Using Real Quaternions to Represent Rotations in 3D (UMAP)

This module raises the general question of representing rotations in three dimensions. It motivates quaternions by referring to Hamilton's search for a four-dimensional system that embeds the real and the complex numbers, and approaches quaternions in three dimensions through the corresponding vectors and matrices. Quaternions are applied to rotating shape representations in computer graphics. An appendix is included, which proves the theorems that connect quaternion and matrix implementations of rotation. Pascal procedures are also included to implements an abstract data type for quaternions.

Table of Contents:

INTRODUCTION

MATRICES AND ROTATION

QUATERNIONS

VECTORS AND REAL QUATERNIONS

QUATERNIONS AND ROTATION

SEQUENCES OF ROTATIONS

EXTRACTING THE COMPONENTS OF A ROTATION

ROTATION ABOUT AN ARBITRARY AXIS

APPLICATION: ROTATING SHAPE REPRESENTATIONS

CONCLUSION

EXERCISES

SAMPLE EXAM

SOLUTIONS TO THE EXERCISES

SOLUTIONS TO THE SAMPLE EXAM

APPENDIX

REFERENCES

ACKNOWLEDGMENTS

ABOUT THE AUTHOR