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Preliminary Mathematics
Published in P.K. Jayasree, K Balan, V Rani, Practical Civil Engineering, 2021
P.K. Jayasree, K Balan, V Rani
A polygon is a plane figure which is surrounded by a fixed number of straight line segments closing in a loop. There are two types of polygons—regular and irregular polygons. A regular polygon has equal sides and equal angles. A polygon with unequal sides and unequal angles is called an irregular polygon.
Areas of common shapes
Published in John Bird, Science and Mathematics for Engineering, 2019
A polygon is a closed plane figure bounded by straight lines. A polygon which has: (a) 3 sides is called a triangle – see Figure 12.1(a)(b) 4 sides is called a quadrilateral – see Figure 12.1(b)(c) 5 sides is called a pentagon – see Figure 12.1(c)(d) 6 sides is called a hexagon – see Figure 12.1(d)(e) 7 sides is called a heptagon – see Figure 12.1(e)(f) 8 sides is called an octagon – see Figure 12.1(f)
Geometry
Published in Dan Zwillinger, CRC Standard Mathematical Tables and Formulas, 2018
A polygon is regular if all its sides are equal and all its angles are equal. Either condition implies the other in the case of a triangle, but not in general. (A rhombus has equal sides but not necessarily equal angles, and a rectangle has equal angles but not necessarily equal sides.)
Morphological analysis of nanoparticle agglomerates generated using DEM simulation
Published in Particulate Science and Technology, 2022
Another method to obtain the volume of agglomerate is convex hull method. It is defined as the smallest convex polygon enclosing any shape. The convex hull is the smallest polyhedral formed by joining the coordinates of the center of the particles around the boundary as displayed in Figure 3(c). “Divide and conquer based Quickhull algorithm” as implemented in MATLAB® is used to get the convex hull (Barber, Dobkin, and Huhdanpaa 1996). Volume of the convex hull thus obtained is used as the volume of agglomerate in Equation (6). The porosity of the agglomerate under study is found to be using this method. It can be noted that porosity obtained from radius of gyration is high as compared to that of convex hull method. This is because radius of gyration method gives large volume of agglomerate which leads to high porosity in comparison to convex hull method. Similar trend of difference in porosity with these methods are noted in Dadkhah, Peglow, and Tsotsas (2012).
Describing the geometric difference of architectural forms in three primary shapes of circle, triangle and square
Published in Journal of Asian Architecture and Building Engineering, 2022
One of the reasons that shape analysis is delicate, despite the introduction of powerful statistical modelling techniques, is that not all variations between shapes are necessarily significant. That is to say, minor perturbations caused by noise have little effect on shape discrimination. The first operation we introduced to ignore the small shape details is the convex hull (CH). The convex hull (CH) of a polygon (P) is a conventional problem in computational geometry and is defined as the smallest convex polygon enclosing the entire polygon. Many studies handle the computing matter of CH; a representative algorithm can be referred to the study by Chen and Rokne(1992). Probable losses of crucial features along the concave part covered by CH, such as dents, indentations and cavities caused by dominant protrusions of steeple spire, can be handled by multiplying convexity as the ratio of the area of P to CH.
Sparsest packing of two-dimensional objects
Published in International Journal of Production Research, 2021
Tatiana Romanova, Alexander Pankratov, Igor Litvinchev, Sergiy Plankovskyy, Yevgen Tsegelnyk, Olga Shypul
The adjusted quasi phi-function for two non-convex polygons. Let two non-convex polygons and be given. Each convex polygon is defined by its vertices and each convex polygon is defined by its vertices . The adjusted phi-function for two non-convex polygons and can be defined as where , is the adjusted phi-function for two convex polygons and of the form (8), .