2.4 Surveying Mathematics, Measurements, and Computations

The math the PS exam actually uses

NCEES lists technical computations and applicable software under Professional Survey Practices, and construction and as-built calculations under Areas of Practice. The exam supplies a reference handbook, so it tests whether you can set up and check a computation, not whether you memorized every constant. Carry units and watch the bearing-versus-azimuth convention.

Coordinate geometry: latitudes and departures

For a course of horizontal distance D at azimuth Az (measured clockwise from north):

• Departure (east-west component): dE = D x sin(Az)
• Latitude (north-south component): dN = D x cos(Az)

Then N2 = N1 + dN and E2 = E1 + dE. The inverse problem recovers a course from two coordinates:

• Distance = sqrt((dN)^2 + (dE)^2)
• Azimuth = atan2(dE, dN), placed in the correct quadrant

Worked example: from point A (N 5000.00, E 5000.00), a course of 300.00 ft at azimuth 60 degrees gives dE = 300 x sin(60) = 259.81, dN = 300 x cos(60) = 150.00, so B is N 5150.00, E 5259.81.

Traverse closure and precision

A closed traverse should return to its start, so the sum of latitudes and the sum of departures should each be zero.

1. Sum all latitudes (closure in N) and all departures (closure in E).
2. Linear misclosure = sqrt((sum dN)^2 + (sum dE)^2).
3. Precision ratio = perimeter / linear misclosure, expressed as 1:N.

Worked example: if sum dN = +0.08 ft and sum dE = -0.06 ft over a 2,000 ft perimeter, linear misclosure = sqrt(0.08^2 + 0.06^2) = 0.10 ft, and precision = 2000 / 0.10 = 1:20,000.

Adjusting a traverse

The compass (Bowditch) rule distributes misclosure to each course in proportion to its length, assuming angles and distances are equally reliable. The correction to a course's latitude equals the total latitude misclosure times (course length / perimeter), with the sign that drives the sum toward zero; departures are corrected the same way. When you have redundant, weighted observations, a least squares adjustment is preferred because it weights by reliability and yields residuals and precision estimates. Always hunt for blunders before adjusting; adjustment spreads random error, it does not fix a wrong backsight.

Area by coordinates

With corner coordinates, the coordinate (shoelace) method gives area:

Area = 0.5 x |sum of (E_i x N_{i+1} - E_{i+1} x N_i)|

Work around the polygon consistently and close back to the first point. The absolute value removes the sign that depends on direction of travel.

Horizontal curves

Route and construction work use circular curves. With radius R and central angle delta:

• Arc length L = R x delta, with delta in radians (degrees x pi/180).
• Tangent T = R x tan(delta/2).
• Long chord LC = 2R x sin(delta/2).
• External E = R x (sec(delta/2) - 1); middle ordinate M = R x (1 - cos(delta/2)).

The degree of curve (arc definition) is the central angle subtending a 100 ft arc: D = 5729.58 / R. Worked example: R = 1000 ft, delta = 30 degrees. L = 1000 x (30 x pi/180) = 523.6 ft; T = 1000 x tan(15) = 268.0 ft.

Error and significant figures

Random errors combine in quadrature. The expected error of a sum of independent measurements with errors e1, e2, ... is sqrt(e1^2 + e2^2 + ...). Repeating an observation n times reduces the random error of the mean by a factor of sqrt(n). Report results to a precision consistent with the measurement; do not claim hundredths of a foot from a method good only to tenths.

Computation checklist

1. Convert bearings to azimuths consistently and confirm the quadrant.
2. Carry units; never mix feet and U.S. survey feet silently.
3. Compute misclosure and precision before trusting coordinates.
4. Remove blunders, then adjust by an appropriate method.
5. Round only at the end, to a defensible precision.

Bearings, azimuths, and angles

Keep the angular conventions straight. A bearing is an acute angle from north or south toward east or west (N 30 deg E); an azimuth is measured clockwise from north through a full 360 degrees. Convert by quadrant: a bearing of S 40 deg W is azimuth 180 + 40 = 220 deg. Interior angles of a closed polygon sum to (n - 2) x 180 deg; a five-sided traverse should sum to 540 deg, and the angular misclosure is the difference between the measured sum and that theoretical value. Distribute small angular misclosure equally among angles before computing latitudes and departures, after confirming no single angle holds a blunder.

Stadia and trigonometric leveling

Though modern work uses EDM and GNSS, the exam can still test the geometry. Trigonometric leveling finds an elevation difference from a measured slope distance S and vertical angle alpha: the vertical component is S x sin(alpha), and the elevation of the target is instrument elevation + height of instrument + S x sin(alpha) - height of target, with curvature and refraction corrections over long sights. The horizontal distance is S x cos(alpha). Carrying the instrument and target heights correctly is the common error point.

Differential leveling arithmetic

Differential leveling computes elevations with backsights and foresights: the height of instrument (HI) = known benchmark elevation + backsight, and the next point elevation = HI - foresight. Worked example: benchmark at 100.00 ft, backsight 4.20 ft gives HI = 104.20 ft; a foresight of 6.55 ft to the turning point gives elevation 104.20 - 6.55 = 97.65 ft. The arithmetic check is that the sum of backsights minus the sum of foresights equals the net elevation change. A loop should close on its starting benchmark within the allowable misclosure for the order of leveling.

Units and the survey foot trap

A recurring blunder is mixing the U.S. survey foot and the international foot, which differ by 2 parts per million. Over short distances this is negligible, but on State Plane coordinates in the hundreds of thousands of feet it produces real shifts. NCEES has moved toward the international foot in modernized datums while legacy data use the survey foot, so always confirm which foot a coordinate or scale factor assumes. Likewise, never mix metric and U.S. customary values silently; convert explicitly and state the unit on the deliverable.