This work presents a mathematical framework for approximating smooth axisymmetric reflective surfaces using piecewise circular arcs with approximate C¹ continuity. The construction enforces position and slope matching at segment boundaries, enabling direct implementation via standard CNC circular interpolation (G02/G03 commands). Key theoretical results include: - Existence and uniqueness theorems for arc-based approximations with geometric constraints - Explicit curvature-mismatch formulation governing approximation error - Rigorous bounds on surface and slope errors with proven convergence rates - Validation via numerical simulation for representative concentrator geometries The framework applies to parabolic, elliptical, and general conic-section profiles, with potential relevance to solar concentrators, optical manufacturing, and precision reflective systems. Manufacturing-specific implementations and process parameters are withheld pending intellectual property protection. Full mathematical proofs, error analysis, and simulation results are documented in the accompanying material.
Federal University of Petroleum Resource Effurun (Fri,) studied this question.