Acute Triangulations Of The Surfaces Of Convex Solids

By a triangulation of the surface of a convex solid, we mean a set of(full) triangles covering the surface, in such a way that the intersection of any two triangles is either empty or consists of a vertex or of a full edge. A triangle is called geodesic if all its edges are segments, i.e., shortest paths between the corresponding vertices. We are interested only in geodesic triangulations, all the members of which are, by definition, geodesic triangles. An acute(resp. non-obtuse) triangulation of the surface of a convex solid is a geodesic triangulation such that the angles of all geodesic triangles are smaller(resp.not greater) than??2. The minimal number of triangles in all acute(resp. non-obtuse)triangulations of the surface of a convex solid is called the size of acute(resp. non-obtuse)triangulations of the surface.In this thesis we investigate non-obtuse and acute triangulations of the surfaces of four types of Archimedean solids and the surfaces of four types of solids of revolution.The problem about acute triangulations of the surfaces of convex solids belongs to frontier research area in Discrete and Combinatorial Geometry, and the results to be obtained will not only be of great significance to promote the development of the theory of acute triangulations of three-dimensional space, but also have positive impacts on the basic theory and technological development of computer science.In Chapter 1, we discuss non-obtuse and acute triangulations of the surfaces of four kinds of Archimedean solids, and get the sizes of non-obtuse and acute triangulations.The sizes of non-obtuse and acute triangulations of the truncated tetrahedral surface are 10 and 12 respectively; the sizes of non-obtuse and acute triangulations of the cuboctahedral surface are 8 and 12 respectively; the sizes of non-obtuse and acute triangulations of the truncated cuboctahedral surface are 8 and 12 respectively. Furthermore, the surface of truncated icosahedron admits at least 10 and at most 14 non-obtuse triangles, and at least 12 and at most 14 acute triangles.In Chapter 2, we consider non-obtuse and acute triangulations of the surfaces of solids of revolution. Firstly, we discuss non-obtuse and acute triangulations of the surfaces of any bounded cylinder, circular cone and truncated circular cone, and get the following results.1. All the sizes of non-obtuse triangulations of the surfaces of any cylinder, circularcone and truncated circular cone are 8. Furthermore, they can not be triangulated into less than 20 acute triangles.2. Let ?? be the circumference of the bottom of the cylindrical surface, ?? be the height of the cylinder. If ?? : ?? ∈(0,2·cot7??36+3·tan7??3618], then the cylindrical surface can be triangulated into 20 acute triangles; if ?? : ?? ∈(2·cot7??36+3·tan7??3618, ∞), then the cylindrical surface can be triangulated into 32 acute triangles.3. The sizes of acute triangulations of the surfaces of any circular cone and truncated circular cone are both 20.By rotating a regular ??-gon about its symmetry axis, we get a solid of revolution,which we can call a regular ??-gonal revolution. There are two types of symmetry axises,let ????be the surface of a regular ??-gonal revolution whose rotational axis is a diagonal of the ??-gon; otherwise, denoted by ????.Also in Chapter 2, we discuss non-obtuse and acute triangulations of the surface of any regular ??-gonal revolution. The main results are following.1. The size of non-obtuse triangulations of the surface of any regular ??-gonal revolution is 8.2. The size of acute triangulations of ??4is 12; the size of acute triangulations of????(?? ≥ 6) is 16.3. The size of acute triangulations of ????is 20.