1-17: Left-hand quick sketch (left) and right-hand quick sketch (right)
Figure 1-18: A simple perspective drawing
Figure 1-19: Cubes in perspective view (left) and cubes in orthogonal view (right)
Figure 1-20: The grid method
Figure 1-21: A contour-line drawing (left) and a regular drawing (right)
Figure 1-22: Positive space (left) and negative space (right)

Chapter 2: Extracting Graphical Structures—A Scientific Point of View

Figure 2-1: Curve drawn at different resolutions
Figure 2-2: Boolean operations
Figure 2-3: Primary object forms
Figure 2-4: CSG tree
Figure 2-5: A 3D fractal
Figure 2-6: Koch curve
Figure 2-7: Bounding box (left) and bounding polygon (right)
Figure 2-8: Global triangulation (left) and local triangulation (right)
Figure 2-9: The quadtree approach (left) and the triangulation approach (right)
Figure 2-10: Progressive refinement of a triangulation
Figure 2-11: A T-bone (left) and a fixed T-bone (right)
Figure 2-12: Re-triangulation with a single triangle partitioning
Figure 2-13: Re-triangulation with a regional repartitioning
Figure 2-14: A 3D perspective drawing
Figure 2-15: Some 3D primary forms (left) and their projections in 2D space (right)
Figure 2-16: An oval structure
Figure 2-17: Partitioning a polygon in different ways
Figure 2-18: Coarse polygon (top left), after adding more points (top right), adding more points still (bottom left), and the final polygon (bottom right)
Figure 2-19: A graphic and its variations
Figure 2-20: Static GSs in a human face
Figure 2-21: Dynamic graphical structures in a human body
Figure 2-22: Individual vs. grouped GSs
Figure 2-23: Grouped GSs can be used to control negative space
Figure 2-24: Base polygon (top left), second level (top right), third level (bottom left), and fourth level (bottom right)
Figure 2-25: Base oval (top left), second level (top right), third level (bottom left), and fourth level (bottom right)
Figure 2-26: Base rectangle (top left), second level (top right), third level (bottom left), and fourth level (bottom right)
Figure 2-27: A triangulation locks all DOFs.
Figure 2-28: Key points (top left) used to lock the area of interest, more key points added (top right), result of adding new points (bottom left), result of one more level down in the image hierarchy (bottom right)

Chapter 3: Drawing with the ABC Method

Figure 3-1: Various angles
Figure 3-2: Angles in a clock
Figure 3-3: Using fingers for angles
Figure 3-4: Two popular ways to create a reference line
Figure 3-5: Test your ability to detect parallel lines.
Figure 3-6: Locating a point by using distances (left) and by using reference lines (right)
Figure 3-7: Multireference checking strategy
Figure 3-8: Traditional proportion measurement (left) and diagonal-line measurement (right)
Figure 3-9: Sketching an oval
Figure 3-10: Find points on an oval.
Figure 3-11: Copying a triangle
Figure 3-12: Duplicating a simple polygon
Figure 3-13: A complicated polygon (top left), polygon simplification (top right), duplicated simple polygons (bottom left), and refinement to get the desired result (bottom right)
Figure 3-14: Basic shapes are sketched without explanations (left), and detail is added directly (right).
Figure 3-15: A scene and its triangulation (left), a coarse level of the triangulation (center), and a refinement treatment (right)
Figure 3-16: Connecting the centers of individual objects to create a virtual anatomical framework
Figure 3-17: Using grouped graphical structures (left) and using reference lines (right)
Figure 3-18: Initial polygons (top left), refinement (top center), simple shapes to control details (top right), duplicated polygons (bottom left), duplicated partitioning (bottom center), and all details added (bottom right)
Figure 3-19: Picture to copy (top left), starting polygons (top center), anatomical knowledge used (top right), further partitioning (bottom left), final result (bottom center), and more dynamic connections (bottom right)
Figure 3-20: 3D head to draw (top left), simple polygons to start (top right), more polygons generated (bottom left), and detail added (bottom right)
Figure 3-21: A figure quick sketch (left) and simple polygons used to guide drawing of details (right)

