Two-Bearing Resection

Shoot a bearing to a known feature and reverse it: that back azimuth is a line FROM the known feature THROUGH you. Do it to a second known feature and you have two lines, and where they cross is your position. The method works identically with a compass on a quadrangle and with a total station on control monuments, and the only care it needs is the declination correction when the bearings are magnetic. The strength-of-fix output is what keeps it honest. When the two lines cross near square the fix is sharp; when they cross at a shallow angle a small bearing error slides the intersection a long way along the lines, because the positional error scales as one over the sine of the intersection angle. Under about thirty degrees the fix should not be trusted, and the answer is a third known point -- whose three back-lines will not meet at a point but will form a small triangle, the cocked hat, whose size is the honest statement of how good the fix is. From known points at (1000, 5000) and (3000, 5400), azimuths of 315 and 45 degrees give back azimuths of 135 and 225, which cross square at (1800, 4200). Move so both features lie nearly in the same direction and the same one-degree uncertainty is worth three or four times as much ground. A field method; the survey of record and a checked closure govern anything that matters.

back azimuth = observed azimuth + declination + 180 (mod 360); solve (Ax + t sin backA, Ay + t cos backA) = (Bx + s sin backB, By + s cos backB); positional error scales as 1 / sin(intersection angle).

Two-bearing resection by back-azimuth intersection, by name -- public plane surveying and land-navigation practice. The strength of fix is reported as the intersection angle, whose sine divides the positional error; under about 30 degrees a third known point is required. A field method; the survey of record and a checked closure govern.

The intersection is plane trigonometry on the user's own observed bearings and published control coordinates.

Estimate. AHJ and licensed professional govern.

Field names used by the API: ax, ay, azimuth_to_a_deg, bx, by, azimuth_to_b_deg, declination_deg, east, north, distance_a_ft, distance_b_ft, intersection_angle_deg

• Back azimuth the observed azimuth plus 180 degrees, after the declination correction; it is a line FROM the known point THROUGH the occupied oneplane surveying practice
• Strength of fix positional error scales as 1 / sin(intersection angle); below about 30 degrees, use a third pointland-navigation practice
• Cocked hat three back-lines form a small triangle rather than meeting at a point; its size states the fix qualityland-navigation practice