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Coordinate Systems Example

Easy DifficultyFE Surveying

Find the horizontal distance and bearing between two points given their (E, N) or (x, y) coordinates in a plane coordinate system.

Concept

In state plane or other plane coordinate systems, points are given as (E, N) (easting, northing) or (x, y). The distance between two points is the straight-line distance: d = (E_{2} - E_{1})^{2} + (N_{2} - N_{1})^{2} . The bearing (direction) from point 1 to point 2 is found from the arctangent of Δ E /Δ N, with the quadrant determined by the signs of Δ E and Δ N . d = (E_{2} - E_{1})^{2} + (N_{2} - N_{1})^{2} Notation: E = easting, N = northing, d = horizontal distance. Bearing is usually stated as angle from north (or south) and east (or west), e.g. N 36°52' E.

Two points are given in a local plane coordinate system (units in feet): Point A has coordinates (200, 400) and Point B has coordinates (500, 800). Assume (E, N) order: first value easting, second northing. Find: Horizontal distance from A to B. Bearing from A to B (direction from A toward B).

Point A: E = 200 ft, N = 400 ft Point B: E = 500 ft, N = 800 ft

Δ E = 500 - 200 = 300 ft, Δ N = 800 - 400 = 400 ft d = (E_{2} - E_{1})^{2} + (N_{2} - N_{1})^{2} d = 30 0^{2} + 40 0^{2} = 250, 000 = 500 ft

Δ E = 300 > 0 and Δ N = 400 > 0, so the direction is northeast. Angle from north: θ = arctan (300/400) = arctan (0.75) \approx 36.9° . θ = arctan (\frac{∣ E _{2} - E _{1} ∣}{∣ N _{2} - N _{1} ∣}) Bearing from A to B: N 36°54' E (or ~N 37° E).

(1) Horizontal distance: d = 500 ft . (2) Bearing from A to B: N 36°54' E.

d = (Δ E)^{2} + (Δ N)^{2}; bearing from Δ E, Δ N