Chapter 4: Drawing Simple Objects

Figure 4-1: Your paper needs to be placed vertically.
Figure 4-2: Examples of key points
Figure 4-3: Examples of key points
Figure 4-4: Existing lines (top left), vertical and horizontal lines (top right), lines forming basic shapes (middle), lines connecting key points (bottom left), and reference lines at an interesting spot (bottom right)
Figure 4-5: Original polygon (left), simplified polygon (center), and further simplified version (right)
Figure 4-6: Ovals in a cup
Figure 4-7: Acquiring the ratio via midpoints
Figure 4-8: Beach ball (left), ovals in the ball (center), and reference lines (right)
Figure 4-9: Duplicating key points and reference lines (left), sketching ovals (center), and the final result (right)
Figure 4-10: Tissue box (left), distorted edges (center), and reference lines (right)
Figure 4-11: Duplicating key points with reference lines (left) and the final drawing of the box (right)
Figure 4-12: A cup (left), major structures (center), and reference lines (right)
Figure 4-13: Sketching the main part (left), drawing reference lines and key points (center left), sketching all parts (center right), and the final drawing (right)
Figure 4-14: A complicated object
Figure 4-15: A very abstract version (left), a simple version (center), and a complicated version (right)
Figure 4-16: The eye to draw (left), reference lines and key points (center left), coarsest triangulation (center right), and refinement (right)
Figure 4-17: Sketch of the coarsest triangulation (left), sketch after refinement (center left), sketch before being finalized (center right), and the final drawing product (right)
Figure 4-18: Nose to draw (left), starting mesh (center), and refined mesh (right)
Figure 4-19: Sketch of the coarsest triangulation (left), sketch after refinement (center), and the final drawing (right)
Figure 4-20: Mouth to draw (left), starting mesh (center), and refined mesh (right)
Figure 4-21: Sketch of the coarsest triangulation (left), sketch after refinement (center), and the final drawing (right)
Figure 4-22: Key points in the face (left), a sketch duplicating the structure (center), and the final draw of the outline (right)

Chapter 5: Drawing Complicated Objects

Figure 5-1: Spray bottle (left), with bounding rectangles and quadrilaterals (center), and decomposition (right)
Figure 5-2: Bounding polygons (left), detail added (center), and final drawing (right)
Figure 5-3: Rose (left) and bounding boxes of some petals (right)
Figure 5-4: Starting mesh (top left), refinement (top right), one more refinement (bottom left), and final drawing (bottom right)
Figure 5-5: An abstract tree structure
Figure 5-6: A realistic tree drawing using a fractal concept
Figure 5-7: Skeleton in a flower (left) and skeleton in a tripod (right)
Figure 5-8: Repeated decorative feature on furniture (top left), circle format in the Space Needle (right), and lots of geometric patterns on a computer motherboard (bottom left)
Figure 5-9: Selective drawing of hair
Figure 5-10: Lightbulb (left), partitioning (center), and further analysis of the top part (right)
Figure 5-11: Outline of the three parts of the lightbulb (left), sketch of all the structures (center), and final drawing (right)
Figure 5-12: Avoid using nonexisting corner points.
Figure 5-13: A piece of fabric (left), partitioning (center), and further structural analysis (right)
Figure 5-14: Outline of initial partitioning (left), more key points added (center), and final drawing (right)
Figure 5-15: A shoe (left), partitioning (center), and further structural analysis (right)
Figure 5-16: Initial partitioning (top left), more key points allocated (top right), sketch of all the structures (bottom right), and the final drawing (bottom left)
Figure 5-17: A lamp (left), initial partitioning (center left), further refinement (center right), and local refinement of the base (right)
Figure 5-18: Initial partitioning (left), more key points allocated (center left), sketch of all the structures (center right), and the final drawing (right)
Figure 5-19: A V sign (left), initial partitioning (center left), adding key points for the other three fingers (center right), and controlling the details of fingers (right)
Figure 5-20: Initial partitioning (left), key points for the other three fingers (center left), sketching all fingers (center right), and the final drawing (right)
Figure 5-21: A tripod (left), key points for the three legs (center), and refinement for the rest of the major structures (right)
Figure 5-22: Duplicating the key points of the three legs (left), adding more key points and reference lines (center left), sketching all the major structures (center right), and the final drawing (right)
Figure 5-23: A giant bird-of-paradise (left) and its simple framework (right)
Figure 5-24: Loose sketch of the structures (left) and selectively drawn detail (right)
Figure 5-25: Lego toy (left), key points for the major body (center left), key points for cylinders (center right), and ovals to extract (right)
Figure 5-26: Draw the main body part with a few key points (left), add key points to control cylinders (center left), and sketch all cylinders (center right). The final drawing (right).
Figure 5-27: Toy dinosaur

