Calculation for Multidimensional Topological Relations in 3D Cadastre Based on Geometric Algebra

Shi Shi,Zhang Zhang,Wang Wang,Yin Yin,Hu Hu,Chen Chen

doi:10.3390/ijgi8110469

Abstract

Maintaining topological consistency is a crucial issue for 3D cadastral modeling as this helps to represent the cadastral boundary clearly and accurately. As a result, 3D cadastral data models are mainly built on the basis of topological models that allow topology to be expressed clearly. However, topological models in Euclidean space cannot directly represent objects’ geometric information. As geometric information is important in 3D spatial analysis, 3D cadastral data models based on topological models cannot realize topological calculation and analysis. Previous research has proved that geometric and topological information for cadastral objects can be integrated and represented by conformal geometric algebra (CGA) expressions. This paper aims to realize 3D topological analysis in the cadastral field using CGA’s advantages in geometric relations computation. A calculation framework is designed on the basis of the outer product to achieve the purpose of multidimensional unity for 3D cadastral topological analysis in this paper. A calculation framework of topological relations between a boundary point (or a boundary line) and a cadastral parcel is developed. A total of 13 types of topological relations between a boundary point and a cadastral parcel and 48 types of topological relations between a boundary line and a cadastral parcel are obtained. The study indicates that the advantages of CGA in multidimensional unified representation and calculation can be used to solve problems encountered by topological models in Euclidean space.

Highlights

Three dimensional topological analysis plays an important role in verifying the topological consistency of 3D cadastral data, updating cadastral parcels, and ensuring the accuracy of cadastral data [1,2,3,4,5,6,7,8,9,10]
Cadastral objects need to be represented in the form of topological models and solid models in these hybrids 3D models, respectively, which increases the complexity of 3D cadastral models
We mainly focus on the problem that topological models cannot realize spatial analysis in Euclidean space

Summary

Introduction

Three dimensional topological analysis plays an important role in verifying the topological consistency of 3D cadastral data, updating cadastral parcels, and ensuring the accuracy of cadastral data [1,2,3,4,5,6,7,8,9,10]. Topological models in Euclidean space cannot realize 3D spatial analysis because they do not directly represent objects’ geometric information [26]. Geometric information for cadastral elements (boundary lines and boundary faces) cannot be directly expressed by their geometric components with lower dimensions in Euclidean space. This leads to difficulties for topological models to realize topological analysis and calculation in 3D space. To solve these problems, Shi and Wang developed different kinds of hybrid 3D cadastral data models based on both topological models and solid models [27,28]. Cadastral objects need to be represented in the form of topological models and solid models in these hybrids 3D models, respectively, which increases the complexity of 3D cadastral models

Objectives

Methods

Discussion

Conclusion