# Affine Maps, Euclidean Motions and Quadrics

The assertion that this is a *textbook* may need some nuance, the book being xi+411 pages long. Since quite often readers loose impetus after 150 pages, such a thick book could discourage many. However those 400 pages can be estimated as 200 of real course. Indeed, the last 100 pages include appendices that are themselves mini-courses in *Linear Algebra and Quadratic Forms,* and maybe almost that many pages could be counted for the lists of exercises suggested at the end of every chapter, and the many examples fully worked out along the lessons either as illustrations of previous results or as motivation for proofs that follow. For instance, there is a 7 page section titled *Clarifying Examples* prior to the classification of affinities in arbitrary dimension. In addition, there are some sections that could be marked as optional, either because they include material better understood with some extra knowledge (as *Projective Geometry*), or because they deal with the recovering of classical results. Here one cannot help praising the succinct 5 page introduction where the author summarizes *Euclid’s axiomatic of Affine Geometry* and explains how it motivates the text. All these are of course added bonuses, but it is good to separate them from the main content of the text. Thus we are left with say 180 pages of *Affine and Euclidean Geometry,* which cover the usual: *Affine spaces, subspaces and their operations; frames; affine maps and invariants, classification; Euclidean affine spaces and Euclidean motions, classification; real affine and Euclidean quadrics, classifications*. Very little need to explain further this list, but even running the usual path, several qualities distinguish this text from others.

First, concerning the global view, there is a neat scheme of action: to give the definitions of the objects to study and discuss their classification problems. Here, although the approach is algebraic, the geometric meaning of the corresponding classification results (canonical forms, canonical equations, tables and lists) is always stressed, with careful choices of terminology. Second, the job is done thoroughly without fault. It is not that common to find: (1) a *full classification of affinities* as given here, (2) the clear distinction between *a quadric and its equations* and the exact description of their relationship. Third, there are separate sections for every classification question in the low dimensional cases, sometimes preceding as preparation the general result. Fourth, it is quite clear that this is a *real life course,* that is, a text written for and from true teaching. Thus any professor will easily find the way to adapt the text to particular whims, discarding technicalities or lightening some lessons. Also, students will find a self-contained book containing all they need to catch the matter: full details and many solved and proposed examples.

All in all the text is a highly recommendable choice for a course on *Affine Geometry,* and fills some gaps in the existing literature.

**Submitted by Jesus M. Ruiz |

**14 / Feb / 2012