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
Description:
CD, no of pages 101 , 31643, ماجستير رياضيات 2/2024
