You are here

Analytic Projective Geometry

Eduardo Casas-Alvero
European Mathematical Society
Publication Date: 
Number of Pages: 
EMS Textbooks in Mathematics
[Reviewed by
Mark Hunacek
, on

Projective geometry, like Euclidean geometry, can be developed both from a synthetic (axiomatic) and analytic point of view. In the two-dimensional case of projective planes, for example, three simple and pleasingly symmetric axioms suffice: one that guarantees the existence of four distinct points, no three of them collinear; one that establishes that two distinct points lie on a unique line; and one that states that two distinct lines intersect in a single point (thereby establishing that there are no parallel lines). Analytically speaking, one can define a projective plane using homogenous coordinates with entries from a field \(F\) (or, more generally, a division ring \(D\)): the “points” in the geometry are ordered triples of elements of \(D\), not all of them \(0\), with the added requirement two such triples are considered equal if one of them is a non-zero scalar multiple of the other. Lines can also be defined to be similar ordered triples, although to avoid confusion it is common to use different notation, say \([x,y,z]\) for points and \(\langle a,b,c\rangle \) for lines. Incidence of a point with a line means, in this notation, that \(ax + by + cz = 0\), a condition that is clearly well-defined because the left-hand side is \(0\_ if and only if any non-zero scalar multiple of it is.

If a coordinate-free analytic approach is required, then one can define a projective plane to be a three-dimensional vector space over \(F\) (or module over \(D\)), with the points being the one-dimensional subspaces and lines being the two-dimensional ones, with incidence being simple set containment. Relative to a fixed basis, it is then fairly easy to see how to create coordinates for the points and lines (for the latter, use the analogue of the vector cross-product formula to find a vector that is the “orthogonal complement” of the subspace).

These ideas generalize without too much difficulty to the case of projective spaces, rather than planes. In the analytic approach, a projective space of dimension n is now a vector space of dimension \(n + 1\); we then get not only points and lines but higher-dimensional analogues of planes. (At least, this is the way I learned the definition; as we’ll shortly see, the definition in this book is somewhat different.)

Both the analytic and synthetic approaches have their advantages. The analytic approach, for example, allows for applications of projective geometry, both to other areas of mathematics such as algebraic geometry and elliptic curve theory (if you’re going to define curves by polynomial equations, having things to plug into the polynomials is an obvious advantage), and to areas outside of mathematics such as the theory of computer graphics.

The book under review is an introduction to the purely analytic theory of projective spaces over the fields of real and complex numbers. It is a thick book, weighing in at a hefty 600+ pages, and, as befits a book written by a professor at the Universitat de Barcelona in Spain and published by the European Mathematical Society, has a decidedly European feel to it. The details that need to be included are included, but the book is written for serious and well-prepared European undergraduates and does not engage in a great deal of hand-holding. Definitions are stated in fairly general and abstract terms.

Thus, for example, the author’s definition of a projective space over a field \(k\) is somewhat more abstract than the one I provided earlier in this review; he defines it as a triple \((P, E,\pi)\), where \(P\) is a set, \(E\) is a finite-dimensional vector space over \(K\) of dimension at least 2, and \(\pi\) is a mapping from \(E – \{0\}\) onto \(P\) satisfying \(\pi(a)=\pi(b)\) if and only if \(a\) is a nonzero scalar multiple of \(b\). It is interesting to note that the first indented entry that follows this definition is a commutative diagram rather than a drawing of anything with points and lines in it; in fact, after a drawing is given on page 25 to illustrate the theorem of Pappus, we then go for almost 40 pages until we see another one. As I said, this is a serious book.

It is clear even from the previous discussion that linear algebra appears front and center in an analytic development of projective spaces. Indeed, even as far back as 1952, it was said in the preface to Baer’s classic Linear Algebra and Projective Geometry that the disciplines of linear algebra and projective geometry are “identical”. Linear algebraic methods are used throughout this book, and an essential prerequisite for a reader is completion of what the author refers to as a “standard course” in linear algebra. By this he means more than the kind of introductory course that most American undergraduates take: the course he refers to should have covered such topics as diagonalization, duality and bilinear forms. There are also occasions where some familiarity with the rudiments of affine geometry from the linear-algebraic standpoint is assumed. The linear-algebraic definition of an affine space (in terms of a vector space acting on a set) is quickly reviewed without proofs in chapter 3, but some prior exposure to these ideas would be helpful in following the presentation.

