论文标题
对界面和无界平面上圆圈2D填充的数学分析
Mathematical analysis of 2D packing of circles on bounded and unbounded planes
论文作者
论文摘要
本文涵盖了分析和广义公式和复发关系的数学推导，以找出在平面区域中刻有或包装在圆形区域（包括圆形弧（包括扇形，半圆圈和四分之一圆圈））和直线的圆圈的半径。使用分析公式和复发关系获得的半径值通过与使用MATLAB代码获得的半径相比进行了验证。与启发式策略和优化技术不同，本文用于包装循环的方法是确定性的。用于切线圆圈平面堆积的分析公式可以被推广，并用于在3D空间中堆积球形，并在球形表面上的圆圈包装，这与非点电荷的分布相似。相同的圆圈的堆积密度，从外部切线，最密集的六边形和无限平面上的堆积密度已得到制定和分析。这项研究为数学上解决了2D容器中圆圈密集堆积的问题，在空隙（四面体和八面体）中的球形堆积并在晶体平面上找到平面密度。
This paper encompasses the mathematical derivations of the analytic and generalized formula and recurrence relations to find out the radii of n umber of circles inscribed or packed in the plane region bounded by circular arcs (including sectors, semi and quarter circles) & the straight lines. The values of radii obtained using analytic formula and recurrence relations have been verified by comparing with those obtained using MATLAB codes. The methods used in this paper for packing circles are deterministic unlike heuristic strategies and optimization techniques. The analytic formulae derived for plane packing of tangent circles can be generalized and used for packing of spheres in 3D space and packing of circles on the spherical surface which is analogous to distribution of non-point charges. The packing density of identical circles, externally tangent to each other, the most densely packed on the regular hexagonal and the infinite planes have been formulated and analysed. This study paves the way for mathematically solving the problems of dense packing of circles in 2D containers, the packing of spheres in the voids (tetrahedral and octahedral) and finding the planar density on crystallographic plane.
