Let P be a set of n points in the plane. We compute the value of \(\theta \in [0,2\pi )\) for which the rectilinear convex hull of P, denoted by \(\mathcal {RH}_{P}({\theta })\), has minimum (or maximum) area in optimal \(O(n\log n)\) time and O(n) space, improving the previous \(O(n^2)\) bound. Let \(\mathcal {O}\) be a set of k lines through the origin sorted by slope and let \(\alpha _i\) be the sizes of the 2k angles defined by pairs of two consecutive lines, \(i=1, \ldots , 2k\). Let \(\Theta _{i}=\pi -\alpha _i\) and \(\Theta =\min \{\Theta _i :i=1,\ldots ,2k\}\). We obtain: (1) Given a set \(\mathcal {O}\) such that \(\Theta \ge \frac{\pi }{2}\), we provide an algorithm to compute the \(\mathcal {O}\)-convex hull of P in optimal \(O(n\log n)\) time and O(n) space; If \(\Theta < \frac{\pi }{2}\), the time and space complexities are \(O(\frac{n}{\Theta }\log n)\) and \(O(\frac{n}{\Theta })\) respectively. (2) Given a set \(\mathcal {O}\) such that \(\Theta \ge \frac{\pi }{2}\), we compute and maintain the boundary of the \({\mathcal {O}}_{\theta }\)-convex hull of P for \(\theta \in [0,2\pi )\) in \(O(kn\log n)\) time and O(kn) space, or if \(\Theta < \frac{\pi }{2}\), in \(O(k\frac{n}{\Theta }\log n)\) time and \(O(k\frac{n}{\Theta })\) space. (3) Finally, given a set \(\mathcal {O}\) such that \(\Theta \ge \frac{\pi }{2}\), we compute, in \(O(kn\log n)\) time and O(kn) space, the angle \(\theta \in [0,2\pi )\) such that the \({\mathcal {O}}_{\theta }\)-convex hull of P has minimum (or maximum) area over all \(\theta \in [0,2\pi )\).

令P为平面中n个点的集合。我们计算\（\ theta \ in [0,2 \ pi）\）的值，其中P的直线凸包由\（\ mathcal {RH} _ {P}（{\ theta}）\）表示在最佳\（O（n \ log n）\）时间和O（n）空间中具有最小（或最大）面积，从而改善了先前的\（O（n ^ 2）\）界限。让\（\ mathcal {ö} \）是一组 ķ线过原点排序由斜率和让\（\阿尔法_i \）是2的尺寸ķ角度通过对接连的两行所定义，\（ⅰ = 1，\ ldots，2k \）。设\（\ Theta _ {i} = \ pi-\ alpha _i \）和\（\ Theta = \ min \ {\ Theta _i：i = 1，\ ldots，2k \} \）。我们得到：（1）给定一个\（\ mathcal {O} \）使得\（\ Theta \ ge \ frac {\ pi} {2} \），我们提供了一种计算\（\ mathcal {在最佳\（O（n \ log n）\）时间和O（n）空间中P的O} \）-凸包 ；如果\（\ Theta <\ frac {\ pi} {2} \），则时间和空间复杂度为\（O（\ frac {n} {\ Theta} \ log n）\）和\（O（\ frac {n} {\ Theta}）\）。（2）给定一个\（\ mathcal {O} \）使得\（\西塔\ GE \压裂{\ PI} {2} \） ，我们计算和保持的边界\（{\ mathcal {ö}} _ {\ THETA} \）的船体-凸 P为\（ \ theta \ in [0,2 \ pi} \）在\（O（kn \ log n）\）时间和O（kn）空间中，或者如果\（\ Theta <\ frac {\ pi} {2} \ ），在\（O（k \ frac {n} {\ Theta} \ log n）\）时间和\（O（k \ frac {n} {\ Theta}）\）空间中。（3）最后，给定一个\（\ mathcal {O} \）使得\（\ Theta \ ge \ frac {\ pi} {2} \），我们以\（O（kn \ log n） \）时间和O（kn）空间，角度\（\ theta \ in [0,2 \ pi } \），使得P的\（{\ mathcal {O}} _ {\ theta} \）-凸包在所有\（ \ theta \ in [0,2 \ pi）\）。