The Geometry of Planar Polygons in the Euclidean Space

Abstract

This thesis delves closely into the interesting field of polygons in 3R 3 - dimensional space. We set out on an expedition to determine the basic components of three-dimensional geometry, such as lines, planes, and their distances. Equipped with these tools, we examine the subtleties of polygons, delving into their particular types, computing their areas by the application of the Shoelace Formula, and examining the generation issue of orthonormal bases for planes. A key component of this research is the mapping of planar polygons from R 3 to R 2 and vice versa, which acts as a bridge between the 2D and 3D realms. Next, we explore the interesting idea of polygon-polygon overlapping, providing a foundation for classifying various scenarios of overlap. The thesis tackles point inclusion methods in closed planar polygons, extending beyond simple visualization. We carefully assess three different approaches: directed ray, global, and ray tracing, which provide powerful tools for locating a point in a polygon. Finally, we round up our investigation with the intriguing topic of 2D and 3D planar polygon smoothing. We present area-conserving smoothing approaches utilizing edge and single node relaxation techniques to achieve results that are both mathematically sound and visually stunning. This thesis provides a comprehensive investigation of polygons in R 3 , providing experts and individuals to obtain an improved understanding of their characteristics, relationships, and interaction in three dimensions