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Geometry of Complex Numbers: Circle Geometry, Moebius Transformation, Non-Euclidean Geometry

Hans Schwerdtfeger
Dover Publications
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The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition.

[Reviewed by
Mark Hunacek
, on

One of the nice things about geometry is that it can be approached in several different ways. You can do it synthetically, from axioms, either quite rigorously (Axiomatic Geometry by Lee) or not-so-rigorously, allowing appeal to pictures (Geometry for College Students, by Isaacs). Or you can take a more analytic approach, letting linear algebra do the work for you, as in Kaplansky’s Linear Algebra and Geometry: A Second Course. You can also, in the spirit of Klein’s Erlangen Programme, study geometry from the point of view of its transformations; a recent book along these lines is Transformational Geometry by Umble and Han.

Or, you can do what the book under review does, and study it from the perspective of complex numbers. After all, a point in the Euclidean plane can be represented by an ordered pair of real numbers, which on the one hand is just a vector, but which on the other hand can be identified with a complex number. All that additional multiplicative structure that the field of complex numbers enjoys can then be put to good use to study geometry, an approach which results in all sorts of interesting things. The idea is fruitful enough, in fact, that not only does the book under review exploit it, but so do other books (see, e.g., Hahn’s Complex Numbers and Geometry, or Complex Numbers from A to … Z by Andrica and Andreescu) and journal articles (e.g., Shastri’s Complex Numbers and Plane Geometry in the January 2008 issue of the journal Resonance). It turns out that many interesting theorems of plane Euclidean geometry can be proved using the complex numbers, including both fairly well-known results (e.g., the existence and properties of the nine-point circle) and also results that seem to be somewhat more obscure (e.g., if one starts with an arbitrary convex quadrilateral and draws an external square on each side, then the two line segments connecting the centers of the opposite squares are equal in length and perpendicular to one another).

The book under review also exploits the properties of the complex numbers to obtain facts about geometry, but is not really comparable to the references cited in the previous paragraph. It eschews the basic theorems of basic plane geometry in favor of more sophisticated ideas, including non-Euclidean geometry. It is written at a somewhat more demanding level than these other references, and the prerequisites for reading it are not trivial: a student should certainly have had a previous course in linear algebra and be comfortable dealing with matrices, should know what a group is, and be familiar as well with the basic facts of real analysis in the plane.

The book is divided into three large chapters, each chapter further divided into sections, which are themselves further divided into subsections. Chapter I is entitled “Analytic Geometry of Circles”, and includes fairly detailed discussions of things like the inversion function, stereographic projection and the cross-ratio of complex numbers.

Chapter 2 is on Moebius transformations, also known as linear fractional transformations; these are the transformations of the extended complex plane of the form \(z \mapsto (ax+b)/(cz+d)\), where \(a, b, c\) and \(d\) are complex numbers and \(ad - bc \neq 0\). These mappings have strong geometric content (it can be shown, for example, that the set of all circles and lines is invariant under any such transformation) and most books on complex analysis spend at least some time discussing them; few, however, address them in the kind of detail that Schwerdtfeger does. (One that does is Needham’s Visual Complex Analysis, correctly described in this column as a “special book” that is “genuinely unique”.) Schwerdtfeger addresses a number of topics connected to Moebius transformations that are not found in the average text, including iterated transformations and applications to projective geometry (specifically, projectivities and perspectivities); projective geometry is referred to briefly but not developed in depth.

The next chapter discusses non-Euclidean geometry (specifically, the Poincaré disc model of hyperbolic geometry, and elliptic and spherical geometry) in connection with various subgroups of the group of all Moebius transformations. This approach, of course, is very much in the spirit of Klein’s Erlangen Programme, which is mentioned in passing.

The original text was published in 1962. For this Dover edition, the author added a supplemental bibliography and some appendices bringing things up to date, at least as of 1979. The bibliography is pretty extensive but still, of course, quite dated, and a number of entries are to papers that are not in English. There is a short (one and a half pages long) and not altogether satisfactory Index; for example, as noted above, Klein’s Erlangen Programme is mentioned in the text, but none of these three words, or anything close to them, appears in the Index.

