Making Images with Mathematics (Undergraduate Topics in Computer Science)

✍ Scribed by Alexei Sourin

• Publisher: Springer
• Year: 2021
• Tongue: English
• Leaves: 256
• Edition: 1st ed. 2021
• Category: Library

✦ Synopsis

This textbook teaches readers how to turn geometry into an image on a computer screen. This exciting journey begins in the schools of the ancient Greek philosophers, and describes the major events that changed people’s perception of geometry. The readers will learn how to see geometry and colors beyond simple mathematical formulas and how to represent geometric shapes, transformations and motions by digital sampling of various mathematical functions.

Special multiplatform visualization software developed by the author will allow readers to explore the exciting world of visual immersive mathematics, and the book software repository will provide a starting point for their own sophisticated visualization applications.

Making Images with Mathematics serves as a self-contained text for a one-semester computer graphics and visualization course for computer science and engineering students, as well as a reference manual for researchers and developers.

✦ Table of Contents

Preface
Aim of the Book
Book Organization
Acknowledgments
Contents
1 From Ancient Greeks to Pixels
1.1 Drawing with Computer
1.1.1 How We See the World
1.1.2 Displaying Images
1.1.3 Computer Monitors
1.1.4 Storing Images in Computers
1.2 From “Earth Measuring” to Computer Graphics
1.2.1 Evolution of Geometry
1.2.2 Computer Graphics and Beyond
1.3 We Need Digits to Draw with Computer
1.3.1 2D Cartesian Coordinates
1.3.2 Polar Coordinates
1.3.3 3D Cartesian Coordinates
1.3.4 Cylindrical Coordinates
1.3.5 Spherical Coordinates
1.3.6 Visualization Pipeline
1.4 Geometric Algebra
1.4.1 Geometric Modeling
1.4.2 Rendering Geometry to Images
1.4.3 Mathematical Functions in Geometric Modeling
1.5 Summary
References
2 Geometric Shapes
2.1 Sampling Geometry
2.2 Points
2.3 Curves
2.3.1 Straight Line
2.3.2 Circle
2.3.3 Ellipse
2.3.4 Plethora of Curves
2.3.5 Three-Dimensional Curves
2.4 Surfaces
2.4.1 Plane
2.4.2 Polygons
2.4.3 Bilinear Surfaces
2.4.4 Quadrics
2.4.5 Making Surfaces by Sweeping Curves
2.5 Solid Objects
2.5.1 Defining Solids by Parametric Functions
2.5.2 Constructive Solid Geometry by Functions
2.6 Summary
References
3 Transformations
3.1 Mathematics of Transformations
3.2 Matrix Representation of Affine Transformations
3.2.1 Homogeneous Coordinates
3.2.2 Identity Transformation
3.2.3 Scaling and Reflection
3.2.4 Shear
3.2.5 Rotation
3.2.6 Translation
3.3 Composition of Transformations
3.3.1 Rotation About a Point
3.3.2 Scaling and Reflection About Points and Lines in 2D Space
3.3.3 Deriving Matrix of an Arbitrary 2D Affine Transformation
3.3.4 Rotation About an Axis
3.3.5 Reflections About Any Point, Axis, or Plane in 3D Space
3.3.6 Deriving Matrix of an Arbitrary 3D Affine Transformation
3.3.7 Matrix Algebra Laws
3.3.8 Definition of Sweeping by Transformation Matrices
3.4 Projection Transformations
3.4.1 Implementations of Projection Transformations
3.4.2 Classifications of Projection Transformations
3.4.3 Parallel Orthographic Projections
3.4.4 Axonometric Parallel Projections
3.4.5 Perspective Projections
3.4.6 Axonometric Perspective Projections
3.4.7 Projections on Any Plane
3.4.8 Viewing Frustum
3.4.9 Stereo Projection
3.5 Summary
References
4 Motions
4.1 Animating Geometry
4.1.1 Motion of Points
4.1.2 Animating Shape Definitions
4.1.3 Time-Dependent Affine Transformations
4.1.4 Shape Morphing
4.2 Physically Based Motion Specifications
4.2.1 Motion with Constant Acceleration
4.2.2 Motion Under Gravity
4.2.3 Rotation Motion
4.3 Summary
References
5 Adding Visual Appearance to Geometry
5.1 Illumination
5.2 Lighting
5.2.1 Ambient Reflection
5.2.2 Diffuse Reflection
5.2.3 Specular Reflection
5.2.4 Phong Illumination
5.3 Shading
5.3.1 Flat Shading
5.3.2 Gouraud Shading
5.3.3 Phong Shading
5.4 Shadows and Transparency
5.5 Textures
5.6 Colors by Functions
5.7 Summary
References
6 Putting Everything Together
6.1 Interactive and Real-Time Rendering
6.2 Positioning the Observer
6.2.1 Direction Cosines
6.2.2 Fixed Angles
6.2.3 Euler Angles
6.3 Fast Rendering of Large Scenes
6.3.1 Hierarchical Representation
6.3.2 Current Transformation Matrix
6.3.3 Logical and Spatial Organizations
6.3.4 Bounding Boxes
6.3.5 Level of Detail
6.4 Summary
References
7 Let’s Draw
7.1 Programing Computer Graphics and Visualization
7.2 Drawing with OpenGL
7.2.1 Introduction to OpenGL
7.2.2 Interaction with GLUT
7.2.3 Drawing Parametric Shapes with OpenGL
7.2.4 Animation and Surface Morphing with OpenGL and GLUT
7.2.5 Interactive Solid Modeling with OpenGL and GLUT
7.3 Drawing with POV-Ray
7.3.1 The Persistence of Vision Ray Tracer
7.3.2 Function-Based Shape Modeling with POV-Ray
7.4 Drawing with VRML/X3D
7.4.1 Introduction to VRML
7.4.2 Introduction to X3D
7.4.3 Function-Based Extension of VRML
7.4.4 Function-Based Extension of X3D
7.5 Drawing with Shape Explorer
7.6 Summary
References
Index