Two Ways to Measure an Irregular Field: An Error Bound for Baseline Offsets Against Fan Triangulation

An irregular field can be measured without instruments beyond a rope in two classic ways: by perpendicular offsets from a baseline, summed as trapezoids, or by fan triangulation from a single corner station. This study propagates stated reading errors (standard deviation 0.05 m per reading, worst case 0.10 m) through both methods, together with the discretisation error caused by replacing a curved boundary with straight segments, for a 100 m baseline with depths of 20 m, 50 m, and 100 m and boundaries ranging from straight to strongly wavy. The two methods are far closer than their reputations suggest. On the strongly wavy 100 m by 50 m field at 10 m spacing, offsets give a total root-sum-square bound of 0.1381% and triangulation gives 0.1428%; at 20 m spacing both fail a 1% worst-case test chiefly through discretisation (about -0.7959%), while at 5 m spacing both sit near 0.05% to 0.07%. The selection rule that emerges is therefore about spacing and rope length, not about method loyalty: choose the spacing from the waviness first (20 m only for straight or gently wavy boundaries, 10 m for strongly wavy ones, 5 m or finer when a worst-case guarantee below about 0.25% is required), then prefer offsets whenever the fan would demand ropes longer than the rope in hand — triangulation from one corner of the 100 m by 50 m field needs a rope reaching 108.87 m, and 138.6 m when the depth is 100 m. The single exception is the narrow field (depth 20 m), where triangulation's root-sum-square bound of 0.1581% beats offsets at 0.1953%, at the price of a 101.47 m rope. This study is simulated, not measured.