We begin with a brief review of topology on Rn, to set the stage, before embarking on the actual tour of the hyperbolic plane.

## Hyperbolic geometry of the olfactory space.

We will examine several realizations of H and explore the connections between them. We will classify the isometries of H, which are the maps from H to H preserving both angle and length, and close by considering area and trigonometry in H. Having successfully completed this module, you will be able to demonstrate knowledge and understanding of:. The instructor will give two lectures per week following a structured week-by week programme, based mainly on Anderson's book.

There will be two timetabled meeting per week, in which the lecturer will outline the topic for that week and be available to assist the students. The students are also expected to meet once a week by themselves, for a self-study tutorial session. They will each take responsibility for leading the session, based on guidance from the lecturer, supplemented as necessary.

Figure 3: A crochet from Daina Taimina [3]. You can clearly see that the parallel postulate does not hold here because there are three lines that go through a point and none of them intersect with the given line on the bottom.

Definition of Distance Although it may be a bit surprising at first, the definition of metric, or measure of distance, is different for each type of geometry. In Euclidean geometry, the distance is given as. Note that in euclidean geometry, the shortest distance between two points is a line as postulated by Euclid. However, in hyperbolic space, the shortest distance between two points is the arc from point to point that lie on the unique semicircle centered at a point on the boundary as shown in Figure 5; if we parameterize the arc as ,.

See [2]. Figure 5: Hyperbolic geodesic pictured as a euclidean semi-circle centered at a point on x.

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Furthermore, projective geometry is independent from the theory of parallels; if not, the consistency of projective geometry would be questioned. Projective geometry has simple origins from Renaissance artists who portrayed the rim of a cup as an ellipse in their paintings to show perspective. Projective geometry can be thought of in this way:. Light rays coming from each point of the scene are imagined to enter his eye, and the totality of these lines is called a projection. From there, the idea eventually entered the academic mathematical community where it was incorporated into the study of linear algebra and other areas of mathematics.

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Projective geometry is the study of invariants on projections — properties of figures which are not modified in the process of projection [11]. Projective geometry is more more general than both euclidean and hyperbolic geometries and this is what Klein uses to show that noneuclidean geometries are consistent. Image found here. Klein gives a general method of constructing length and angles in projective geometry, which he believed to be the fundamental concept of geometry. The bottom-up methods are easier to visualize and to deal with applications of hyperbolic geometry. Moreover, projective geometry is consistent outside of geometry [10].

Distance in general The different geometries we get from projective geometry come from the the projection of the fundamental conic. This idea is illustrated below in Figure 7. An isometry is a way of transforming a figure in a given space without changing its angles or lengths.

## Hyperbolic Geometry | SpringerLink

Every isometry in hyperbolic space can be written as a linear fractional transformation in the following form:. You can think of a complex coordinate in the same way that you think about on the euclidean plane where. So these isometries take triangles to triangles, circles to circles and squares to squares. Recall, our visualizations of hyperbolic space using the upper-half plane model from Figure 4 A , then the fundamental conic is the real line and the fuchsian groups are the isometries acting on.

By using projective geometry to construct noneuclidean geometries, Klein has created a proof that is indisputable. Henri Poincare A discussion of the history of hyperbolic geometry cannot leave out Henri Poincare after which the Poincare disk model is named See Figure 4 B.

## MA448 Hyperbolic Geometry

Originally, he began his studies in differential equations, then stumbled upon hyperbolic geometry naturally in his work. He wrote in his book Science and Methods that:.

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