A Multi-dimensional Unified Concavity and Convexity Detection Method Based on Geometric Algebra

The detection of concavity and convexity of vertices and edges of three-dimensional (3D) geometric objects is a classic problem in the field of computer graphics. As the foundation of other related graphics algorithms and operations, scholars have proposed many algorithms for determining the concavity and convexity of vertices and edges. However, existing concavity and convexity detection algorithms mainly focus on vertices and not on concavity and convexity detection methods for edges of 3D geometric objects. Furthermore, existing algorithms often require different detection methods when dealing with two-dimensional (2D) planar geometric objects and 3D spatial geometric objects. This means that the algorithm structure of those algorithms becomes very complex when dealing with concavity and convexity judgments involving both planar polygon vertices and 3D geometric object edges. To solve the above problems, this paper proposes a multi-dimensional unified concave convex detection algorithm framework for geometric objects taking advantage of geometric algebra in multi-dimensional unified expression and calculation. The method proposed in this article can not only achieve concavity and convexity detection of planar polygon vertices and 3D geometric object vertices based on unified rules, but also further achieve concavity and convexity detection of 3D geometric object edges on this basis. By unifying the framework and detection rules of different dimensional geometric object concavity detection algorithms, the complexity of synchronous detection algorithms for planar polygon vertices and 3D geometric object vertices and edges concavity can be effectively simplified.