二维方向长度框架的整体刚度
Global rigidity of 2-dimensional direction-length frameworks
摘要：
二维方向-长度框架是平面中的点的集合，这些点通过成对约束链接，成对约束固定连接某些点对的线段的方向或长度。我们把它表示为一对$(G,p)$，其中$G=(V;D,L)$是一个“混合”图，而$p:V\to{\mathbb R}^2$是$V$的一个点配置。如果每个满足相同约束条件的方向长度框架$(G,q)$都可以通过$180^\circ$的平移或旋转从$(G,p)$中得到，则它是全局刚性的。我们表明，问题的特征时，通用框架$(G,p)$是全球刚性的情况下，可以减少到$G$属于一个特殊的家庭的“方向不可约”的混合图，并证明了{每}通用实现的方向不可约混合图$G$是全球刚性的当且仅当$G$是2-连通，方向平衡和冗余刚性。
英文摘要：
A 2-dimensional direction-length framework is a collection of points in the plane which are linked by pairwise constraints that fix the direction or length of the line segments joining certain pairs of points. We represent it as a pair $(G,p)$, where $G=(V;D,L)$ is a `mixed' graph and $p:V\to{\mathbb R}^2$ is a point configuration for $V$. It is globally rigid if every direction-length framework $(G,q)$ which satisfies the same constraints can be obtained from $(G,p)$ by a translation or a rotation by $180^\circ$. We show that the problem of characterising when a generic framework $(G,p)$ is globally rigid can be reduced to the case when $G$ belongs to a special family of `direction irreducible' mixed graphs, and prove that {every} generic realisation of a direction irreducible mixed graph $G$ is globally rigid if and only if $G$ is 2-connected, direction-balanced and redundantly rigid.
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