Turning is the most popular machining operation. The quality of the product may be determined using a variety of metrics, such as the surface generation method and the surface roughness of the product. This work uses cutting variables to obtain the best surface quality through a mathematical model. The suggested surface generation in this work results from deriving it using the Bezier technique, with degree (5th) having six chosen control points. One of the critical indicators of the quality of machined components is the surface roughness created during the machining process. Surface roughness improvement via machining process parameter optimization has been extensively researched. The Taguchi Method and actual tests were employed for evaluating the surface quality of complicated forms; regression models with three different variables for the cutting process, such as cutting speed, depth of cut, and feed rate, were also used. According to the experimental findings, the most significant effect of feed rate on the surface roughness is approximately (40.9%), and the more minor effect of depth of cut on the surface roughness is almost (16.23%). In addition, the average percentage error is 4.93%, the maximum error is 0.14 mm, and the minimum error is -0.143 mm for the prediction using the regression equation.