Geometric Modeling with Splines: An Introduction

Part I: Introduction

Chapter List

Chapter 1: Review of Basic Concepts

In the sections that follow we provide a short presentation of some of the basic material which will be needed in various other chapters of the book. If the reader is familiar with the contents of some sections, he may prefer to skip those sections.

1.1 Vector Analysis

We shall provide a brief review of some necessary concepts and manipulation techniques. Many of these concepts are general and not dependent on any particular vector space. The concept of cross product, however is defined only in .

Definition 1.1

A vector space V is defined over a set of elements, the vectors, that have two operations, addition + : V V ? V, and scalar multiplication : V ? V which satisfy the following rules:

If u, v, w ? V and if r, , then

  1. r u + s v ? V ;

  2. There is an element 0 ? V such that for all v ? V, 0 + v = v + 0 = v ;

  3. r( u + v) = r u + r v;

  4. ( r + s) u = r u + s u.

Definition 1.2

A finite subset of vectors C in V is called independent if for every choice of n less than or equal to...

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