formula.expert

Deflection Angle to a Point on a Circular Curve

Also known as deflection angle stakeout · curve staking deflection · chord stakeout angle · deflection per station · laying out a horizontal curve · inscribed angle curve layout

δ=ℓ2R\delta = \frac{\ell}{2R}

Worked example: R = 200 m, 20 m of arc → delta = 2.8648 deg — press Try an example to run it live, then adjust anything.

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See also
Horizontal Curve Long Chord — The same half-angle geometry taken all the way to the PT — and the formula for the chord you tape once this angle is turned.Horizontal Curve Length from Degree of Curve — The total arc L this arc distance ℓ is a running part of, and the Δ the deflections must add up to half of.Radius from Degree of Curve (Arc Definition) — Where R comes from when the plan gives a degree of curve instead of a radius.

Learning zone

A horizontal curve is not staked by measuring out to its centre — on a 600 m radius that point is in somebody's field. It is staked from the PC, with the instrument sighting back down the tangent and every point on the curve turned off that one reference line. The angle to turn comes from the inscribed-angle theorem: the angle between a tangent and a chord is half the central angle the chord subtends. A point ℓ of arc along the curve subtends a central angle of ℓ/R, so the deflection to it is δ = ℓ/2R. On a 400 m radius curve, the first full 20 m station sits at 20/800 = 0.025 rad, which is 1°25′57″.

The method carries its own proof, which is why field crews trust it. Run the deflections all the way out to the PT, where ℓ equals the full curve length L, and δ must come to exactly Δ/2 — half the intersection angle read off the plan. If the last deflection does not close on that figure, the curve data is wrong before a single stake goes in the ground. What the formula does not give you is the distance to tape. Deflections are computed from arc length; the chainman measures the straight chord between consecutive stakes, which is shorter by 2R sin δ against 2Rδ. On a flat curve with 20 m subchords the difference is under a millimetre and nobody cares; on a tight ramp or a rail turnout it is real, and the old field tables of chord corrections exist for exactly that reason.

Deflection Angle to a Point on a Circular Curve
δ=ℓ2R\delta = \frac{\ell}{2R}
Where
  • δ\delta= Deflection angle (°)
  • ℓ\ell= Arc distance from the PC (m)
  • RR= Curve radius (m)
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