Separating and intersecting spherical polygons: computing machinability on three-, four-, and five-axis numerically controlled machines

We consider the computation of an optimal workpiece orientation allowing the maximal number of surfaces to be machined in a single setup on a three-, four-, or five-axis numerically controlled machine. Assuming the use of a ball-end cutter, we establish the conditions under which a surface is machinable by the cutter aligned in a certain direction, without the cutter's being obstructed by portions of the same surface. The set of such directions is represented on the sphere as a convex region, called the visibility map of the surface. By using the Gaussian maps and the visibility maps of the surfaces on a component, we can formulate the optimal workpiece orientation problems as geometric problems on the sphere. These and related geometric problems include finding a densest hemisphere that contains the largest subset of a given set of spherical polygons, determining a great circle that separates a given set of spherical polygons, computing a great circle that bisects a given set of spherical polygons, and finding a great circle that intersects the largest or the smallest subset of a set of spherical polygons. We show how all possible ways of intersecting a set of n spherical polygons with v total number of vertices by a great circle can be computed in O(vn log n) time and represented as a spherical partition. By making use of this representation, we present efficient algorithms for solving the five geometric problems on the sphere.

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