Current tool path computation in the CAM algorithms approximates the surface by piecewise linear interpolation. In the case of three-axis machining on a CNC machine the tool will exactly reproduce this computed tool path. However in the case of five-axis simultaneous machining the real tool path on the CNC machine will not follow the linear approximation computed by the conventional CAM algorithm. A new CAM algorithm is proposed which approximates the surface to be machined by a piecewise curved approximation. This curve represents the real tool path followed on the five-axis machine. This piecewise curved approximation is further optimized by formulating the tool path computation as the generation of a grid based on a variational smoothness penalty function. This new algorithm considerably improves the accuracy and reduces the number of blocks and machining time.