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Abstract:
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This project treats a classic problem in combinatorial geometry: the study of obtuse angles (angles larger than pi/2 radians) in a set of points on the plane. The main question is: What is the minimum number of obtuse angles that are determined by any set of n points? This problem was studied by Conway, Croft, Erdös and Guy in their work "On the distribution of values of angles determined by coplanar points" in the year 1979, where they could determine bounds on the minimum number of obtuse angles among all point clouds with n points. This thesis presents a computational approach to the problem. In the first place, these proofs of Conway et al. are discussed. Furthermore, a program to verify the good quality of these bounds has been implemented. Also the expected number of obtuse angles in a random point set, which was determined by E. Langford in 1970, is confirmed by means of experiments. The central part of the project studies the effect of geometrical transformations on the number of obtuse angles in a given point cloud. The considered functions include: the Möbius transformation, the exponential function, and the shear transformation, among others. We present experimental results on the question: Which of the considered transformations tend to decrease the number of obtuse angles, if applied to a random point set? Also, we study the degree of reduction of obtuse angles, which measures by how much the number of obtuse angles can be decreased if a specific transformation is applied to a point cloud. Finally, we combine some of the considered transformations in order to generate point clouds with few obtuse angles. The different experiments in this project have been carried out with the program MATLAB. |
