Finding a Minimum Set of Axial Lines Representing an Entire Building's Geometry

An axial line is a geometrical line element that represents a convex space and its links to other convex spaces in a built environment. Axial lines are connected to other axial lines to show the spatial relationships of connectivity between convex spaces of building interiors. Because this connectivity describes the geometries of buildings, an axial map that consists of axial lines has been used to analyze a spatial configuration of linear two-dimensional shapes. For finding the axial line maps that represent particular buildings, scholars and researchers have used all-line maps due to its intuitive and convenience to generate by computer programs. However, the existing greedy and subset algorithms in computer programs for finding axial line maps cannot be guaranteed to identify the real minimum of axial lines from an all-line axial map. Due to the computational difficulty of the set cover problem, they have used approximation algorithms, which hardly guarantee to find a minimum set of axial lines in buildings. Thus, this project aims to overcome this difficulty and fully represent the geometry of built spaces. If the minimum set of axial lines can be found algorithmically, then an axial line map that has the fewest and longest lines can be achieved. This method can be guaranteed to find the minimum set of axial lines in a building. This project will devise an algorithm for finding the minimal set of axial lines that can represent a building’s geometry in two dimensions. The algorithm will use linear programming to obtain a minimal set of axial lines. To minimize the number of axial lines that represent the entire building’s geometry, a linear programming problem will be established in which a set of axial lines represents the entire building’s geometry. The axial lines must have at least one intersection with every extension line of the wall edges to the sides of the reflex angles. If a solution to this linear programming problem exists, it will be guaranteed to be an optimum. However, some solutions of this general linear programming problem may include isolated lines, which are undesirable for an axial line analysis. To avoid isolated axial lines, this project will state a new formulation by adding a constraint to the original formulation. By examining the modified linear programming problem in various two-dimensional building maps and spatial layouts, this project will demonstrate that the proposed algorithm can guarantee a minimum set of axial lines to represent a two-dimensional geometry. This modified linear programming problem will prevent isolated axial lines in the process of axial line reduction. This project will deliver a novel method for finding a minimum set of axial lines representing an entire building’s geometry. This method will help scholars and professionals to effectively analyze spatial configurations in the built environment and will help to promote interdisciplinary strategies for integrating spatial characteristics into spatial configurations.