Orienting the workpiece in such a way as to minimize the number of setups in a 4-axis or a 5-axis Numerical Control (NC) machine is formulated as follows: Given a set of spherical polygons (that are representations of curved surfaces visible to a 3-axis NC machine), find a great circle (the 4th axis) or a band (the 4th and the 5th axes) containing a great circle that intersects the polygons maximally. While there are potentially infinitely many solutions to this problem, a sphere is partitioned into O(N2) regions based on the N polygons. Within each of these regions, it is shown that it requires O(NlogN) time to determine maximum intersections and all the solutions are congruent. Central projection mapping is employed so as to present the algorithms in the plane.

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