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 $ α\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：本文研究加权区域问题。在加权区域问题中，路径长度定义为各区域内子路径的权重之和，其中子路径的权重为其欧几里得长度乘以取决于该区域的权重系数 $ α\geq 0 $。我们研究了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，并证明即使该问题的这一高度受限版本也无法在有理数代数计算模型（ACMQ）内求解。在积极方面，我们给出了可在ACMQ内计算的最短路径方程。此外，我们还提供了矩形区域边界上（或内部）源点对应的最短路径图中区域间分界线的方程。