Coordinate Systems, Latitudes and Departures (FE Civil Surveying)

NCEES FE Civil spec: Surveying, D. Coordinate systems (e.g., state plane, latitude/longitude). FE Reference Handbook: Civil Engineering chapter, Surveying section (latitudes and departures).

What the FE Civil Spec Covers

Section D tests how survey positions are expressed: plane rectangular coordinates (northing and easting), state plane coordinates, and geographic latitude and longitude. In practice nearly every question reduces to latitudes and departures: go forward from a known point using a distance and direction, or inverse between two known points to get distance and bearing. Traverse closure and precision are also fair game.

Key Concepts

Plane coordinates: N (northing, y) and E (easting, x). Azimuths are measured from the N axis, which is why the trig is swapped compared with math class.
State plane coordinate systems project the curved earth onto a flat grid for each state or zone, using Lambert conformal conic projections for zones that run east-west and transverse Mercator for zones that run north-south.
Grid vs ground: grid distance = ground distance times the combined factor (scale factor times elevation factor).
Latitude/longitude: angular positions on the ellipsoid. One degree of latitude is about 69 miles (111 km); one degree of longitude shrinks with the cosine of the latitude.

Core Formulas

For a line of length L and azimuth Az:

Lat = L cos (A z) Dep = L sin (A z)

N_{2} = N_{1} + Lat E_{2} = E_{1} + Dep

Inverse between two points:

L = Δ N^{2} + Δ E^{2} bearing angle = tan^{- 1} \frac{Δ E}{Δ N}

Closure of a traverse with perimeter P:

e = (\sum Lat)^{2} + (\sum Dep)^{2} precision = 1 : \frac{P}{e}

Compass (Bowditch) rule correction for line i:

c_{Lat, i} = - \sum Lat \frac{L _{i}}{P} c_{Dep, i} = - \sum Dep \frac{L _{i}}{P}

Where It Is in the FE Reference Handbook

The Civil Engineering chapter, Surveying section, shows the latitude and departure sign convention (north and east positive). The trig itself is in the Mathematics chapter. State plane details are not tabulated, so questions give you any factors you need.

Worked Example

Point A has coordinates N 5000.00 ft, E 2000.00 ft. Line AB has a grid distance of 412.50 ft and an azimuth of 137°15'. Find the coordinates of B and the bearing of AB.

Step 1, decimal azimuth. 137°15' = 137.25°. This lies between 90° and 180°, so the line points southeast: latitude negative, departure positive.

Step 2, latitude and departure.

Lat = 412.50 cos 137.2 5^{\circ} = - 302.91 ft

Dep = 412.50 sin 137.2 5^{\circ} = + 280.01 ft

Step 3, coordinates of B.

N_{B} = 5000.00 - 302.91 = 4697.09 ft E_{B} = 2000.00 + 280.01 = 2280.01 ft

Step 4, bearing. SE quadrant: bearing angle = 180° minus 137°15' = 42°45', so the bearing is S 42°45' E. Check by inverse: the square root of 302.91² + 280.01² returns 412.50 ft.

If 412.50 ft had been a ground distance with a combined factor of 0.99992, the grid distance would be 412.50 times 0.99992 = 412.47 ft, and that value would go into Step 2.

Exam Strategy

Build a small table with columns for line, azimuth, length, latitude and departure. Even for one line, writing the sign you expect next to each component (from the quadrant) before computing catches most errors, because the calculator will give the sign automatically and you can compare.

For inverse problems, find the bearing angle from the absolute values of ΔE and ΔN, then attach the quadrant letters from their signs: ΔN positive is N, negative is S; ΔE positive is E, negative is W. A line with ΔE = minus 150 m and ΔN = plus 260 m is in the NW quadrant, with a bearing angle of about 30°.

Precision questions often give closure errors in latitude and departure separately. Combine them with the Pythagorean theorem first; dividing the perimeter by only one of them overstates the precision.

Common Mistakes

Using sine for latitude. With azimuths from north, latitude uses cosine.
Taking the calculator's inverse tangent at face value. It only returns an angle between minus 90° and 90°; use the signs of ΔN and ΔE to pick the quadrant.
Mixing ground and grid distances in the same computation.
Computing precision as e divided by P and reporting a decimal instead of a 1:x ratio.

Practice Problems

Try these from the free PolyCalc FE Civil problem bank. Each one has a full worked solution.

What is the distance between coordinates (0, 0) and (3, 4), with both coordinates given in meters? See the worked solution or practice the Surveying deck
Point A is at E = 500 m, N = 1000 m and point B is at E = 800 m, N = 1400 m. The distance and bearing from A to B are: See the worked solution or practice the Surveying deck
A survey line starts at E = 2000 m, N = 3000 m, is 250 m long, and has an azimuth of 120°. The ending coordinates are most nearly: See the worked solution or practice the Surveying deck
A line has ΔE = −150 m and ΔN = +260 m. Its quadrant bearing is most nearly: See the worked solution or practice the Surveying deck
A closed traverse with a perimeter of 800 m has a closure error of +0.08 m in northing and −0.06 m in easting. The precision of the traverse is most nearly: See the worked solution or practice the Surveying deck

Related Calculators and Examples

Frequently Asked Questions

What are latitude and departure in surveying?

Latitude is the north-south component of a traverse line (length times cosine of the azimuth, positive north). Departure is the east-west component (length times sine of the azimuth, positive east). Adding them to a starting northing and easting gives the next point.

What is a combined scale factor?

In a state plane coordinate system, ground distances are reduced to grid distances by multiplying by a combined factor, which is the grid scale factor times the elevation factor. The combined factor is usually very close to 1, such as 0.99992.

How is traverse precision reported?

Compute the linear misclosure as the square root of the sum of the squared latitude and departure errors, then divide the traverse perimeter by it. A result such as 1:8,000 means one unit of error per 8,000 units of distance.

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