Error Controlled Cubic Spline Interpolation in CNC Technology

Traditional CNC technology mostly uses the method of increasing the degree of interpolation polynomial when constructing $C^2$ continuous NURBS curves, but this often leads to the appearance of Runge phenomenon in interpolation curves. Alternatively,the method of adding boundary conditions at the endpoints can often make it difficult to control the error range of the interpolation curve. This article presents a $C^2$ continuous cubic B-spline curve interpolation method,which achieves $C^2$ continuity of the interpolation curve when the interpolation polynomial is cubic. At the same time, this article also studies the corresponding error control methods.