The Beacon Geometry

Ada audits the beacon geometry — one position solve, three residual checks

foundations
math-foundations
ble
indoor-positioning
intermediate
Ada ADA · CALCULATION AUDIT

Foundations · optional mathematics and physics

The Beacon Geometry

The positioning simulator fixes beacons at (0, 0), (5, 0), and (0, 5) and feeds three RSSI-derived distances of 1.78 m, 4.47 m, and 2.82 m into circle algebra, landing a best-fit point near (0.82, 2.02) with an average residual of just 0.28 m. Yet at a fixed 2 m range the chapter’s own RSSI can swing the estimate from 1.4 m to 5.0 m. This audit reproduces the solve and asks whether that tidy point can be trusted when the RSSI spread dwarfs the 0.28 m residual.

Companion to the chapter Indoor Positioning with BLE Beacons — every number here comes from that chapter.

One position solve, three residual checks, ~4 minutes

The physics problem is not the geometry; it is the radio evidence feeding the geometry. Before trusting the point estimate, reproduce the algebra and then ask how much the RSSI-derived distances can move.

Position from the three chapter distances

The simulator starts with beacons at (0, 0), (5, 0), and (0, 5), with distances 1.78 m, 4.47 m, and 2.82 m.

d1^2 = 1.78^2 = 3.1684; d2^2 = 4.47^2 = 19.9809; d3^2 = 2.82^2 = 7.9524
x = (d1^2 - d2^2 + 25) / 10 = (3.1684 - 19.9809 + 25) / 10 = 0.81875 m
y = (d1^2 - d3^2 + 25) / 10 = (3.1684 - 7.9524 + 25) / 10 = 2.0216 m

Residuals and RSSI spread

  • Beacon 1 residual: |sqrt(0.81875^2 + 2.0216^2) - 1.78| = 0.40 m.
  • Beacon 2 residual: |sqrt((0.81875 - 5)^2 + 2.0216^2) - 4.47| = 0.17 m.
  • Beacon 3 residual: |sqrt(0.81875^2 + (2.0216 - 5)^2) - 2.82| = 0.27 m.
  • Average residual: (0.40 + 0.17 + 0.27) / 3 = 0.28 m, which is good only for these three supplied distances.
  • The chapter's fixed-distance RSSI example spans 5.0 m - 1.4 m = 3.6 m; relative to the real 2 m point, the near estimate is 0.6 m low and the far estimate is 3.0 m high.

The calculation explains why the point looks neat while the claim remains approximate. Clean circle algebra gives a best-fit location near (0.82 m, 2.02 m), but the chapter's own RSSI spread is much larger than the residual for the sample distances, so the release record should keep uncertainty with the position.

Every number above is taken from the chapter's own beacon-geometry example and re-derived step by step.