We propose a novel algorithm for determining whether a given point lies within a convex polyhedron, achieving a sub-linear computational expected complexity of \(O_{\text {exp}}(N^{1/2})\) , where N represents the number of triangles in the polyhedron’s triangular mesh. In contrast to traditional methods with linear complexity O(N), our approach significantly reduces computational overhead, making it especially effective for large polyhedral models. The algorithm is formulated entirely in projective space, utilizing homogeneous coordinates for the tested points and triangle vertices. By leveraging vector–vector operations optimized for SSE, AVX instructions, and GPU architectures, our method is robust and straightforward, tailored to handle even highly complex convex polyhedra. The efficiency of the approach was validated through theoretical analysis and estimated speed-up calculations, demonstrating its potential to accelerate applications in computer graphics, computational geometry, collision detection, and related fields. Additionally, the simplicity of the proposed algorithm ensures a high potential for broad applicability and supports further advancements in this area.
