By Athanase Papadopoulos
This e-book includes rigorously revised and elevated models of 8 classes that have been offered on the collage of Strasbourg, in the course of geometry grasp sessions, in 2008 and 2009. the purpose of the grasp periods was once to provide to fifth-year scholars and PhD scholars in arithmetic the chance to benefit new themes that lead on to the present study in geometry and topology. The classes have been held via best specialists. the topics handled comprise hyperbolic geometry, three-manifold topology, illustration thought of primary teams of surfaces and of three-manifolds, dynamics at the hyperbolic airplane with functions to quantity concept, Riemann surfaces, Teichmüller conception, Lie teams and asymptotic geometry.
The textual content is addressed to scholars and mathematicians who desire to examine the topic. it will probably even be used as a reference ebook and as a textbook for brief classes on geometry.
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Furthermore, from the fact that the angle sum of each triangle used in this tiling is equal to , it follows that the edges of the triangles in the tiling form a network of straight lines (see Figure 21 (b)). By taking unions of adjacent triangles, 47 Notes on non-Euclidean geometry ˇ A ˇ ˛ (a) ˛ (b) Figure 21. At the vertex A, we have ˛ C ˇ C D . we can find for each integer n > 1 a triangle Tn which is homothetic to T by the homothety factor n. This shows that (1) H) (2). Now we prove that (2) H) (3).
4 (Euclid, Book I, Proposition 17). In any triangle in the neutral plane, the sum of any two angles is Ä . Proof. Let ABC be an arbitrary triangle and let us prove that By C Cy Ä . Let I be the midpoint of the edge BC . We take a point A0 on the line AI such that I is the midpoint of AA0 (Figure 11). The two triangles IBA and ICA0 are congruent, therefore ABC Á A0 CB. The two angles ACB and BCA0 are adjacent and they are on the same side of the line AC , therefore their sum is Ä . Thus, ABC C ACB Ä .
A closed half-plane is the union of an open half-plane with the line that bounds it. It is easy to see that a polygon is convex if and only if it is the intersection of (a finite number of) closed half-planes. The interior angle at any vertex of a convex polygon is < . Conversely, if the interior angle at each vertex of a polygon P is < , then P is convex. An n-gon is a polygon with n vertices. 14 (Angular deficit of a polygon). n 2/ . Â1 C Â2 C Â3 /. From this definition it follows immediately that if we add to a polygon P some extra (fake) vertices which are at the interior of edges, then the angular deficit of P , computed using these extra vertices, is unchanged, since each of the extra angles is equal to .