In this dissertation, a ruled surface in Euclidean space is represented as a curve on a dual unit sphere (DUS) by employing the Klein mapping and the Study mapping. A novel definition of dual spherical spline is proposed and a complete kinematic ruled surface approximation algorithm is developed and tested with turbocharger blade data. More generally, a ruled surface is defined by several control points of a dual spherical spline. It provides an initial prototype for the blade geometry optimization with ruled surface.
Finally, combining the kinematic ruled surface approximation algorithm with the offset theory, a novel design and manu- facturing strategy is proposed. A desired surface is presented as a tool path of the flank milling method with a cylindrical tool in 5-axis CNC machining. It integrates the manufacturing requirements in the design phase, which can reduce the design- cycle time and save the manufacturing cost.