Chapter 6: Drawing Complicated Scenes

Figure 6-1: An example scene (top left), some grouped GSs (top center and top right), sketch of grouped GSs (bottom left), sketch of all items (bottom center), and the final drawing (bottom right)
Figure 6-2: Negative space
Figure 6-3: Three examples of grouping GSs in the same scene
Figure 6-4: A face (left), and overlapping shapes to lock the nose (center) and control the right eye (right)
Figure 6-5: Controlling an irregular shape via overlapping of regular shapes
Figure 6-6: A scene with a virtual skeleton (left) and virtual skeleton assisted by triangulation (right)
Figure 6-7: A zoomed-in area (left), duplicate of polygonal partitioning (left center), added detail in the top area (right center), and negative space in the top area (right)
Figure 6-8: A scene composed of fruit (left), its virtual skeleton (left center), small structures added (right center), and the final drawing (right)
Figure 6-9: A pair of dumbbells (left), sketch of two large cylinder structures (left center), detail structures added (right center), and the final drawing (right)
Figure 6-10: Some leaves (left), some grouped graphical structures (left center), detail structures added (right center), and the final drawing (right)
Figure 6-11: A landscape (top left), overall planning (top right), adding simple shapes for small structures (bottom left), and the final drawing (bottom right)
Figure 6-12: A stovetop setting (top left), overall planning (top right), adding structures (bottom left), and the final drawing (bottom right)
Figure 6-13: Fruit and container (top left), partition for groups (top right), adding structures (bottom left), and the final drawing (bottom right)
Figure 6-14: Messy boats (top left), a quick partition (top right), further partition (bottom left), and the final drawing (bottom right)
Figure 6-15: A pair of women’s shoes (top left), some grouped graphical structures (top right center), detail structures added (bottom left), and the final drawing (bottom right)
Figure 6-16: Group of dancers (top left), partition with virtual skeletons (top right), adding large structures (bottom left), and the final drawing (bottom right)

Chapter 7: Using Advanced Techniques

Figure 7-1: Softness of polynomials
Figure 7-2: Approximating a large number of points with a piecewise parabola (left) and avoiding a sharp-turned curve (right)
Figure 7-3: A curved triangle (left) and a curved polygon (right)
Figure 7-4: Initial step of a curved triangulation (left), refinement of the triangulation (center), and the traditional triangulation (right)
Figure 7-5: A subdivision process of a quadratic Bézier curve (left), and controlling the curve with end points and tangent lines (right)
Figure 7-6: An incorrect sketch of a small digital camera with lens retracted (left) and lens extended (right)
Figure 7-7: The short axis is always parallel to the direction that the cutting plane faces
Figure 7-8: Drawing cross sections with the perspective theory (left) and a new way developed from geometry (right)
Figure 7-9: Shadow map in a portrait drawing
Figure 7-10: Fruit under direct sunlight
Figure 7-11: Apple with reflected textures (left), initial structure on paper (center left), placing reflections (center right), and the final drawing (right)
Figure 7-12: A parking lot scene (top left), vertical and horizontal rhythms in the scene (top right), smaller items being placed (bottom left), and the actual drawing (bottom right)
Figure 7-13: A garden flower
Figure 7-14: Starting with a curved triangle (left), locking structures in the body (center left), placing all structures (center right), and the final drawing (right)
Figure 7-15: Van Gogh’s face (top left), starting with a curved polygon (top center), partitioning with reference lines (top right), locking all major features (bottom left), sketching all features (bottom center), and the final drawing (bottom right)
Figure 7-16: A digital camera (top left), a structural analysis (top center), starting with the body and base of the lens (top right), adding more oval structures (bottom left), sketching all of the features (bottom center), and the final drawing (bottom right)
Figure 7-17: Ring toss toy (left), starting with overall shapes (center left), sketching all major structures (center right), and the final drawing (right)
Figure 7-18: A grouping of fruit in direct sunlight
Figure 7-19: A fruit scene (top left), starting with a couple of curved polygons (top right), sketching all of the major structures (bottom left), and the final drawing (bottom right)
Figure 7-20: Bath faucet and metal shower hose
Figure 7-21: A faucet system (top left), starting with curved polygons (top right), arranging major structures (bottom left), and the final drawing (bottom right)
Figure 7-22: A scene from Yosemite National Park (top left), using rhythm for a quick partitioning (top right), sketching structures roughly (bottom left), and the final drawing (bottom right)

