Measurement and Computations
Domain 3 covers the arithmetic and error analysis behind every survey: traverse closure, coordinate geometry, area, curve geometry, and the judgment to know when a closure is good enough for its purpose.
Error and precision
- Random errors scatter and can be averaged down with redundant observations; systematic errors (a mis-set prism constant, a stretched tape, an unmodeled scale factor) repeat and must be found.
- Accuracy is closeness to truth; precision is repeatability. A precise but biased traverse is still wrong.
- Closure ratio = perimeter / misclosure. A 3,000.00 ft traverse closing 0.30 ft gives 1:10,000 - routine for topographic work, thin for a boundary retracement.
Traverse computations
Latitude = distance × cos(bearing); departure = distance × sin(bearing). Signs follow the quadrant: N and E positive. Compass (Bowditch) rule spreads misclosure proportional to distance; the transit rule spreads by latitude and departure magnitude. Area by coordinates (shoelace): 2A = Σ(xᵢyᵢ₊₁ − xᵢ₊₁yᵢ).
Curves
T = R tan(Δ/2); L = RΔ in radians; LC = 2R sin(Δ/2); M = R(1 − cos(Δ/2)); E = R(sec(Δ/2) − 1). Arc definition: R = 5729.578/D.
Worked example
Stations A(1000.00, 1000.00), B(1400.00, 1000.00), C(1000.00, 1300.00), clockwise. AB: 1000×1000 − 1400×1000 = −400,000 BC: 1400×1300 − 1000×1000 = +820,000 CA: 1000×1000 − 1000×1300 = −300,000 Σ = +120,000, so A = 60,000 sf = 60,000/43,560 = 1.377 acres. If the boundary of that triangle closed 0.30 ft on a 3,000 ft perimeter, the acreage uncertainty is far smaller than the 0.001-acre rounding - but a 1:10,000 closure cannot be used to set an urban lot corner.
Traps
Quadrant sign errors; using sine for latitude; forgetting to halve the shoelace sum; mixing decimal degrees with DMS; applying curvature formulas with Δ in degrees instead of radians.