We study the problem of finding a short path from a start to a goal within a two-dimensional continuous and isotropic terrain that has been discretized into an array of accessible and blocked cells. A classic approach obtains a grid path where each step is along the edge of an accessible cell or diagonally across one. Grid paths suffer from `digitization bias' -- even if two locations have line-of-sight, the minimum travelling cost between them can be greater than the distance along the line-of-sight. In a vertex path, steps are allowed from a cell corner to any other cell corner if they have line-of-sight. While the `digitization bias' is smaller, shortest vertex paths are impractical to find by brute force. Recent research has thus turned to methods for finding short (but not necessarily shortest) vertex paths. To establish the methods' potential utility, we calculate upper bounds on the difference in length between the shortest vertex paths versus the shortest r-constrained ones where an r-constrained path consists of line segments that each traverse at most r rows and at most r columns of cells. The difference in length reduces as r increases -- indeed the shortest vertex paths are at most 1 percent shorter than the shortest 4-constrained ones. This article will be useful to developers and users of short(est) vertex paths algorithms who want to trade path length for improved runtimes in a predictable manner.
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