The tool paths of modern computerized numerically controlled (CNC) manufacturing machines are usually described by means of circular arcs and straight line segments. In a computer-aided design (CAD) environment, however, objects are often designed using B-spline or $B\'ezier$ curves. It is therefore necessary to approximate B-spline or $B\'ezier$ curves by arc spline. Many papers have been written on arc spline approximation using biarc. In programming the CNC machine, the number of arc segments is required to be small since too many arc segments can cause memory overflow. In this thesis, we give a new method for approximating quadratic $B\'ezier$ curves by $G^1$ arc spline which reduces the number of arc segments. Numerical results are given for some typical curves used in CAD to illustrate the efficiency of the algorithm.