Chapter 8: Using Special Techniques for Drawing Human Bodies

Figure 8-1: A human skeleton with major joints highlighted (front view)
Figure 8-2: Muscles in the human body and some landmarks
Figure 8-3: Isaiah (top left), initial polygons based on joints (top center), refined structures (top right), arranged facial features (bottom left), refinement of all detailed structures (bottom center), and the final drawing (bottom right)
Figure 8-4: Static graphical structures in a human face
Figure 8-5: A picture of JFK (top left), initial polygons and feature lines (top center), polygons for eyes and eye-nose triangle (top right), arranged facial structures detail (bottom left), all detailed structures drawn (bottom center), and some minor problems fixed (bottom right)
Figure 8-6: Bad and good examples of drawing a nose
Figure 8-7: Step 1 (left), step 2 (center left), step 3 (center right), and step 4 (right)
Figure 8-8: A 2-minute life figure drawing
Figure 8-9: A 5-minute quick sketching (left), simple shapes re-created (center), and a line drawing (right)
Figure 8-10: Core shadows in a dancer (left) and a quick sketch using core shadows (right)
Figure 8-11: Using shadows to hint at structures
Figure 8-12: A torso as a rectangle (left) and a torso as two ovals (right)
Figure 8-13: Some samples of 2-minute life drawings
Figure 8-14: A picture of a woman (top left), initial lines (top center), locking major structures (top right), refining face and hands (bottom left), drawing all detailed structures (bottom center), and the final artwork (bottom right)
Figure 8-15: A picture of The Thinker (top left), initial polygons (top center), sketching arms and legs (top right), arranging detailed structures (bottom left), drawing face and muscles (bottom center), and the final artwork (bottom right)
Figure 8-16: A picture of Einstein (top left), initial polygons (top center), feature lines, sunglasses, and eye-nose triangle (top right), arranging detailed facial structures (bottom left), drawing all detailed structures (bottom center), and fixing some minor problems (bottom right)
Figure 8-17: A picture of little boy (top left), initial circle, polygon, and feature lines (top center), polygons for eyes, nose, and mouth (top right), arranging detailed facial structures (bottom left), drawing all detailed structures (bottom center), and fixing some minor problems (bottom right)
Figure 8-18: Five 1-minute quick sketches
Figure 8-19: Five 2-minute quick sketches
Figure 8-20: Five 5-minute quick sketches

Development Editor: Gary Schwartz

Technical Editor: James Haldy

Production Editor: Eric Charbonneau

Copy Editor: Sharon Wilkey

Editorial Manager: Pete Gaughan

Production Manager: Tim Tate

Vice President and Executive Group Publisher: Richard Swadley

Vice President and Publisher: Neil Edde

Book Designer: Caryl Gorska

Compositor: Maureen Forys, Happenstance Type-O-Rama

Proofreader: Kim Wimpsett

Indexer: Ted Laux

Project Coordinator, Cover: Katherine Crocker

Cover Designer: Ryan Sneed

Cover Images: Wei Xu, PhD

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Dear Reader,

Thank you for choosing Drawing in the Digital Age: An Observational Method for Artists and Animators. This book is part of a family of premium-quality Sybex books, all of which are written by outstanding authors who combine practical experience with a gift for teaching.

