Now let us come to the real stuff about diagrams. In this chapter we will investigate more advanced applications of tensors and diagrams. In particular, we will link tensor diagrams more closely to geometric theorems and concepts we encountered in previous chapters. 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.
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. 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. In this chapter we will merge two different worlds: CP 1 and RP 2. Both can be considered as representing a real two-dimensional plane. They have different algebraic structures, and they both represent different compactifications of the Euclidean plane: For RP 2 we added a line at infinity.
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For CP 1 we added a point at infinity. Both spaces have different weaknesses and strengths. In the first two parts of the book we learned that RP 2 is very well suited for dealing, for instance, with incidences of lines and points, with conics in their general form, and with cross-ratios. We did not have a proper way to talk about circles, angles, and distances in RP 2.
The previous two chapters introduced CP 1. This space was very good for dealing with cocircularity and also for dealing with angles. However, lines were poorly supported by CP 1. They had to be considered circles with infinite radius, and they were not even projectively invariant objects.
In the previous chapter we laid the foundations for representing Euclidean concepts transformations, angles, distances, orthogonality, cocircularity… in a projective framework. In this chapter we will apply these concepts.
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. 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. While in the last chapter we have focused on measurement aspects and related analytic issues, we will return to more qualitative relations of objects in a Cayley-Klein geometry. In this and the next chapter we will mainly study the projective nature of Cayley-Klein geometries. This chapter focuses on aspects concerning transformations , their projective invariants , and the behavior of measurements under these transformations.
The next chapter treats geometric objects and their elementary geometric properties , including some elementary geometric theorems. Although Cayley-Klein geometries unify several rather different types of geometric playgrounds in a common projective framework, each type of geometry in the sense of Section This is due to the fact that the degeneracies of the fundamental conic of the geometry lead to degeneracies in the relevant geometric constructions.
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For this reason we will sometimes have to perform a special case analysis for the different types of Cayley-Klein geometries. Sometimes one general definition covers a certain effect for instance reflection in all types of geometries.
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Still it may be instructive to consider its particular specialization to certain geometries. Based on the measurement in a Cayley-Klein geometry, we can now define specific geometric objects and relations. For instance a circle may be defined as the set of all points that have a constant distance to a given point.
Being orthogonal may be defined as a certain angle relation between two lines. In each type of a Cayley-Klein geometry the objects and relations will have very specific properties. In this chapter we will deal with aspects of elementary geometry in the context of Cayley-Klein geometries.
Following the spirit of this book, we will again focus on algebraic and geometric representations of geometric primitive operations, on incidence theorems, and on invariance properties. Again we try to present the definitions and statements in a way that they apply as generally as possible to degenerate Cayley Klein geometries.
Still some statements may break down if the geometric configurations or the underlying geometry becomes too degenerate. Since we do not want to spend most of the exposition mainly with pathological degenerate cases, we will base our definitions whenever possible on constructive approaches that allow us to explicitly calculate the objects involved. Let us come to another interesting class of geometric objects: circles.
Speaking about circles in general Cayley-Klein geometries first raises a conceptual issue.
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Euclidean circles have several geometric properties on which one could base a more general definition. Imagine you are a two-dimensional being living in the interior of a real nondegenerate fundamental conic of a Cayley-Klein geometry. All your measurements distances and angles are done with respect to this Cayley-Klein geometry, and you have no knowledge of the fact that your world is embedded in some larger space the projective plane in which the Cayley-Klein geometry is defined. One day your dog, your ruler, and you decide to take a long, long walk always following the same direction.
How would that feel? In a sense it would not feel very exciting, and this is indeed an exciting thing. The three of you simply go on without anything remarkable happening. A person from the outside observing you will see you all getting smaller and smaller as you approach the boundary of the fundamental object. With your legs shrinking, your step size observed from the outside is getting smaller and smaller, too.
Back to mathematics! This chapter is dedicated to several interesting topics in hyperbolic geometry. With our previous knowledge on the real projective plane RP 2 , on the complex projective line CP 1 , and of Cayley-Klein geometries we have an ideal departure point to explain several hyperbolic effects from an elegant and advanced standpoint. Compared to the general considerations in Chapters 20—23 we are now in a somewhat better situation.
When we dealt with general Cayley-Klein geometries we spent a lot of our efforts on the treatment of case distinctions that arose from the various degrees of degeneracy of the fundamental conic. The algebraic structure became easier the less degenerate the fundamental conic was. Now we will deal only with one particular Cayley-Klein geometry, which in addition is nondegenerate. After having mastered the basics of hyperbolic geometry, having learned how to calculate distances and angles, perform transformations, represent it in two different models, a whole world of interesting topics opens, by far more than can be covered here in this book.
We now want to focus on a few of these topics that are of particular beauty and show specific relationships to our projective approaches to hyperbolic geometry. This book now comes to an end, and it became considerably fatter than originally intended. Still there are many interesting, amazing, deep, esthetic topics that we did not touch at all.
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Exceptional configurations. Hence the product of the two reflections is not a rotation but a translation. A bit poetically, one can say, a euclidean translation is a rotation around a point at infinity — by a vanishing angle! The path to geometric algebra. The observant reader will not have missed noting that the correct solution of the exceptional configurations described above proceeded a bit magically.
In particular, finding the correct solution for the first two configurations involved the ideal points and line of the euclidean plane. These are concepts which are not necessarily associated to euclidean geometry. This Cayley-Klein construction can be directly applied when the quadratic form is non-degenerate, and produces most importantly elliptic or spherical and hyperbolic space of any dimension. It also works for euclidean space, but requires a degenerate quadratic form some subspaces consist of points with vanishing norm.
For this post, the relevant object is the projective model of the euclidean plane. Then the points of vanishing norm form a line, the so-called ideal line of the euclidean plane. Parallel lines meet in points of this line. This circumstance allowed us to handle the first two exceptional configurations above, where we sought the intersection of two parallel lines. For this reason, the projective model of euclidean geometry has clear advantages over the traditional approach.
In order to carry this out in a rigorous fashion, its useful to translate things into in the correct algebraic setting. The traditional way of proving that the construction outlined above is correct relies on converting the steps into expressions in vector analysis of the plane. This approach avoids mention of ideal points and line, and is limited to the euclidean setting. Is there a better algebraic tool for the job? In fact, there is a much more comprehensive algebraic structure — which includes vector analysis as a small sub-algebra — such that every step of the above construction can be expressed by a single compact expression.
Furthermore, even though the steps of the construction are made based on the euclidean metric, the resulting expressions also provide a correct representation of the same construction when the underlying metric is elliptic or hyperbolic. The differences that arise express naturally the differences between these metrics.