Least-squares network adjustment (2D)

Adjusts a small horizontal network of distances, angles, and direction sets by weighted least squares, with adjusted coordinates, standard errors, 95% error ellipses, the reference variance, and a chi-square test.

Experimental: not yet fully verified. How results are checked

Computing your own values needs JavaScript and WebAssembly. The worked example below was computed when this page was built, and the documentation under "How we got this" applies.

0.346

The reference variance is 0.346; the chi-square test passed.

Adjusted points
Point	Northing	Easting	Northing standard error	Easting standard error	95% ellipse semi-major axis	95% ellipse semi-minor axis	Ellipse orientation
P	1,300.0046 m	1,200.0110 m	0.0023 m	0.0028 m	0.0175 m	0.0144 m	90.00 deg

Residuals
Observation	Residual	Standardized residual
distance A–P	-0.002 m	-1.29
distance B–P	0.0018 m	1.19
angle B–A–P	2″	0.81
angle P–B–A	-2.4″	-0.97

Degrees of freedom
Chi-square test (95%)
Chi-square
95% bounds
Iterations
Largest standardized residuals

Provenance

Computed by: survey.cogo.least-squares-2d 1.0.0, core 0.1.0
Model: Weighted Gauss-Newton least squares (weights 1/σ²) on linearized distance, angle, and direction equations, each direction set with its own orientation unknown. Reference variance vᵀWv / r; two-tailed χ² test at 95%; 95% error ellipses scaled by √(2 F(0.05; 2, r)), or by 2.4477 without redundancy
Accuracy: Converges to the least-squares solution for well-conditioned networks with approximate coordinates near the truth. Experimental: check it against your adjustment software before relying on it
Notes: 1 shown with the answer
Cites: Ghilani, C. D., and Wolf, P. R., Pearson, Elementary Surveying: An Introduction to Geomatics

Something look off?

Your values

Showing an example. Change anything.

Control points Held fixed: name, northing, easting, like BM1, 5000, 5000 · one per line: point, northing, easting Points to adjust With approximate coordinates, like P1, 5400, 5100 · one per line: point, northing, easting Distancesoptional From, to, horizontal distance, and standard deviation, like BM1, P1, 400.12, 0.01 · one per line: from, to, horizontal distance, standard deviation Anglesoptional Backsight, station, foresight, angle, and standard deviation, like BM2, BM1, P1, 91-15-20, 5 · one per line: backsight, station, foresight, angle, standard deviation

More options 1

Direction setsoptional Station, target, circle reading, and standard deviation, like A, P1, 45-10-30, 3 · one per line: station, target, direction, standard deviation

How we got thisFormula, worked example, sources, and proof

Model: Weighted Gauss-Newton least squares (weights 1/σ²) on linearized distance, angle, and direction equations, each direction set with its own orientation unknown. Reference variance vᵀWv / r; two-tailed χ² test at 95%; 95% error ellipses scaled by √(2 F(0.05; 2, r)), or by 2.4477 without redundancy

Show your work

Degrees of freedom
$observations − unknowns$
4 − 2 = 2
Reference variance
Σ v²/σ² ÷ degrees of freedom
0.692 ÷ 2 = 0.346

The same steps an agent gets from the MCP server with explain: true.

Accuracy: Converges to the least-squares solution for well-conditioned networks with approximate coordinates near the truth. Experimental: check it against your adjustment software before relying on it

Worked example: One new point from two control points. Source: Weighted least squares after Ghilani and Wolf (2021), chapter 16.

You enter

Angles: B, A, P, 303-41-26, 5 P, B, A, 303-41-20, 5
Control points: A, 1000 m, 1000 m B, 1000 m, 1400 m
Distances: A, P, 360.567 m, 0.005 m B, P, 360.551 m, 0.005 m
Points to adjust: P, 1300 m, 1200 m

You get

Reference variance: 0.346
Degrees of freedom: 2
Chi-square test (95%): passed
Chi-square: 0.692
95% bounds: 0.051 to 7.378
Iterations: 2
Largest standardized residuals: distance A–P (-1.29); distance B–P (1.19); angle P–B–A (-0.97)

Review: Not yet independently reviewed by a licensed surveyor.

Last verified: 2026-09-23, when a maintainer last confirmed this tool's sources at the issuer. See the sources ledger.

Status: version 1.0.0, core 0.1.0. See this tool in the verification report.

Changes

2026-09-23, fixed: Survey tools cite Ghilani and Wolf's Elementary Surveying, 16th edition (2021), but their method notes said "Ghilani & Wolf 2018". The notes now give 2021, the edition cited. No results change. Changelog

Checked against: 6 golden test vectors (download the test vectors, each with its source and tolerance). See how results are checked and every source.

Sources

Elementary Surveying: An Introduction to Geomatics, Ghilani, C. D., and Wolf, P. R., Pearson, 16th edition. Chapters 10 (traverse computations), 12 (area), 24 (horizontal curves), 25 (vertical curves), 26 (volumes).

Experimental means this tool has not yet met the stable bar: at least 20 golden vectors, differential tests, and an independent worked example.

Learn the concept: Traverse closure and precision explained