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

Abstract

This article presents a geometric method for positional centralization and multi-angular circular-consistency verification based on orthogonal chords and oriented semi-chords. From an arbitrary interior point P, four semi-chords are measured along two mutually orthogonal local directions. Differences between opposite semi-chords define the signed components Δₓ and Δᵧ of a resultant correction vector R⃗, which determines the magnitude, direction, and sense of the displacement required for centralization. Under the ideal circular model, the first resultant vector gives the exact displacement to the geometric center, while practical implementations may use iterative acquisitions within adopted positional tolerances. After centralization, the four semi-chords must be mutually consistent and the corresponding orthogonal chord lengths Dₓ and Dᵧ must agree within the adopted dimensional tolerance. The accepted 0° acquisition establishes a fixed dimensional reference. The measurement directions are then reoriented to 15.6°, 17.3°, and 39.4° while the geometry and centralized point remain physically fixed. At each orientation, the complete acquisition is repeated to verify preservation of positional symmetry, four-semi-chord consistency, orthogonal dimensional equality, and dimensional invariance relative to the 0° reference. A scalar circular-consistency index Iθ is subsequently evaluated for each additional orientation using the fixed reference established at 0°. Comparison of Iθ with π, through the deviation |Iθ − π|, provides the final quantitative filter within a specified scalar tolerance. The resulting framework combines vector centralization, local geometric consistency, multi-angular dimensional verification, and scalar evaluation without requiring prior knowledge of the center coordinates in a global Cartesian reference frame. Potential applications include dimensional metrology, CNC positioning and alignment, coordinate measuring systems, robotics, geometric control, and automated positioning.