Consider an input consisting of a set of $n$ disjoint triangular obstacles in $\mathbb{R}^3$ and a target point $t$ in the free space, all enclosed by a large sphere $S$ of radius $R$ centered at $t$. An articulated probe is modeled as two line segments $ab$ and $bc$ connected at point $b$. The length of $ab$ can be equal to or greater than $R$, while $bc$ is of a given length $r \leq R$. The probe is initially located outside $S$, assuming an unarticulated configuration, in which $ab$ and $bc$ are collinear and $b \in ac$. The goal is to find a feasible (obstacle-avoiding) probe trajectory to reach $t$, with the condition that the probe is constrained by the following sequence of moves -- a straight-line insertion of the unarticulated probe into $S$, possibly followed by a rotation of $bc$ at $b$ for at most $\pi/2$ radians, so that $c$ coincides with $t$. We prove that if there exists a feasible probe trajectory, then a set of extremal feasible trajectories must be present. Through careful case analysis, we show that these extremal trajectories can be represented by $O(n^4)$ combinatorial events. We present a solution approach that enumerates and verifies these combinatorial events for feasibility in overall $O(n^{4+\epsilon})$ time using $O(n^{4+\epsilon})$ space, for any constant $\epsilon > 0$. The enumeration algorithm is highly parallel, considering that each combinatorial event can be generated and verified for feasibility independently of the others. In the process of deriving our solution, we design the first data structure for addressing a special instance of circular sector emptiness queries among polyhedral obstacles in three dimensional space, and provide a simplified data structure for the corresponding emptiness query problem in two dimensions.

中文翻译：

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计算三维铰接探针的可行轨迹

考虑一个输入，该输入由$ \ mathbb {R} ^ 3 $中的一组$ n $不相交的三角形障碍和一个自由空间中的目标点$ t $组成，它们全部被半径为$ R $的大球体$ S $包围以$ t $为中心。铰接探针建模为连接在点$ b $处的两个线段$ ab $和$ bc $。$ ab $的长度可以等于或大于$ R $，而$ bc $的长度是给定的$ r \ leq R $。该探针最初位于$ S $外部，假定为非铰接配置，其中$ ab $和$ bc $是共线的，而$ b \ ac为ac $。目的是找到一种可行的（避障）探针轨迹以达到$ t $，条件是探针受到以下移动顺序的约束-将未铰接的探针直线插入$ S $，可能接着是$ bc $在$ b $处旋转，最多$ \ pi / 2 $弧度，因此$ c $与$ t $一致。我们证明，如果存在可行的探测轨迹，那么必须存在一组极值的可行轨迹。通过仔细的案例分析，我们表明这些极值轨迹可以用$ O（n ^ 4）$组合事件表示。我们提供了一种解决方案方法，该方法枚举并验证这些组合事件的可行性，并使用$ O（n ^ {4+ \ epsilon}）$空间来计算$ O（n ^ {4+ \ epsilon}）$的总时间是否可行。 \ epsilon> 0 $。考虑到每个组合事件都可以独立于其他事件生成和验证，因此枚举算法是高度并行的。在得出解决方案的过程中，我们设计了第一个数据结构，用于处理三维空间中多面体障碍物之间的圆形扇形空度查询的特殊情况，