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2021 | Book

Making Images with Mathematics


About this book

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

Chapter 1. From Ancient Greeks to Pixels
The chapter explains how we see the world and how computer makes images. Beginning with Ancient Greek Geometry, it travels to modern geometry, introduces the subject of computer graphics and visualization, explains how the graphics pipeline works, and how a geometric point turns into a color spot on a computer screen.
Alexei Sourin
Chapter 2. Geometric Shapes
This chapter presents the mathematical foundations of shape modeling. Curves, surfaces, and solid objects are considered as set of points, which are obtained by sampling various types of mathematical functions. Using the concept of sweeping, many varieties of shapes are defined based on only a few simple foundation principles.
Alexei Sourin
Chapter 3. Transformations
This chapter considers how the same formulas, used for making shapes, can define their transformations. The rationale for using matrix transformations is explained and affine and projection matrix transformations are presented. Generalization of geometric sweeping implemented with matrices is further discussed.
Alexei Sourin
Chapter 4. Motions
This chapter explains how the previously introduced mathematical formulas, defining shapes and transformation matrices, can be extended to time-dependent models of moving shapes. Motions of rigid shapes and shape morphing transformations are considered. Besides pseudo-physical motions, definitions based on Newtonian physics are also introduced.
Alexei Sourin
Chapter 5. Adding Visual Appearance to Geometry
In this chapter, we consider how visual appearance including colors, shadows, material properties and textures can be added to geometry and how its photorealistic appearance can be achieved. The formulas, previously used for defining geometry, now will define variable colors as a new modality of immersion into the world of geometric definitions.
Alexei Sourin
Chapter 6. Putting Everything Together
In this chapter, the ways of making interactive, real-time and immersive visualization environments are considered including technical and physiological design and implementation issues. Still the same transformations, and actually the same basic mathematical principles, will be used in the fast visualization methods.
Alexei Sourin
Chapter 7. Let’s Draw
This chapter introduces to the reader a few commonly used freeware software tools—OpenGL, POV-Ray, VRML, and X3D—which will let the readers apply theoretical principles into practice without requesting expensive hardware and software solutions. Also, the readers will learn how immersive visual mathematics can be implemented using the function-based extensions of VRML and X3D, which allow for defining geometric shapes and their appearances with mathematical functions. Finally, the Shape Explorer tool will be presented to the reader as a multi-platform companion viewer for all the examples used in the book.
Alexei Sourin
Making Images with Mathematics
Prof. Dr. Alexei Sourin
Copyright Year
Electronic ISBN
Print ISBN