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Affine Maps, Euclidean Motions and Quadrics

Agustí Reventós Tarrida
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Springer Undergraduate Mathematics Series
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The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition.

[Reviewed by
Mark Hunacek
, on

Anyone who, like me, is particularly interested in how geometry can be developed using the tools of linear algebra will certainly want to look closely at this book, which gives a linear-algebraic definition of affine and Euclidean geometry and uses that framework to discuss the major theorems of these geometries, with special emphasis on both the structure-preserving transformations on them and on quadric surfaces. This is material that is also discussed in books such as Gallier’s Geometric Methods and Applications but the exposition here seems to be intended for a less sophisticated audience, and much less material is covered in this book than is covered in Gallier.

The prerequisites for this book seem pretty modest: a good one-semester course in linear algebra is certainly necessary, but there is an extensive appendix (about 70 pages long) which is largely devoted to the more sophisticated topics in linear algebra that are used, mostly with proofs included. (These topics include, for example, inner products, bilinear forms, diagonalization, and the spectral theorem; one other section of the Appendix discusses zeroes of polynomials in many variables and proves versions of the Nullstellensatz for quadratic polynomials over the field of real numbers and over an algebraically closed field.)

Affine geometry can be thought of as “Euclidean geometry without measurement” — thus, the concepts of interest in affine geometry relate to incidence and parallelism rather than distance and angles. Some books, such as Kaplansky’s Linear Algebra and Geometry, simply define an affine space as any vector space, with affine subspaces defined as cosets of vector subspaces: “lines” are the cosets of one-dimensional subspaces, and “planes” the cosets of two-dimensional subspaces, etc. This is certainly intuitively appealing, but under this formulation the vector 0 is given undue emphasis; in addition, an affine subspace such as a line is generally not an affine space in its own right. A way to get around this is to define an affine space as a set X (the elements of which are the “points” of the space) on which a vector space V acts (the action of v  V on P  X is denoted P + v) in a manner satisfying certain basic axioms. (So, of course, a vector space V is itself an affine space by taking V = X and the action to be ordinary vector space addition.) Within this framework one can define a line through P, for example, as the set of all points P + v as v ranges over a one-dimensional subspace in V; other properties relating solely to incidence can be similarly defined. One can also, by fixing a “frame” — i.e., a point P in X and an ordered basis for V — establish affine coordinates in X; if the elements of V are also given coordinates relative to the ordered basis, then the action of V on X is just addition of n-tuples. Tarrida describes this set-up in the first chapter of his book, and proves a number of fairly sophisticated theorems of affine geometry, such as the theorems of Thales, Menelaus and Ceva (all of which can, of course, be given synthetic proofs in the spirit of high school geometry, but which are seldom taught there.) The affine versions of the theorems of Desargues and Pappus appear in the exercises.

The next three chapters discuss, in some detail, the affinities and semi-affinities of an affine space. The definition of “affinity” is a bit technical, but the upshot is that affinities are, relative to the affine coordinates described in the preceding paragraph, simply translates of linear mappings; semi-affinities are essentially affinities modulo an automorphism of the underlying field, and these turn out to be precisely the mappings of the affine space that preserve collinearity. (So, for fields like the real numbers that have no nontrivial automorphisms, the notions of semi-affinity and affinity coincide.) An affinity of n-dimensional affine space that in affine coordinates is the mapping u → Au + a can then be realized as an (n + 1) × (n + 1) matrix, which allows Tarrida to use linear algebra in the classification theory described in chapters 3 (for affinities of the plane) and 4 (affinities of arbitrary n-dimensional space). Tarrida’s treatment of affine spaces and affinities is roughly comparable, in scope and level of difficulty, to part I of Snapper and Troyer’s Metric Affine Geometry, although Tarrida presents details of the classification of affinities of an arbitrary n-dimensional affine space that go beyond what Snapper and Troyer (or, for that matter, any other book of which I am aware) do.

