Centroid-Deviation Field Theory: A High-Precision Computational Framework for 3D Geometry Based on Midpoint Calculus形心偏差场论:基于中道微积分的三维几何高精度计算框架

Abstract This paper proposes the Centroid-Deviation Field Theory (CDFT), a novel computational framework for three-dimensional geometry based on the principles of Midpoint Calculus. Traditional methods for calculating geometric properties of irregular objects, such as volume and moment of inertia, face fundamental challenges, including voxelization resolution limits, catastrophic cancellation errors, and difficulties in separating concave and convex structures. This study reveals that the centroid represents the zero-order deviation equilibrium state of a geometry, and the deviation field defined with the centroid as the reference origin completely encodes all geometric information of the object. By decomposing a polyhedron into tetrahedra sharing the centroid as a common vertex, and utilizing the Deviation Conservation Law to ensure the precise cancellation of contributions from shared faces of adjacent elements, this framework enables the simultaneous and parallel computation of volume, centroid, moment of inertia, and principal axes of inertia. Numerical validation demonstrates that the Centroid-Deviation Field method maintains volume accuracy equivalent to the convex hull method while automatically separating concave and convex structures. Furthermore, the computational complexity is reduced from O(N³) to O(N), and memory usage is decreased by two orders of magnitude. This framework transforms “irregular geometric computation” from a “numerical integration problem” into an “algebraic moment problem,” providing new theoretical tools for fields such as core analysis, meteorite measurement, and 3D printing quality inspection. Keywords: Centroid-Deviation Field Theory; Midpoint Calculus; Non-associative Algebra; Geometric Moments; Parallel Computing; Concave-Convex Analysis