GPS Trilateration (2D): Least-Squares Fix & Dilution of Precision

Satellite Constellation

Satellite count (N) 6

Geometry spread (sky arc) 360°

Measurement

Range noise (1σ) 15 m

Fix rate 5 Hz

Live Readout

Fixes taken

DOP (this geometry)

–

Current fix error

0.0 m

RMS error (last 60)

0.0 m

Predicted 1σ (DOP·σ)

0.0 m

GN iterations

–

How it works

Each of N satellites sits at a known position S_i. It reports a noisy range r_i = |P_true − S_i| + ε_i, ε_i ~ N(0, σ²). The receiver position is not read off circle intersections visually — it is solved for numerically by nonlinear least squares (Gauss–Newton), minimizing Σ(|P − S_i| − r_i)²:

residual_i(P) = |P − S_i| − r_i
J_i = (P − S_i) / |P − S_i|        (unit line-of-sight vector)
ΔP  = −(JᵀJ)⁻¹ Jᵀ·residual          (solved each Gauss–Newton step)
P  ← P + ΔP,  repeat to convergence

This is exactly the 2-unknown (x, y) analogue of real GPS, which solves 4 unknowns (x, y, z, clock bias) from ≥4 satellites — dropping the clock-bias term is the one simplification made here so the geometry stays visible in 2D.

Dilution of precision (DOP) quantifies how satellite geometry amplifies range error into position error. With unit line-of-sight vectors forming matrix H, DOP = √tr((HᵀH)⁻¹). Spread satellites across the whole sky (360° arc) and their sight-lines are nearly orthogonal — DOP stays near 1. Cram them into a narrow arc and the sight-lines become nearly parallel, HᵀH becomes ill-conditioned, and DOP rockets up — the same σ now produces a much bigger position error, even though nothing about the noise changed.

Scatter cloud — the last ~150 independent fixes, plotted relative to the true position. Its spread and shape visualize the error covariance; the drawn ellipse is the theoretical 1σ covariance ellipse (DOP·σ scaled), the dots are the empirical fixes — they should match.
Error strip — position error per fix over time, with the predicted DOP·σ bound as a reference line.

True position Estimated (least-squares) fix Measured range circle

Click the sky to move the receiver