In Volume 2, Chapter 12 of Howard Eves' "A Survey of Geometry" he introduces the basic notions of the Geometry of Complex Numbers. Everyone gets a small dose of this in high school but not a lot. Oftentimes the next time one runs into complex numbers is a Complex Analysis course in an undergraduate or graduate course. After reading Eves' treatment in his "Survey" I wish I had had this deeper dig into this topic to better appreciate the analytic course. That is, a deeper understanding of the algebraic and geometric properties of the Inversive plane gives one a better appreciation for the properties of analytic functions.
Like many topics in mathematics, the modern approach in school is often to emphasize breadth at the cost of depth in introducing new concepts. The result is that one can achieve advanced placement standing in Calculus with a very elementary depth of knowledge about plane Geometry. Often concepts are presented without an appreciation for the historical development of that concept. This can make it hard for any but the most advanced to fully grasp the subtleties of the things they learn. Unless students are presented with and then work through a wide assortment of problems they will never run against the "special cases" that make them fully appreciate the results they are learning.
In Eves' Chapter 12, section 6, he looks at the cross ratio of four points in the inversive plane. The cross ratio is shown to have a useful connection to homographies which basically account for all circular transformations in the plane (i.e. transformations that map circles into circles). In fact, the Moebius theorem says that a circular transformation must be a homography or an antigraphy. A homography is a transformation of the inversive plane onto itself defined as z' = (az+b)/(cz+d) where (ad-bc) is not zero. An antigraphy is just the product of a homography and a reflection in the real line. (Replace z with its conjugate.)In his Theorem 12.6.9, Eves establishes the result that the homography determined by the three distinct pairs of finite corresponding points Z1,Z1'; Z2,Z2'; and Z3,Z3' can be expressed in terms of the cross ratio (z'z1',z2'z3') = (zz1,z2z3) where (ab,cd) = (a-c)(b-d)/(a-d)(b-c).
The point of this entry is much simpler than than the (motivating) discussion above. It turns out that in the cross ratio discussion of section 6, Eves draws on the idea of oriented angles and segment measures in a way that is very useful but possibly a little confusing if one is not tuned into the details of the argument. To fine tune the reader's attention I have included a short paper that lays out in detail the definitions of oriented angles and segment measures to make the arguments on cross ratio representations most clear. For anyone interested the paper is Oriented Angles and Segment Lengths.
