When machining a spherical surface along a contour line with a ball-end mill at high feed speed, the finished surface roughness is greatly affected by the tilting angle of the tool axis. In order to obtain fine surface roughness, this paper proposes concepts of optimum tilting angle and minimum surface roughness for spherical surface machining. The computer simulation demonstrates that the surface roughness can be dramatically improved by tilting the tool axis with an optimum tilting angle, as

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... ompared with machining by employing the normal vector on the surface as the tool axis. Furthermore, a mathematical expression has been established by applying the Powell method squares to calculate the optimum tilting angle and the minimum surface roughness under any machining condition . The effectiveness of the estimation for both optimum tilting angle and the minimum surface roughness is also proved, as the estimating error is small enough to satisfy requirements.