7.2 Coordinate Geometry, Latitudes, and Departures

COGO Components and Coordinate Accumulation

Coordinate geometry is the bridge between field observations and a usable survey map. A line described by a distance and direction is useful, but a line expressed as coordinate changes can be closed, checked, adjusted, used for area, and transferred into CAD or GIS. The FS exam may present this as a pure calculation or as a practical workflow question about organizing traverse data in a calculator or spreadsheet.

For azimuths measured clockwise from north, the standard component equations are latitude = distance cos azimuth and departure = distance sin azimuth. Latitude is the north-south component. Departure is the east-west component. In a northing and easting coordinate system, north is positive latitude, south is negative latitude, east is positive departure, and west is negative departure.

Suppose a course is 250.00 ft at azimuth 126 deg 52 min. Convert the angle to decimal degrees or enter it correctly in DMS mode. The latitude should be negative because the azimuth is southeast. The departure should be positive. Approximate values are latitude = 250 cos 126.8667 deg = -150.17 ft and departure = 250 sin 126.8667 deg = +199.87 ft. If the signs do not match the quadrant, fix the setup before moving on.

Coordinate accumulation starts with a known point. If point A has northing 10000.00 and easting 5000.00, then the endpoint of the example course has northing 9849.83 and easting 5199.87. The next course starts from that endpoint. A spreadsheet should calculate the endpoint of each row from the prior row, not from a manually retyped value. That reduces transcription errors and makes closure checks easier.

Distance from coordinates reverses the process. Delta northing is ending northing minus starting northing. Delta easting is ending easting minus starting easting. Distance is the square root of delta northing squared plus delta easting squared. Direction is found from the ratio of departure to latitude, then placed in the correct quadrant. Calculator inverse tangent results must be interpreted carefully because arctangent alone may not know the intended quadrant unless an atan2 function is available.

COGO tasks often mix units. A plan may give feet, chains, meters, or stationing. Convert before summing. One chain is 66 ft, and a station in common U.S. route work is 100 ft. Slopes may appear as percent, ratio, or vertical over horizontal. Keep a unit column when the problem gives more than one unit system.

A practical COGO row layout is:

• Point or course label
• Distance
• Bearing or azimuth
• Decimal azimuth used for calculation
• Latitude
• Departure
• Ending northing
• Ending easting
• Notes for source, adjustment, or check

This structure matches the exam skill because it forces every line to carry direction, magnitude, signs, and coordinates. It also mirrors real production work. If a field note, spreadsheet, or CAD input has a wrong sign, the traverse may look plausible until closure or area exposes the error.