John Vince - Vector Analysis for Computer Graphics

The author(s) has/have asserted their right(s) to be identified as the author(s) of this work in accordance with the Copyright, Designs and Patents Act 1988.

This Springer imprint is published by the registered company Springer-Verlag London Ltd. part of Springer Nature.

The registered company address is: The Campus, 4 Crinan Street, London, N1 9XW, United Kingdom

The first edition of Vector Analysis for Computer Graphics was published in 2007, and after submitting the manuscript to Springer, I came across geometric algebra. Since then, I have thought about it intensely, and written two books on the subject. Meanwhile, what I call traditional vector analysis, still exists, and continues to be taught throughout the world. So, I decided to write a 2nd edition, and here it is.

It has taken over a year to prepare the final files, because the originals had been created using MS Word, and I now use . This has meant retyping every equation and creating all the figures again!

The books main objective is to introduce the subject of vector analysis to readers studying computer graphics, although it will be of interest to a wider readership. It includes most of the original topics, but I have extended various areas, and it still comprises eleven chapters. Chap. introduces dependent and independent equations, which are so important to vector analysis.

Chapter develops the use of the grad differential operator by showing how tangent and normal vectors are computed for various curves and surfaces.

Chapters examines three ways of rotating vectors in and , finishing with quaternions.

I have thought very carefully about what to leave out, and what to include, and really hope that you are happy with the result.

In this chapter I trace the emergence of the name vector, the invention of quaternions, and the branch of mathematics called vector analysis.

Today, Hamilton is recognised as the inventor of quaternion algebra, which became the first non-commutative algebra to be discovered. One can imagine the elation he felt when finding a solution to a problem he had been thinking about for a decade!

The invention provided the first mathematical framework for manipulating vectorial quantities, although this was to be refined by the American theoretical physicist, chemist, and mathematician Josiah Willard Gibbs [18391903]. Although Hamilton had arrived at his invention through an algebraic route, it was obvious to him that quaternions had significant geometric potential, and he immediately started to explore their vectorial and rotational properties.

Unbeknown to Hamiltonand virtually everyone else at the timethe French social reformer, and brilliant recreational mathematician Benjamin Olinde Rodrigues [17951851], had already published a paper in 1840 describing how to represent two successive rotations about different axes, by a single rotation about a third axis [].