A Linear Programming Method for Finding a Minimal Set of Axial Lines Representing an Entire Geometry of Building and Urban Layout

Sung Kwon Jung,Youngchul Kim

doi:10.3390/app10124273

Abstract

This paper devises an algorithm for finding the minimal set of axial lines that can represent a geometry of building and urban layout in two dimensions. Although axial lines are useful to analyze spatial configuration in the Space Syntax, existing methods for selecting axial lines seldom address the optimality of their solutions. The proposed algorithm uses linear programming to obtain a minimal set of axial lines. To minimize the number of axial lines that represent the entire geometry of building and urban layout, a linear programming problem is established in which a set of axial lines represents the entire 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 paper states a new formulation by adding a group of constraints to the original formulation. By examining the modified linear programming problem in various two-dimensional building maps and spatial layouts, this paper demonstrates that the proposed algorithm can guarantee a minimum set of axial lines to represent a two-dimensional geometry. This modified linear programming problem prevents isolated axial lines in the process of axial line reduction.

Highlights

An axial line is a geometrical element that represents a convex space, i.e., a spatial unit comprising the entire geometry of building and/or urban layout
An axial line is defined as the longest line that can be drawn in a divided convex space that links to other axial lines that cover all of the given space without repetition [1]
This paper develops an algorithm for finding a minimal set of axial lines that can represent a geometry of building and urban layout in two dimensions

Summary

Introduction

An axial line is a geometrical element that represents a convex space, i.e., a spatial unit comprising the entire geometry of building and/or urban layout. All-line axial maps are drawn from a set of vertex-to-vertex lines within a given space that do not intersect the wall lines or boundaries of that space [2,3]. The rules for these maps are intuitive and allow the maps to be generated by computer programs. By following these mapping rules, this paper develops a method of axial line reductions for finding the minimal set of axial lines. Batty and Rana [4]

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Methods

Conclusion