A Vertex Concavity-Convexity Detection Method for Three-Dimensional Spatial Objects Based on Geometric Algebra

Pengcheng Yin,Chengyan Wu,Xiying Sun,Jiyi Zhang,Zhifeng Shi,Di Hu

doi:10.3390/ijgi9010025

Abstract

Vertex concavity-convexity detection for spatial objects is a basic algorithm of computer graphics, as well as the foundation for the implementation of other graphics algorithms. In recent years, the importance of the vertex concavity-convexity detection algorithm for three-dimensional (3D) spatial objects has been increasingly highlighted, with the development of 3D modeling, artificial intelligence, and other graphics technologies. Nonetheless, the currently available vertex concavity-convexity detection algorithms mostly use two-dimensional (2D) polygons, with limited research on vertex concavity-convexity detection algorithms for 3D polyhedrons. This study investigates the correlation between the outer product and the topology of the spatial object based on the unique characteristic that the outer product operation in the geometric algebra has unified and definitive geometric implications in space, and with varied dimensionality. Moreover, a multi-dimensional unified vertex concavity-convexity detection algorithm framework for spatial objects is proposed, and this framework is capable of detecting vertex concavity-convexity for both 2D simple polygons and 3D simple polyhedrons.

Highlights

Algorithms for the concavity-convexity detection of vertices of spatial objects are the basis of many graphics algorithms, including test algorithms [1,2,3,4,5], orientation and convexity-concavity determination algorithm for simple polygons [6,7], and computer graphics processing [8,9]
The angle, left-right-point, and vector area methods use the inherent properties of simple polygons for algorithm design, while the cross product, ray, slope, and extremity vertices sequence methods tremendously reduce the algorithmic complexity during the identification of the concavity-convexity of fixed points of simple polygons
This paper introduces geometric algebra into the vertex concavity-convexity detection of spatial objects based on the computation principle of the cross product method for determining the vertex concavity-convexity of a simple polygon

Summary

Introduction

Algorithms for the concavity-convexity detection of vertices of spatial objects are the basis of many graphics algorithms, including test algorithms [1,2,3,4,5], orientation and convexity-concavity determination algorithm for simple polygons [6,7], and computer graphics processing [8,9]. The commonly used vertex concavity-convexity detection algorithms for polygons include the convex hulls [10], angle [11], left-right-point [12], vector area [13], cross product [14], slope [15], and extremity vertices sequence methods [16]. This paper introduces geometric algebra into the vertex concavity-convexity detection of spatial objects based on the computation principle of the cross product method for determining the vertex concavity-convexity of a simple polygon. Taking advantage of the fact that the outer product computation in geometric algebra possesses unified and clear geometric implications among spaces with different dimensionalities, the limitation that the cross product in the Euclidean space can only be applied between vectors is eliminated, and a detection method for the vertex concavity-convexity of simple polygons and polyhedrons that is based on the outer product computation is proposed. Conclusions and discussion are drawn, on the basis of the previous sections and some issues for future study

Basic Idea

Outer Product Operation

CGA Expression for Geometric Objects Based on Outer Product

Case Studies