Geometric Method for Determining Centrality in Circles Based on Orthogonal Chords and Oriented Semi-Chords

Abstract

This article describes a geometric method for determining the center of circular geometries and performing positional centralization based on two orthogonal chords drawn through an arbitrary interior measurement point. The method decomposes these chords into scalar and oriented semi-chords, from which the displacement components between the measurement point and the geometric center are obtained. These components are combined into a resultant correction vector R⃗, which directly determines the direction, sense, and magnitude of the displacement required for centralization. The method is structured in two sequential and complementary stages. In the first stage, the vector formulation is used to determine and apply the correction vector R⃗. The semi-chords are then measured again, and the correction process may be repeated until R⃗ is sufficiently close to the zero vector within the adopted positional tolerances. In the ideal circular model, the first correction vector provides the exact displacement to the center. In the second stage, after positional centralization has been achieved, scalar metric relationships involving the perimeter C, the orthogonal chord lengths Dₓ and Dᵧ, and the constant π are used to evaluate the consistency of the measured geometry with the ideal circular model. A scalar circular-consistency index is defined as I = 2C / (Dₓ + Dᵧ), with the ideal circular relationship characterized by I = π. Equivalent metric relationships provide additional means of comparing the measured quantities with the theoretical circular condition. The vector formulation implicitly establishes a local coordinate system anchored at the measurement point, making the centralization procedure independent of prior knowledge of the center coordinates in a global Cartesian reference frame. The resulting framework integrates local measurement, vector position correction, and subsequent scalar metric verification, with potential applications in dimensional metrology, CNC machine alignment, coordinate measuring systems, robotics, computer vision, geometric control, and automatic positioning.