The book begins with the aforementioned definition of projective spaces and a discussion of the basic facts about them, including the theorems of Pappus and Desargues. (As will be noted a little later in this review, while these are indeed theorems in the context of real and complex projective spaces, they are not provable in the general synthetic theory of projective planes.) The next four chapters address fairly standard issues (the introduction of coordinates, the relationship between projective and affine spaces, the principle of duality, and projective transformations) but, again, the discussion is at a fairly high and theoretical level.

Chapter 6 introduces quadrics, the study of which will occupy the rest of the book (thereby making this text very valuable reading for a student who is planning to proceed to study algebraic geometry). While quadric surfaces may be viewed concretely as sets of points whose coordinates satisfy a second degree polynomial equation, the author here gives a more abstract, coordinate-free presentation in terms of symmetric bilinear forms. The classification of quadrics in both projective and affine spaces is discussed, and then the author proceeds to more advanced topics that are, to my knowledge, not readily found in other books on projective geometry. For example, the idea that the set of quadrics on a projective space can itself be viewed as the set of points of a projective space is discussed and exploited.

The book ends with a pair of appendices: the first, about ten pages long, gives a mathematical discussion of perspective; the second one, about three times as long, uses projective geometry to give models of Euclidean, hyperbolic and elliptic geometry.

In keeping with its intended use as a text, each chapter ends with a collection of exercises. Solutions are not provided, and there is no instructor’s manual.

Back in the days when I was an undergraduate, my college offered no less than five upper-level courses in geometry. Besides projective geometry, there were courses in advanced Euclidean geometry, foundations of geometry, geometric transformations, and “higher analytic geometry”, which basically was a course in the use of vector methods in geometry. These days seem to be gone forever. Now, despite the importance and usefulness of the subject, most American universities do not offer undergraduate courses devoted entirely to projective geometry; the average mathematics major is exposed to this subject, if at all, only briefly and in connection with other material.

Thus, it seems unlikely that I will ever be in the position of selecting a textbook for a course on projective geometry. If I were to have this opportunity, however, I would be inclined to select one (like Casse’s Projective Geometry: An Introduction) that was not devoted exclusively to the analytic aspects of the theory, if only because there is an interesting and beautiful connection between the analytic and synthetic theory, arising from the question of when an axiomatically-defined projective plane can be represented in homogenous coordinates over a field. The answer turns out to be that this is the case if and only if Pappus’ theorem holds in the plane. Likewise, an axiomatically-defined plane can be represented by homogenous coordinates over a division ring if and only if Desargues’ theorem holds in the plane. These connections between geometry and algebra are results that I think should be discussed in any course on projective geometry, but are necessarily omitted here because of the author’s focus only on the analytic theory (and his decision to consider only real and complex spaces). This is, of course, a matter of personal taste, but I mention it only to point out that the subject matter of this text necessarily results in the exclusion of a substantial and beautiful portion of the theory.

In compensation for this, however, this book contains (as the previous description of topics should make clear) a great deal of material, far more than could be reasonably be expected to be covered in a single semester undergraduate course. And, as I indicated above, there are some topics covered here that are not easy to find in the existing textbook literature. Thus, although not a book that I would likely use as a text, this is nonetheless an excellent reference source for people interested in the subject.

Mark Hunacek ( teaches mathematics at Iowa State University. 

  • Projective spaces and linear varieties
  • Projective coordinates and cross ratio
  • Affine geometry
  • Duality
  • Projective transformations
  • Quadric hypersurfaces
  • Classification and properties of quadrics
  • Further properties of quadrics
  • Projective spaces of quadrics
  • Metric geometry of quadrics
  • Three projective classifications
  • Appendix A. Perspective (for artists)
  • Appendix B. Models of non-Euclidean geometries
  • Bibliography
  • Symbols
  • Index