Staking the Curve the Crew Could Not Occupy

Consistency check (arc definition): Da = 5729.578 / R = 5729.578 / 500.00 = 11.4592° = 11°27'33". Matches the plan. (A chord-definition value for the same radius would be about 11°27'26" - not what the plan says.)

Elements:

• Tangent: T = R tan(Δ/2) = 500.00 × tan 24°15'00" = 500.00 × 0.450455 = 225.23 ft
• Length: L = R Δ(rad) = 500.00 × 0.846485 = 423.24 ft
• Long chord: LC = 2R sin(Δ/2) = 1000.00 × 0.410719 = 410.72 ft
• Middle ordinate: M = R(1 − cos(Δ/2)) = 500.00 × (1 − 0.911774) = 44.11 ft
• External: E = R(sec(Δ/2) − 1) = 500.00 × (1.096763 − 1) = 48.38 ft

Stations:

• PC = 12+34.56
• PI = PC + T = 1234.56 + 225.23 = 14+59.79
• PT = PC + L = 1234.56 + 423.24 = 16+57.80

Sanity check: T = 225.23 and E = 48.38 satisfy E = T tan(Δ/4) = 225.23 × 0.214855 = 48.39. The curve is short enough that a 50-ft station interval is appropriate; the final partial station carries 23.24 ft of arc.

Correct procedure: occupy the PC, backsight the tangent (the 200-ft point is fine), and set each station by turning the total deflection from the back tangent, not an incremental angle from the last chord - incremental turns accumulate instrument and pointing error with every setup. Set the PT independently by the long chord (410.72 ft) and check it by turning Δ/2 = 24°15'00" from the back tangent. Reserve the last station so the final chord is computed to the PT rather than chained from the previous station, preventing error from piling up at the closure.

Where the error came from - the chord/arc confusion. The plan's 50-ft stations are arc lengths. The chord subtending 50.00 ft of arc on a 500-ft radius is:

c = 2R sin(θ/2), θ = 50.00/500.00 rad = 0.10000 rad c = 1000.00 × sin(0.05000) = 49.979 ft

The crew chained 50.00 ft instead - 0.021 ft too long per full station. Over eight full stations that is about 0.17 ft of overrun, which explains most of the 0.21 ft at the PT. The remainder is ordinary taping and pointing error. The measured long chord, 410.83 ft against the computed 410.72 ft, is consistent: a slightly larger subtended angle gives a slightly longer chord, and 0.17 ft of curve is about 0.17 ft of chord at this flat curvature.

The closing check the crew skipped: eight stations at 2°51'53" each plus 1°19'54" for the 23.24-ft final arc should total 24°14'58" ≈ Δ/2. Turning that deflection at the PT would have exposed the error in the field.

Do not accept it - but keep it in proportion.

Why it matters here:

• The error is systematic, not random. Every intermediate station is roughly 0.02 ft per station off, growing to 0.21 ft at the PT, so the staked alignment is not the designed alignment anywhere - it is progressively wrong. That is different from a single random 0.2-ft shot.
• The alignment controls the pavement edge, shoulder, ditch, and any structure set from it. The equipment shed that swallowed the PI was presumably sited from the design; a centerline shifted 0.2 ft is small for an earthwork quantity but not for a structure tie, a guardrail, or a utility crossing, which are typically held to ±0.05 to ±0.10 ft.
• "The ROW still fits" is not the acceptance criterion. The plans are the acceptance criterion.

Next steps:

1. Compute the coordinates of the PC, PT, and each station from the plan geometry in the project coordinate system.
2. Stake the PT from the PC by the long chord (410.72 ft), turn Δ/2 = 24°15'00" as an independent check, and confirm the PT against the computed coordinates from the project control.
3. Restake intermediate stations by total deflection from the back tangent, chaining the true chords (49.979 ft for full stations), with the final partial chord computed to the PT.
4. Verify instrument and EDM setup - prism constant, atmospheric correction, and the backsight distance - before blaming the geometry.
5. Document the original staking, the discovery, the correction, and the as-staked coordinates, so the contractor and the county have a defensible record of what was actually set.