Closest Pair Algorithms in 2D space,a commentary on complexity and reductions

One of the most challenging problems in computational geometry is closest pair of points given n points. Brute force algorithms[1] and Divide and conquer[1] have been verified and the lowest complexity of attributed to latter class of algorithms, with worst case being for the former being . We propose a method of partitioning the set of n-points based on the least area rectangle that can circumscribe these points

Exploring the Patterns of Resident Resettlement in Rural and Suburban Areas and Their Influence on the Passenger Trip Generation

The region is a complex multi-element and heterogeneous dynamic system. Regional public transport systems are characterized by heterogeneity and incompleteness of information about the processes that take place in them. The factors determining the demand for public passenger transport services, the structure of suburban population transit and the economic consequences of changing this structure have not yet been sufficiently investigated. There is no quantitative assessment of the control factors’ influence on the results of the passenger transport system functioning in suburban traffic, which complicates the effective management in this field.In order to determine the mutual influence of objects with unclear functional connection, one of the methods of computational geometry is considered, which allows quantifying and obtaining tabulated functions of various components and parameters of settlements' interaction and centers of gravity. The numerical characteristics obtained by the described method are further used in the modeling of the processes of efficient functioning and collaboration of urban passenger transport with other external transport types, in particular at interchanges.As a result of the proposed operations, we obtain a tabulated function of the investigated parameter for the area of spatial interaction of infrastructure objects and visualize the calculated parameters in the form of graphical dependencies or in the form of a three-dimensional surface. Thus, using the method of computational geometry, with constructing a Voronoi diagram and executing the Delaunay triangulation, it is possible to obtain tabulated characteristic parameters for inhomogeneous zones of mutual influence where infrastructure objects with unclear functional connection are located.

Review of A Short Course in Computational Geometry and Topology by Herbert Edelsbrunner

Computational geometry and topology are huge branches of mathematics. Focussing on concepts that lead to computation is one strategy to provide a concrete conceptual basis for ideas that hold in a more general context. Indeed, this short book gives an introduction to a surprisingly broad range of ideas that can serve as a good introduction to geometry and topology (even broadly conceived) for undergraduates.

High-throughput Calculation of Atomic Planar Density for Compounds

A large collection of element-wise planar densities for compounds obtained from the Materials Project is calculated using brute force computational geometry methods. We demonstrate that the element-wise max lattice plane densities can be useful as machine learning features. The methods described here are implemented in an open-source Mathematica package hosted at https://github.com/sgbaird/LatticePlane.

Prevention of hazards induced by a Radiation Fireball through Computational Geometry and Parametric Design

Radiation Fireballs are singular phenomena which involve severe thermal radiation and consequently, they need to be duly assessed and prevented. Although the radiative heat transfer produced by a sphere is relatively well known, the shadowing measures implemented to control the fireball’s devastating effects have frequently posed a difficult analytical instance, mainly due to its specific configuration. In this article, since the usual solving equations for the said cases are impractical, the authors propose a novel graphic-algorithm method that sorts the problem efficiently for different kinds of obstructions and relative positions of the fireball and the defenses. Adequate application of this method may improve the safety of a significant number of facilities exposed to such risks.

Computational Geometry of Period-3 Hyperbolic Components in the Mandelbrot Set

A parametric theoretical boundary equation of a period-3 hyperbolic component in the Mandelbrot set is established from a perspective of Euclidean plane geometry. We not only calculate the interior area, perimeter and curvature of the boundary line but also derive some relevant geometrical properties. The budding point of the period-3k component, which is born on the boundary of the period-3 component, and its relevant period-3k points are theoretically obtained by means of Cardano’s formula for the cubic equation. In addition, computational results are presented in tables and figures to support the theoretical background of this paper.