A Static Moment for a Polygon and Its Applications

Abstract

The application of geometry of roofs (straight skeletons) and, related to that, Voronoi diagrams for polygons to solve optimization tasks such as determination of the routes of surveys of regions with minimal static moments with respect to the sides of polygons is discussed. A special interpretation of the notion of static moment with respect to the segment of line and a polygon with respect to the border is explained and an appropriate theorem is formulated and proved. Examples of potential employment of these notions have been indicated.

Volume Optimization of Solid Waste Landfill Using Voronoi Diagram Geometry

“…As a theoretical basis, the research results of Koźniewski (Koźniewski, 2004a(Koźniewski, , 2004b(Koźniewski, , 2016(Koźniewski, , 2018 regarding roof geometry and Koźniewski et al (Koźniewski et al, 2013) concerning, among others, applications of Voronoi diagrams for embankment modeling were adopted. The source setting of roof geometry lies in the area of teaching classical descriptive geometry (Grochowski, 2018) and roof geometry (Koźniewski, 2016).…”

“…The roof geometry and Voronoi diagrams found applications in many elds ranging from photogrammetry with (semi) automatic reconstruction of city models (Bernd et al;, Dikaiakou et al, 2003Flamanc et al, 2003;Laycock and Day, 2003;Swift et al, 2003), reconstruction of roads from satellite images (Akel, 2005;Thom, 2005), in cartography (Haunert and Sester, 2004;Wang et al, 2011), in the analysis of the morphology of materials with grain structure (Akel, 2005), in medicine for the representation, reconstruction and visualization of human organs ( Barequet et al, 2003), in the topographic analysis of the area, computer graphics in the eld of object shape recognition and analysis (Lawlor, 2004), in transport of readymixed concrete (Koźniewski et al, 2017), and in designing of earthwork organization (Koźniewski, 2018).. The solution presented in this paper is a new example of applying the above-mentioned concepts in the practical eld of designing the shape of embankments.…”