Curve staking
Stake a circular curve by tangent offsets and by chord offsets
A 700 ft radius curve is laid out with a tape alone: first by rectangular offsets from the back tangent, then by successive offsets from the prolonged chord, with the approximate and exact formulas compared.
Given
- A simple circular curve with R = 700.00 ft and Δ = 22°00′00″, curving to the right.
- Station of the PC = 18+40.00.
- The site is flat and open. No total station is available; the curve is to be set with a tape and a right-angle prism.
- Stakes are wanted at every full station and at the PT.
- All distances are US survey feet.
Required
- The arc length and the station of the PT.
- A table of tangent distances and tangent offsets for each staked point, measured from the PC along the back tangent.
- The equivalent chord-offset values for the alternative method.
- A comparison of the exact tangent offset with the common approximation.
Work it through yourself before reading on — the solution below shows every step, so there is no way to skim it without giving the answer away.
Worked solution
Elements and stationing
The elements follow from R and Δ as usual. The tangent distance is needed here not for stationing but because it bounds how far the offsets can run: the last tangent distance must be less than T, or the point being set lies beyond the PI.
The PT station is the PC station plus the arc length, giving a final sub-arc of 8.78 ft.
Δ/2 = 11°00′00″
T = 700.00 × tan(11°00′00″) = 700.00 × 0.194380 = 136.07 ft
L = 700.00 × 0.383972 = 268.78 ft
C = 1400.00 × sin(11°00′00″) = 267.13 ft
PT = 18+40.00 + 268.78 = 21+08.78
Sub-arcs: 60.00 ft to 19+00.00, then 100.00, 100.00, and 8.78 ft to the PTGeometry of the tangent-offset method
Set the zero of the tape at the PC and run it along the back tangent. For a point that is an arc distance l around the curve, the deflection from the tangent is δ = l/(2R) and the chord from the PC is c = 2R sin δ. Resolving that chord into components along and perpendicular to the tangent gives the two numbers the tape needs.
The tangent distance x is measured along the tangent from the PC; the offset y is measured at right angles from that point to the curve. Because the offset is perpendicular to the tangent and not to the chord, the pair (x, y) is exact, not an approximation.
δ = l / (2R) = l / 1400.00 radians
c = 2R sin δ = 1400.00 sin δ
x = c cos δ (distance along the tangent)
y = c sin δ (offset perpendicular to the tangent)
equivalently y = R − √(R² − x²)Tangent-offset table
Each row is computed from its own arc distance; the values do not accumulate, so an error in one row does not propagate into the next. That independence is the main practical advantage of the method.
The last column shows the widely quoted approximation y ≈ x²/(2R). It is close for short offsets and degrades quickly. At station 19+00 it is right to 0.00 ft; at the PT it is 1.85 ft short, which would put the stake well off the curve.
| Station | Arc from PC (ft) | Deflection δ | Chord from PC (ft) | Tangent distance x (ft) | Offset y (ft) | Approximation x²/2R (ft) |
|---|---|---|---|---|---|---|
| 19+00.00 | 60.00 | 2°27′20″ | 59.98 | 59.93 | 2.57 | 2.57 |
| 20+00.00 | 160.00 | 6°32′53″ | 159.65 | 158.61 | 18.21 | 17.97 |
| 21+00.00 | 260.00 | 10°38′26″ | 258.51 | 254.06 | 47.73 | 46.11 |
| 21+08.78 (PT) | 268.78 | 11°00′00″ | 267.13 | 262.22 | 50.97 | 49.12 |
The chord-offset alternative
When the tangent runs out of usable ground, the curve can be carried forward from chord to chord instead. Set the first point by a tangent offset from the PC, then prolong each chord by its own length and swing the next offset from the end of that prolongation.
The first offset, taken from the tangent, is c²/(2R). Every offset after that, taken from a prolonged chord, is c²/R, twice as large, because the prolonged chord is already deflected by one chord angle where the tangent was not. Getting that factor of two wrong is the classic error in this method.
First sub-chord (60.00 ft arc): c₁ = 1400.00 sin(2°27′20″) = 59.98 ft
first tangent offset = c₁²/(2R) = 3597.60 / 1400.00 = 2.57 ft
Full-station chord: c = 2R sin(l/2R) = 1400.00 sin(4°05′33″) = 99.91 ft
successive chord offset = c²/R = 9982.01 / 700.00 = 14.26 ft
Last sub-chord (8.78 ft arc): c₄ = 8.78 ft, offset = c₄²/R = 0.11 ftChoosing between them in the field
Tangent offsets are self-checking and independent, but they need clear ground alongside the tangent for the full 262 ft and offsets that reach almost 51 ft. If a fence line or a cut slope is in the way, the method dies.
Chord offsets need only a short working strip that follows the curve, but the errors accumulate: an error in the first point carries into every point after it. Use them for the far end of a long curve, and always close on an independently set PT.
Answer
- L = 268.78 ft, PT = 21+08.78.
- Tangent distances and offsets from the PC: 59.93 and 2.57 ft at 19+00.00; 158.61 and 18.21 ft at 20+00.00; 254.06 and 47.73 ft at 21+00.00; 262.22 and 50.97 ft at the PT.
- Chord-offset values: first offset from the tangent 2.57 ft on a 59.98 ft sub-chord; then 14.26 ft on each 99.91 ft full-station chord; then 0.11 ft on the closing 8.78 ft sub-chord.
- The approximation y = x²/2R is adequate only for the first stake; by the PT it is 1.85 ft in error.
Check
Every offset must satisfy the circle equation y = R − √(R² − x²). At x = 158.61 ft: 700.00 − √(490,000 − 25,157.1) = 700.00 − 681.79 = 18.21 ft, which is the tabulated value.
At the PT the tangent distance and offset must reproduce the long chord: √(262.22² + 50.97²) = √(68,759.3 + 2597.9) = 267.13 ft, equal to C = 2R sin(Δ/2) = 267.13 ft.
The PT can be located from the tangent distance alone: x = T(1 + cos Δ) = 136.07 × 1.927184 = 262.22 ft and y = T sin Δ = 136.07 × 0.374607 = 50.97 ft, both reproducing the last table row from elements that never entered it.
Chord-offset cross-check: the sum of the sub-chords 59.98 + 99.91 + 99.91 + 8.78 = 268.58 ft must fall just short of the arc length 268.78 ft. It does, by 0.20 ft.
More curve staking
All in this category- Stake a simple circular curve by deflection angles from the PCA 1150.00 ft radius curve with a 28°42′00″ central angle is solved for every element, stationed through the PI, and taken to a full deflection-angle and chord table for staking from the PC.
- Recover a curve radius from a measured long chord and middle ordinateAn existing curve with no plan record is measured with a tape: the long chord between two identifiable points and the middle ordinate at its center. Radius and central angle follow in closed form.
- Fit a horizontal curve to a required external distanceAn obstruction near the PI forces the curve to stay at least 65.00 ft out from the intersection point. The minimum radius is solved from the external distance, rounded to a design value, and stationed.