Separating bichromatic point sets in the plane by restricted orientation convex hulls

We explore the separability of point sets in the plane by a restricted-orientation convex hull, which is an orientation-dependent, possibly disconnected, and non-convex enclosing shape that generalizes the convex hull. Let R and B be two disjoint sets of red and blue points in the plane, and O be a set of k >= 2 lines passing through the origin. We study the problem of computing the set of orientations of the lines of O for which the O-convex hull of R contains no points of B. For k = 2 orthogonal lines we have the rectilinear convex hull. In optimal O (n log n) time and O(n) space, n = vertical bar R vertical bar + vertical bar B vertical bar. we compute the set of rotation angles such that, after simultaneously rotating the lines of O around the origin in the same direction, the rectilinear convex hull of R contains no points of B. We generalize this result to the case where O is formed by k >= 2 lines with arbitrary orientations. In the counter-clockwise circular order of the lines of O, let alpha(i) be the angle required to clockwise rotate the ith line so it coincides with its successor. We solve the problem in this case in O(1/Theta . N log N) time and O(1/Theta . N) space, where Theta = min {alpha(1), ..., alpha(k)} and N = max{k, vertical bar R vertical bar + vertical bar B vertical bar}. We finally consider the case in which O is formed by k = 2 lines, one of the lines is fixed, and the second line rotates by an angle that goes from O to pi. We show that this last case can also be solved in optimal O(n log n) time and O(n) space, where n = vertical bar R vertical bar + vertical bar B vertical bar.