Jürgen Richter-Gebert

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Cayley-Klein Geometries

We started out developing projective geometry for two reasons: It was algebraically nice and it helped us to get rid of the treatment of many special situations that are omnipresent in Euclidean geometry. Then, to express Euclidean geometry in a projective setup, we needed the help of complex numbers, our special points I and J, cross-ratios, and Laguerre’s formula. We now come to another pivot point in our explanations: We will see that our treatment of Euclidean geometry in a projective framework is only a special case of a variety of other reasonable geometries. One might ask what it means to be a geometry in that context. For us it means that there are notions of points, lines, incidence, distances, and angles with a certain reasonable interplay. Besides Euclidean geometry, among those geometries there are quite a few prominent examples, such as hyperbolic geometry, elliptic geometry, and relativistic space-time geometry.

Configurations, Theorems, and Bracket Expressions

This section is devoted to several specific examples of theorems and configurations in projective geometry. Clearly, our considerations in Chapter 5 demonstrated that there is an infinite variety of incidence theorems in real projective geometry.

More on Bracket Algebra

The last chapter demonstrated that determinants (and in particular multihomogeneous bracket polynomials) are of fundamental importance in expressing projectively invariant properties. In this chapter we will alter our point of view. What if our “first-class citizens” were not the points of a projective plane but the values of determinants generated by them? We will see that with the use of Grassmann-Plucker relations we will be able to recover a projective configuration from its values of determinants.

Conics and Their Duals

So far, we have dealt almost exclusively with situations in which only points and lines were involved. Geometry would be quite a pure topic if these were the only objects to be treated. Large parts of classical elementary geometry deal with constructions involving circles. The most elementary drawing tools treated by Euclid (the straightedge and the compass) contain a tool for generating circles. In a sense, so far we have dealt with the straightedge alone. Unfortunately, circles are not a concept of projective geometry. This can easily be seen by observing that the shape of a circle is not invariant under projective transformations. If you look at a sheet of paper on which a circle is drawn from a skew angle, you will see an ellipse. In fact, projective transformations of circles include ellipses, hyperbolas, and parabolas. They are subsumed under the term conic sections, or conics, for short. Conics are the concept of projective geometry that comes closest to the concept of circles in Euclidean geometry. It is the purpose of this section to give a purely projective treatment of conics. Later on, we will see how certain specializations provide interesting insights into the geometry of circles.

Complex Numbers: A Primer

So far, almost all our considerations have dealt with real projective geometry. The main reason for this is that we wanted to stay with all our considerations as concrete and close to imagination as possible. Nevertheless, almost all considerations we have made so far carry over in a straightforward way to other underlying fields.

The Complex Projective Line

Now we will study the simplest case of a complex projective space: the complex projective line. We will see that even this case has already very rich geometric interpretations. The close relation of complex arithmetic operations to geometry allows us to express geometric properties by nice algebraic structures. In particular, this case will be the first example of a projective space in which we will be able to properly deal with circles.

Calculating with Points on Lines

In the previous section we have seen that cross-ratios form a universal link from projective geometry to the underlying coordinate field. In this chapter we want to elaborate more on this topic. We will see that one can recover the structure of the real numbers from the purely geometric setup of the real projective plane.

Projective d -space

Different topic! So far, we have dealt almost exclusively with projective geometry of the line and of the plane. We explored the tight and very often elegant relationships between geometric objects and their algebraic representations.

Lines and Cross-Ratios

At this stage of this monograph we enter a significant didactic problem. There are three concepts that are intimately related and that unfold their full power only if they play together. These concepts are performing calculations with geometric objects, determinants and determinant algebra, and geometric incidence theorems. The reader should understand that in a beginner’s text that makes few assumptions about prior knowledge, these concepts must be introduced sequentially. Therefore we will sacrifice some mathematical beauty for clarity of exposition. Still, we highly recommend that the following chapters be read (at least) twice, so that the reader may obtain an impression of the interplay of the different concepts.