Discrete Coordinate System with Uniformity and Isotropy for Honeycomb Structure

Abstract

This paper presents an architecture of a discrete coordinate system that features uniformity and isotropy aiming to describe regular hexagon tilings. The hexagon tilings, or honeycomb structures, are widely applied for formulizations of robotic system control: navigation mapping, multi-robot motion planning, and area segmentation of sensing images, for example. With implementing an ordinary Cartesian coordinate system or an oblique coordinate system over the honeycomb structure, extra treatments are unavoidable for the computation scheme, such as re-scaling in certain directions or case-division with respect to belonging quadrants. The proposing coordinate system, named “H'Cl? (HoneyComb by Projection),” can be constructed with a simple mapping of a three dimensional cube tessellation on a projecting plane. The framework of H Cl? shows the preferable characteristics of uniformity and isotropy over the whole coordinate surface. The theoretical background of H'Cl? and its construction process are presented in this paper with considering handedness and other possible variations. The consistent computation environment developed over H‘Cl’ is also described, including a distance function and a conversion method with the Cartesian system.