Given the date of publication, it is not surprising that the book has a decidedly old-fashioned “vibe” to it, and I suspect that students of the current generation will not find it to be an easy read. For one thing, the exposition is terse. In addition, the use of upper case German letters to denote matrices is cumbersome, and sometimes leads to unfortunate equations like (see page 48) \(\mathfrak{Z’S’ESZ}=0\), which looks almost like hieroglyphics.

While many topics treated in the book are definitely valuable things for an undergraduate to know, there are also some topics presented here (e.g., pencils of circles) that struck me as somewhat peripheral to a modern undergraduate mathematics education. (Of course, I suppose it can be said that quite a lot of geometry is now peripheral to a modern undergraduate mathematics education, but even at universities that offer one or even two semesters of undergraduate geometry, one doesn’t hear the phrase “pencils of circles” a lot.) For all these reasons, I doubt this book will find much use as a primary textbook for an undergraduate course; however, there is a lot of interesting mathematics in this book that is not readily found elsewhere, so as a reference for instructors, or perhaps as supplemental reading in a course on complex analysis, it might prove quite useful.

Mark Hunacek ( teaches mathematics at Iowa State University. 


  § 1. Representation of Circles by Hermitian Matrices
    a. One circle
    b. Two circles
    c. Pencils of circles
  § 2. The Inversion
    a. Definition
    b. Simple properties of the inversion
  § 3. Stereographic Projection
    a. Definition
    b. Simple properties of the stereographic projection
    c. Stereographic projection and polarity
  § 4. Pencils and Bundles of Circles
    a. Pencils of circles
    b. Bundles of circles
  § 5. The Cross Ratio
    a. The simple ratio
    b. The double ratio or cross ratio
    c. The cross ratio in circle geometry
  § 6. Definition: Elementary Properties
    a. Definition and notation
    b. The group of all Moebius transformations
    c. Simple types of Moebius transformations
    d. Mapping properties of the Moebius transformations
    e. Transformation of a circle
    f. Involutions
  § 7. Real One-dimensional Projectivities
    a. Perpectivities
    b. Projectivities
    c. Line-circle perspectivity
  § 8. Similarity and Classification of Moebius Transformations
    a. Introduction of a new variable
    b. Normal forms of Moebius transformations
    c. "Hyperbolic, elliptic, loxodromic transformations"
    d. The subgroup of the real Moebius transformations
    e. The characteristic parallelogram
  § 9. Classification of Anti-homographies
    a. Anti-homographies
    b. Anti-involutions
    c. Normal forms of non-involutory anti-homographies
    d. Normal forms of circle matrices and anti-involutions
    e. Moebius transformations and anti-homographies as products of inversions
    f. The groups of a pencil
  § 10. Iteration of a Moebius Transformation
    a. General remarks on iteration
    b. Iteration of a Moebius transformation
    c. Periodic sequences of Moebius transformations
    d. Moebius transformations with periodic iteration
    e. Continuous iteration
    f. Continuous iteration of a Moebius transformation
  § 11. Geometrical Characterization of the Moebius Transformation
    a. The fundamental theorem
    b. Complex projective transformations
    c. Representation in space
  § 12. Subgroups of Moebius Transformations
    a. The group U of the unit circle
    b. The group R of rotational Moebius transformations
    c. Normal forms of bundles of circles
    d. The bundle groups
    e. Transitivity of the bundle groups
  § 13. The Geometry of a Transformation Group
    a. Euclidean geometry
    b. G-geometry
    c. Distance function
    d. G-circles
  § 14. Hyperbolic Geometry
    a. Hyperbolic straight lines and distance
    b. The triangle inequality
    c. Hyperbolic circles and cycles
    d. Hyperbolic trigonometry
    e. Applications
  § 15. Spherical and Elliptic Geometry
    a. Spherical straight lines and distance
    b. Additivity and triangle inequality
    c. Spherical circles
    d. Elliptic geometry
    e. Spherical trigonometry
  1. Uniqueness of the cross ratio
  2. A theorem of H. Haruki
  3. Applications of the characteristic parallelogram
  4. Complex Numbers in Geometry by I. M. Yaglom