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

Abstract

Detecting the concavity and convexity of three-dimensional (3D) geometric objects is a well-established challenge in the realm of computer graphics. Serving as the cornerstone for various related graphics algorithms and operations, researchers have put forth numerous algorithms for discerning the concavity and convexity of such objects. The majority of existing methods primarily rely on Euclidean geometry, determining concavity and convexity by calculating the vertices of these objects. However, within the realm of Euclidean geometric space, there exists a lack of uniformity in the expression and calculation rules for geometric objects of differing dimensions. Consequently, distinct concavity and convexity detection algorithms must be tailored for geometric objects with varying dimensions. This approach inevitably results in heightened complexity and instability within the algorithmic structure. To address these aforementioned issues, this paper introduces geometric algebra theory into the domain of concavity and convexity detection within 3D spatial objects. With the algorithms devised in this study, it becomes feasible to detect concavity and convexity for geometric objects of varying dimensions, all based on a uniform set of criteria. In comparison to concavity-convexity detection algorithms grounded in Euclidean geometry, this research effectively streamlines the algorithmic structure.