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.

翻译：传统的数控技术在构造$C^2$连续的NURBS曲线时，大多采用提高插值多项式次数的方法，但这往往会导致插值曲线出现龙格现象。或者，在端点处添加边界条件的方法通常难以控制插值曲线的误差范围。本文提出了一种$C^2$连续的三次B样条曲线插值方法，该方法在插值多项式为三次时即可实现插值曲线的$C^2$连续性。同时，本文还研究了相应的误差控制方法。