标题： 二维方向-长度框架的全局刚性 显示英文标题

标题： Global rigidity of 2-dimensional direction-length frameworks

摘要： 二维方向-长度框架是平面上由成对约束连接的点集合，这些约束固定了某些点对之间线段的方向或长度。 我们将其表示为一对 $(G,p)$，其中$G=(V;D,L)$是一个“混合”图，$p:V\to{\mathbb R}^2$是$V$的点配置。 如果每个满足相同约束的方向-长度框架$(G,q)$都可以通过对$(G,p)$进行平移或绕$180^\circ$旋转得到，则它是全局刚性的。 我们证明了当一个一般框架 $(G,p)$ 是全局刚性的条件可以简化为当 $G$ 属于一个特殊的“方向不可约”混合图家族的情况，并证明了一个方向不可约混合图 $G$ 的一般性实现 {每个} 是全局刚性的当且仅当 $G$ 是2-连通的、方向平衡的并且冗余刚性的。 显示英文摘要

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.