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 Course: Seminar Topic: Seminar Description: The object of this seminar is to explore Poincare's model of hyperbolic geometry as a perfectly consistent alternative to the usual Euclidean geometry we learn in school (all those theorems about congruent triangles ABC!). If you think about it, one is free to define a "geometry" as any set of consistent axioms about "points", "straight lines" and so on. The important point is that the axioms must be consistent, i.e. one must never be able to prove a statement and its negation from this set of axioms. Also, let us try to select a set of axioms that are not too unrealistic, in the sense that the axioms correspond to the physical behaviour of objects in the real world. For example we wish in some sense that "light travels in a straight line" in our geometry. Imagine a world composed of the interior of a circle C such that the velocity of light at any point inside the circle is equal to the distance of that point from the circumference. Now, it can be proved that rays of light will take the shape of circular arcs with each end perpendicular to the circumference of C. In such a geometry, the properties of "straight lines" defined as light rays will be different from properties of straight lines in usual Euclidean geometry. For example, the axiom of Euclidean geometry that "there is exactly one line parallel to a given line and passing through a point not on the line" will not be true because there will be infinitely many "straight lines" through any point that do not intersect a given "straight line". This is Poincare's model of hyperbolic geometry. We present a simple program using C++ and OpenGL that shows the "straight lines" of this geometry. It was interesting writing the program, since the axioms of the geometry had to be essentially written in C++!