The book contains eleven chapters: The first chapter distinguishes between scalar and vector quantities, which is reasonably straightforward. The second chapter introduces vector represen- tation, starting with Cartesian coordinates and concluding with the role of direction cosines in changes in axial systems. The third chapter explores how the line equation has a natural vector interpretation and how vector analysis is used to resolve a variety of line-related, geometric problems. Chapter 4 repeats Chapter 3 in the context of the plane.
At this point in the book, the reader has enough knowledge to tackle some standard problems encountered in ray tracing, such as reflections (Chapter 5) and intersections (Chapter 6).
Quaternions are the subject of Chapter 7, which is where I show how the clever combination of a scalar and a vector creates an object capable of rotating points about an arbitrary axis.
Chapter 8 introduces the idea of differentiating vector quantities, which are needed later on when we tackle bump mapping. Chapter 9 shows how vector analysis is used to describe projections, especially when the projection plane is oblique to the viewer’s line of sight. Chapter 10 examines Gouraud and Phong shading as well as bump mapping, all of which utilize vectors as part of their algorithms. Finally, the book concludes with a short chapter on motion.
After reading this book, the reader should have a good understanding of how to employ vector analysis in solving a variety of geometric problems. I have found that the diagram used to summarize a problem’s geometry often determines whether the solution drops out in a few lines or runs over several pages. For example, finding the formula for spherical interpolation involves the sine rule. And as you will discover in Chapter 7, the solution is extremely simple. However, if you start with the wrong diagram, the proof explodes into an algebraic nightmare!