In parts II and II of their book, Snapper and Troyer then proceed to introduce metric structures into these affine spaces, as does Tarrida in chapter 5 of his text. There is, however, one major difference: Snapper and Troyer use arbitrary bilinear forms over general fields, while Tarrida restricts attention to positive definite inner products over the real numbers. Thus, Tarrida deals only with Euclidean spaces, whereas Snapper and Troyer’s discussion involve spaces that are very much non-Euclidean. Tarrida’s book thus provides a more direct and focused, albeit less general, introduction to the topics covered. The basic idea here is that an inner product on the underlying vector space of an affine space A allows one to define the norm of a vector in that space, and so, given two points P and Q in A, one may define the distance between them as simply the norm of the (unique) vector in V that acts on P to give Q. With the concept of a distance defined on A, it is then a routine matter to define a Euclidean motion as a distance-preserving map: Tarrida does this in chapter 6 and gives examples. Chapter 7 then provides a classification theory of Euclidean motions in dimensions 1, 2 and 3, both algebraically and geometrically. It is proved, for example, that any non-identity motion of the Euclidean plane is a translation, rotation, reflection (or “symmetry”, as they are called here), or glide reflection.

One could, I suppose, argue that a lot of this could be accomplished with much less algebraic machinery. Books like Martin’s Transformation Geometry, for example, classify isometries in the plane and space (and also discuss wallpaper and frieze groups, which Tarrida does not) using purely geometric methods that extend high school geometry and require less mathematical background to understand. However, this kind of reasoning is inherently less rigorous than the algebraic approach adopted by Tarrida, and also fails to give the student an appreciation of the interplay between linear algebra and geometry. And, of course, affine geometry is useful in other contexts as well, as are various non-Euclidean geometries which can also be treated by the methods of linear algebra. (The preface to Snapper and Troyer’s book gives a very nice justification for using linear algebra to study both metric and non-metric geometries.)

The remaining two chapters of Tarrida’s book talk about real quadrics (sets of points in affine space which, in affine coordinates relative to a fixed frame, are the zeros of a quadratic polynomial) and their classification, both with respect to affinities (chapter 8) and the more restrictive set of Euclidean motions (chapter 9). Because of the close connection between multi-variable quadratic polynomials and matrices, the discussion here largely involves matrix computations and is a good introduction to the theory of real quadratic forms, developed in considerable detail.

Tarrida clearly intends for this book to be used primarily as a text, rather than as a reference. It was obviously written with the needs of students in mind. The formalism of affine geometry may take a little while to become accustomed to, but the exposition here is clear and detailed, with enough diagrams and geometric motivation provided to help the student understand the intuitive meaning of results. There is also a good supply of worked examples and end-of-chapter exercises. (No solutions to any of the exercises appear in the book, which is rare for books in the Springer Undergraduate Mathematics Series; in fact, Springer’s webpage description for this series touts solutions to some or all of the exercises as a feature of the series. Personally, I think the absence of solutions is, from the standpoint of an instructor looking to assign homework problems, a good thing.)

There is also a reasonably good bibliography, but for some reason the only reference to Artin’s Geometric Algebra was to a French version, even though an English version exists (but is incredibly expensive). The bibliography also lists a number of texts on projective geometry, and Tarrida explicitly notes that projective geometry provides an alternate way of looking at some of the material developed in the text. I thought it was a shame that there was no chapter introducing projective geometry from the linear-algebraic perspective, but I suppose all authors must draw the line somewhere and it is certainly not unreasonable to draw it here.

If I have one complaint about the book at all (other than the three blank pages I found scattered throughout the text, which I hope is not a widespread problem but, rather, an isolated glitch with my particular copy), it is with the index, which could be beefed up; a person reading a book in which Euclidean motions are one of the main topics should not have to hunt for the term “rotation” in the index (I did eventually discover it, but only under “Euclidean motions” in the E’s).

Mark Hunacek ( teaches mathematics at Iowa State University.