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 $$\mathcal {O}$$ O be a set of $$k\ge 2$$ k ≥ 2 lines passing through the origin. We study the problem of computing the set of orientations of the lines of $$\mathcal {O}$$ O for which the $$\mathcal {O}$$ O -convex hull of R contains no points of B. For $$k=2$$ k = 2 orthogonal lines we have the rectilinear convex hull. In optimal $$O(n\log n)$$ O ( n log n ) time and O(n) space, $$n = \vert R \vert + \vert B \vert $$ n = | R | + | B | , we compute the set of rotation angles such that, after simultaneously rotating the lines of $$\mathcal {O}$$ 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 $$\mathcal {O}$$ O is formed by $$k \ge 2$$ k ≥ 2 lines with arbitrary orientations. In the counter-clockwise circular order of the lines of $$\mathcal {O}$$ O , let $$\alpha _i$$ α 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 $$O({1}/{\Theta }\cdot N \log N)$$ O ( 1 / Θ · N log N ) time and $$O({1}/{\Theta }\cdot N)$$ O ( 1 / Θ · N ) space, where $$\Theta = \min \{ \alpha _1,\ldots ,\alpha _k \}$$ Θ = min { α 1 , … , α k } and $$N=\max \{k,\vert R \vert + \vert B \vert \}$$ N = max { k , | R | + | B | } . We finally consider the case in which $$\mathcal {O}$$ O is formed by $$k=2$$ k = 2 lines, one of the lines is fixed, and the second line rotates by an angle that goes from 0 to $$\pi $$ π . We show that this last case can also be solved in optimal $$O(n\log n)$$ O ( n log n ) time and O(n) space, where $$n = \vert R \vert + \vert B \vert $$ n = | R | + | B | .
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This article presents a system to enable access to those Information Systems at the National Autonomous University of Mexico (UNAM) that are related to Biodiversity and the Environment. The system in question associates existing Geographic Information Systems (GIS’s) as well as standard relational databases in a federation, allows the contents of the individual GIS (or relational databases) to be consulted in a manner transparent to the user, and permits the exports of the underlying systems’ data under the corresponding set of permissions. Our approach is based upon three principles: compliance with international standards, reliance upon Open Source Software in implementation, and usage of servers of proven reliability and robustness.
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