This chapter is an introduction to the principal ideas covered in this book. It gives an informal treatment of these topics. Precise, unambiguous definitions, careful algebra, and the description of well honed estimation algorithms is postponed until chapter 2 and the following chapters in the book. Throughout this introduction we will generally not give specific forward pointers to these later chapters. The material referred to can be located by use of the index or table of contents.

Introduction – the ubiquitous projective geometry

We are all familiar with projective transformations. When we look at a picture, we see squares that are not squares, or circles that are not circles. The transformation that maps these planar objects onto the picture is an example of a projective transformation.

So what properties of geometry are preserved by projective transformations? Certainly, shape is not, since a circle may appear as an ellipse. Neither are lengths since two perpendicular radii of a circle are stretched by different amounts by the projective transformation. Angles, distance, ratios of distances – none of these are preserved, and it may appear that very little geometry is preserved by a projective transformation. However, a property that is preserved is that of straightness. It turns out that this is the most general requirement on the mapping, and we may define a projective transformation of a plane as any mapping of the points on the plane that preserves straight lines.