Duality and Dimensionality Reduction Discrete Line Generation Algorithm for a Triangular Grid

Lingyu Du,Qiuhe Ma,Jiahao Li,Rui Wang,Jin Ben

doi:10.3390/ijgi7100391

Lingyu Du, Qiuhe Ma + Show 3 more

Open Access

https://doi.org/10.3390/ijgi7100391

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Abstract

Vectors are a key type of geospatial data, and their discretization, which involves solving the problem of generating a discrete line, is particularly important. In this study, we propose a method for constructing a discrete line mathematical model for a triangular grid based on a “weak duality” hexagonal grid, to overcome the drawbacks of existing discrete line generation algorithms for a triangular grid. First, a weak duality relationship between triangular and hexagonal grids is explored. Second, an equivalent triangular grid model is established based on the hexagonal grid, using this weak duality relationship. Third, the two-dimensional discrete line model is solved by transforming it into a one-dimensional optimal wandering path model. Finally, we design and implement the dimensionality reduction generation algorithm for a discrete line in a triangular grid. The results of our comparative experiment indicate that the proposed algorithm has a computation speed that is approximately 10 times that of similar existing algorithms; in addition, it has better fitting effectiveness. Our proposed algorithm has broad applications, and it can be used for real-time grid transformation of vector data, discrete global grid system (DGGS), and other similar applications.

Highlights

Vector and raster are two fundamental spatial data models, each having advantages for various applications
The discretization of geometric elements forms the basis of vector data discretization
Representative achievements include Sphere Quad Tree [11] and Quaternary Triangular Mesh (QTM) [12], which are mainly used in spatial data indexing [13] and map generalization models [14]

Summary

Introduction

Vector and raster are two fundamental spatial data models, each having advantages for various applications. Representative achievements include Sphere Quad Tree [11] and Quaternary Triangular Mesh (QTM) [12], which are mainly used in spatial data indexing [13] and map generalization models [14] Both hexagonal and diamond grids can be formed by triangular grid polymerization. Zhang proposed the full-path algorithm, in which, according to the geometric relationship between the vector line and grid cell, all grid cells crossed by the vector line are selected one by one [2]; the discrete line generated by this algorithm is ideal, but the process is complex, and computation speeds are slow. We solve the model using the dimensionality reduction method to theoretically simplify the problem

Definition of Duality and Weak Duality

Application of Our Algorithm in DGGS