# Vectors and Matrices for Geometric and 3D Modeling

## Vectors and Matrices for Geometric and 3D Modeling

#### Michael E. Mortenson

Overview

Vectors are perhaps the most important mathematical objects used in modeling and animation. They have the properties of magnitude and direction, and provide visual understanding of model construction and analysis. Matrices are natural and hardworking partners of vectors.

This work presents lessons on vectors and matrices in geometric and 3D modeling—the mathematics at the foundation of computer graphics applications. The lessons appear as chapters, generally organized from introductory to more complex topics. Within each chapter, there is a similar order of elementary-to-advanced discussion.

Here are topics that are usually published in briefer form in more advanced texts as part of their supporting mathematics. In this work, vectors and matrices are the main subjects. This text offers an easier-to-understand introduction to the main ideas behind vectors and matrices, stripped of formalism, and leading directly to geometric modeling. It demonstrates the relationships between vectors, matrices, basis vectors and barycentric coordinates, all of which are not usually seen in ordinary texts. This text also discusses how these concepts are applied to produce curves and surfaces, and how they facilitate the analysis of spatial relationships.

For those readers beginning studies in geometric and 3D modeling, animation, CGI, or CAD/CAM, this work serves as an introduction to vectors and matrices, and provides a good start to understanding how they are applied. For instructors, this book can be a primary text or supplement to more advanced or specialized texts on geometric and 3D modeling.

Features

• More than 150 illustrations support visual understanding of the content.
• 100+ exercises and extended solutions enhance the classroom environment.
• A comprehensive range of topics offers an in-depth look at the math underlying 3D modeling and animation courses.
• Linear algebra and calculus are not prerequisites.

Michael E. Mortenson does independent research and writes on topics in geometric and 3D modeling. He is a former research scientist with a major aerospace corporation, and the author of several successful textbooks, including Geometric Modeling, Third Edition, Mathematics for Computer Graphics Applications, 2e, Geometric Transformations for 3D Modeling, 2e, and 3D Modeling, Animation, and Rendering: An Illustrated Lexicon. These works have garnered over 2400 citations. He is a graduate of the UCLA School of Engineering.

• 1. Vectors PDQ
• 2. The Visual Vector
• 3. The Algebraic Vector
• 4. Matrix Basics
• 5. Special Matrices
• 6. Basis Vectors
• 7. Barycentric Coordinate Systems
• 8. Translation and Rotation
• 9. More Transformations
• 10. Vector-Defined Geometric Objects I
• 11. Vector-Defined Geometric Objects II
• 12. Spatial Relationships
• Solutions