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I hope you see all that reflected in these pages. I’d be very interested to hear your comments and get your feedback on how we’re doing. Feel free to let me know what you think about this or any other Sybex book by sending me an email at nedde@wiley.com. If you think you’ve found a technical error in this book, please visit http://sybex.custhelp.com. Customer feedback is critical to our efforts at Sybex.

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Acknowledgments

First and foremost, I would like to thank my parents for fostering my artistic dreams when I was a little child. This book originated from those small dreams.

I am grateful to my wife, Lin, for her continuous support for this book project.

Thanks also go to my lovely daughter, Alicia, and my clever son, Daniel, for being the first users of the new method I describe in this book. Their initial learning success validated that my method is easy to learn, and even kids can pick it up without difficulty.

I offer profound thanks to the wonderful team at Wiley, especially Mariann Barsolo and Jenni Housh for establishing the book project and working on the contract quickly, Gary Schwartz for polishing my scripts as well as turning my “Chinenglish” into English, James Haldy for spotting and fixing technical issues, Pete Gaughan for solving template issues, and Eric Charbonneau for making the book ready for publication. Also contributing to the book you now hold were copy editor Sharon Wilkey, proofreader Kim Wimpsett, and compositor Maureen Forys. Thanks again to all for a job exceptionally well done.

This book is the product of years of research and art learning. I’d like to thank members of the faculty and staff of the Art Institute of California at San Diego for their assistance in my art education and writing about art, particularly the following individuals (in alphabetical order): Jack Beduhn, Christian Bradley, Rebecca Browning, Dr. Alicia Butters, Elizabeth Erickson, Harry Hamernik, Jason Katsoff, Wattana Khommarath, Jack Madi, Dr. Mary McDermott, Dzu Nguyen, Lena Pham, Grace Piano, Dr. Kim Varey, and Grady Williams.

I also want to thank the students at the Art Institute of California at San Diego for attending my workshops and giving me positive feedback over the past two years. Without their involvement, my method could still be in the research stage instead of being a published book. Special thanks go to my students Alvin Revilas and Blake Fox for their proofreading of my research articles related to this book.

Of course, I cannot forget to thank Susan Varnum, Dr. Edward Abeyta, and Robin Wittman of the University of California San Diego Extension for their efforts to set up a new art class to teach my method in the winter of 2012.

I owe debts of gratitude to faculty, staff, and students at Zhejiang University, especially Dr. Weidong Geng and Ms. Xiao Li for inviting me to lecture their multimedia students on my drawing method in the summer of 2011.

I’d like to thank Alex Reed for her language assistance in preparing the first draft of this book.

Finally, I want to thank you, the reader, for picking up this book and taking an exciting drawing journey with me!

About the Author

Wei Xu, PhD, is a computer scientist, mathematician, and artist. Currently, he serves as faculty or adjunct/visiting faculty at several universities including the Art Institute of California at San Diego, the University of California San Diego Extension, and Zhejiang University (China). He is also the president and cofounder of Geomy Entertainment, a video game consulting and mobile app development firm.

Being a computer scientist and video game veteran, Dr. Xu offers a wide range of video game production classes for college students and technical training for game companies. As an artist, he has been hosting drawing workshops at the Art Institute of California at San Diego since 2009. Starting January 2012, he will officially offer drawing classes to teach his new drawing system, the ABC method, at both the Art Institute of California at San Diego and the University of California San Diego Extension.

Dr. Xu obtained his doctoral degree in computer graphics and applied mathematics from the University of Texas at Austin. He also holds an MS degree in applied math and computer graphics from ZheJiang University, China, and a BS in computational mathematics from Fudan University, China. He has served as the lead engineer for game technology R&D at Sony Computer Entertainment America, San Diego. Prior to joining Sony, he was a senior computer graphics software engineer at the Schlumberger Austin Technology Center.

AN OBSERVATIONAL METHOD FOR ARTISTS AND ANIMATORS