In this paper, we consider the Weighted Region Problem. In the Weighted Region Problem, the length of a path is defined as the sum of the weights of the subpaths within each region, where the weight of a subpath is its Euclidean length multiplied by a weight $ \alpha \geq 0 $ depending on the region. We study a restricted version of the problem of determining shortest paths through a single weighted rectangular region. We prove that even this very restricted version of the problem is unsolvable within the Algebraic Computation Model over the Rational Numbers (ACMQ). On the positive side, we provide the equations for the shortest paths that are computable within the ACMQ. Additionally, we provide equations for the bisectors between regions of the Shortest Path Map for a source point on the boundary of (or inside) the rectangular region.

翻译：在本文中，我们考虑加权区域问题。在加权区域问题中，路径的长度定义为各区域内子路径权重之和，其中子路径的权重为其欧氏长度乘以依赖于该区域的权重参数α≥0。我们研究通过单个加权矩形区域的最短路径确定问题的一个受限版本。我们证明，即使在有理数上的代数计算模型(ACMQ)框架下，这一高度受限的问题版本也是不可解的。从正面来看，我们提供了可在ACMQ模型下计算的最短路径方程。此外，我们还给出了矩形区域边界上（或内部）源点的最短路径图中各区域间平分线的方程。