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 $\mathcal{O}$ be a set of $k \geq 2$ lines passing through the origin. We study the problem of computing the set of orientations of the lines of $\mathcal{O}$ for which the $\mathcal{O}$-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, $n = \vert R \vert + \vert B \vert$, we compute the set of rotation angles such that, after simultaneously rotating the lines of $\mathcal{O}$ 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 $\mathcal{O}$ is formed by $k \geq 2$ lines with arbitrary orientations. In the counter-clockwise circular order of the lines of $\mathcal{O}$, let $α_i$ be the angle required to clockwise rotate the $i$th line so it coincides with its successor. We solve the problem in this case in $O(1/Θ\cdot N \log N)$ time and $O(1/Θ\cdot N)$ space, where $Θ= \min \{ α_1,\ldots,α_k \}$ and $N=\max\{k,\vert R \vert + \vert B \vert \}$. We finally consider the case in which $\mathcal{O}$ is formed by $k=2$ lines, one of the lines is fixed, and the second line rotates by an angle that goes from $0$ to $π$. We show that this last case can also be solved in optimal $O(n\log n)$ time and $O(n)$ space, where $n = \vert R \vert + \vert B \vert$.

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