NSPS CST Level IV cheat sheet

Ellipsoid vs. geoid: the geoid is an equipotential surface of Earth's gravity field approximated by mean sea level; ellipsoid heights (h) and orthometric heights (H) relate by h = H + N, where N is the geoid undulation.

State Plane Coordinate System zones use either Lambert Conformal Conic (east-west extent zones) or Transverse Mercator (north-south extent zones); scale error is zero along the standard parallels (LCC) or central meridian (TM) and grows with distance from them.

Universal Transverse Mercator divides Earth into 60 zones of 6 degrees longitude; the central meridian of zone n is at longitude (6n − 183) degrees, and a false easting of 500,000 m is applied at the central meridian.

Grid-to-ground (combined) factor = ellipsoid (elevation) factor × projection (scale) factor; elevation factor ≈ R / (R + H) using mean Earth radius R ≈ 6,371,000 m and orthometric height H.

Traverse closure: linear error of closure = square root of (ΔE² + ΔN²); relative precision is that closure divided into the total traverse length, expressed as 1:N.

Resection (three-point problem) determines an occupied station from angles or distances to three known points; the danger circle is the locus of occupied positions giving no unique solution, so avoid setups where the instrument lies on the circle through all three known points.

GNSS RTK horizontal accuracy is typically quoted as 8 mm + 1 ppm horizontally and 15 mm + 1 ppm vertically (RMS); ppm error is multiplied by baseline length in km, so a 5 km baseline adds about 5 mm horizontal.

Differential leveling misclosure tolerance commonly follows the form C × √K, where K is loop length in km or miles and C is a project-specified constant; misclosure is distributed proportional to accumulated distance from the starting benchmark.

Coordinate rotation by angle θ: E′ = E·cos θ + N·sin θ and N′ = −E·sin θ + N·cos θ for a clockwise rotation about the origin; translation is applied after rotation.

Least-squares adjustment minimizes the sum of squared residuals; the normal equations are (AᵀWA)x = AᵀWL, and the a posteriori variance factor is σ₀² = (VᵀWV) / (n − u) with n observations and u unknowns.

Area by coordinate (shoelace) method: 2A = Σ(Eᵢ·Nᵢ₊₁ − Eᵢ₊₁·Nᵢ) around the closed figure, with the final point wrapping to the first; the result's sign indicates traverse direction.

Horizontal curve formulas: tangent T = R·tan(Δ/2), curve length L = R·Δ (Δ in radians), external E = R·(sec(Δ/2) − 1), and middle ordinate M = R·(1 − cos(Δ/2)).

Metadata required for delivered spatial data commonly includes coordinate reference system, datum and epoch, units, accuracy statement, source and date, and a lineage or processing description.

Error propagation for a sum or difference adds variances; for a product or quotient add relative variances: (σz/z)² = (σx/x)² + (σy/y)² for z = x·y or x/y with independent inputs.

A plat or boundary description must close mathematically; check that bearings and distances return to the point of beginning within the stated precision before recording, and reconcile area against the deed or record area.