A High-Order, Optimization-Free, Tangent Continuous Approximation of Conic Sections Using Cubic Bezier Curves

Abstract

This paper presents a high-order, optimization-free method for approximating conic sections using cubic Bézier curves. By matching endpoints and tangents while analytically determining free parameters via midpoint interpolation, the method achieves unprecedented accuracy and efficiency. For elliptic arcs, it delivers tenth-order convergence with a maximum absolute error of just $1.2 \times 10^{-3}$. Parabolic arcs are reconstructed exactly with machine-level accuracy ($3.55 \times 10^{-15}$ error). The approach maintains computational efficiency, processing all cases in under 1.5 seconds without requiring optimization or rational forms. Its combination of mathematical simplicity, superior accuracy, and rapid execution makes it ideal for CAD applications where both precision and performance are critical. The robustness of the proposed method under geometric transformations and seamless scalability to 3D surfaces further demonstrate its practical value for industrial applications.