Curve staking
Degree of curve on plan data: arc definition against chord definition
A plan labels a curve D = 3°30′00″ with Δ = 24°15′00″ but does not say which definition of degree of curve applies. Both radii are computed and the two sets of elements compared.
Given
- An old highway plan sheet labels a horizontal curve D = 3°30′00″, Δ = 24°15′00″.
- The title block does not state whether the degree of curve is by the arc definition or the chord definition.
- The alignment is stationed in 100 ft stations. All distances are US survey feet.
- Arc definition: D is the central angle subtended by a 100 ft arc.
- Chord definition: D is the central angle subtended by a 100 ft chord.
Required
- The radius implied by each definition.
- The full set of curve elements under each definition.
- The size of the discrepancy, and whether it matters for this curve.
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
Radius from the arc definition
Under the arc definition a 100 ft arc subtends D. Arc length equals radius times central angle in radians, so 100 = R D(rad), and the radius is a fixed constant divided by D in degrees.
That constant, 100 × 180/π = 5729.5780, is worth memorising. It is the whole of the arc definition.
100 = R × D(rad)
R = 100 / D(rad) = 100 × (180/π) / D(deg)
R(arc) = 5729.5780 / 3.5 = 1637.02 ftRadius from the chord definition
Under the chord definition a 100 ft chord subtends D. Half the chord is R sin(D/2), so the radius comes from a sine rather than from the arc constant.
The chord definition always gives the larger radius, because a chord is shorter than the arc it subtends: to make a 100 ft chord subtend the same angle as a 100 ft arc, the circle must be slightly bigger.
50 = R sin(D/2)
R(chord) = 50 / sin(1°45′00″) = 50 / 0.030539 = 1637.28 ft
Difference: 1637.28 − 1637.02 = 0.26 ft, about 16 parts per 100,000Elements under each definition
Both curves have the same Δ = 24°15′00″, so both use the same trigonometric functions of the half angle 12°07′30″. Only the radius differs, and every element scales with it.
The arc-length row deserves care. Under the arc definition the curve length carried in the stationing is the true arc, and it is exactly 100Δ/D = 100 × 24.25 / 3.5 = 692.86 ft. Under the chord definition the length carried in the stationing is conventionally the same 100Δ/D figure, made up of nominal 100 ft chords, while the true arc on that slightly larger circle is 692.96 ft.
| Element | Arc definition (R = 1637.02 ft) | Chord definition (R = 1637.28 ft) | Difference (ft) |
|---|---|---|---|
| Tangent T | 351.69 | 351.75 | 0.06 |
| True arc length | 692.86 | 692.96 | 0.10 |
| Stationed length 100Δ/D | 692.86 | 692.86 | 0.00 |
| Long chord C | 687.70 | 687.80 | 0.10 |
| Middle ordinate M | 36.52 | 36.52 | 0.00 |
| External E | 37.35 | 37.36 | 0.01 |
Which definition is on the plan
Three practical tests settle it. First, the era and the agency: railroads and older highway departments used the chord definition; most modern highway work uses the arc definition. Second, the plan data itself, if any two elements are tabulated: compute D from the tabulated radius both ways and see which one reproduces the printed D. Third, the recovered monumentation: at 0.26 ft of radius difference this curve will not distinguish the two, but a sharp curve will.
The sensitivity scales roughly with the cube of D. At D = 3°30′00″ the two radii differ by 0.26 ft; at D = 10° they differ by 6.1 ft; at D = 20° by 49 ft. On flat highway curves the definition is a bookkeeping question. On sharp curves it is a boundary question.
D(chord) implied by R = 1637.02 ft: 2 arcsin(50/1637.02) = 3°30′02″
D(arc) implied by R = 1637.28 ft: 5729.5780/1637.28 = 3°29′58″Effect on the field work
For staking this curve the choice changes the tangent distance by 0.06 ft and the long chord by 0.10 ft. Neither is detectable with the tolerances of a construction stake, so either radius will build the same road.
For retracement the choice changes the position of the radius point by 0.26 ft, and it changes the arc length used to compute the PT station by 0.10 ft. Over a string of curves that accumulates. Record on the computation sheet which definition was assumed.
Answer
- Arc definition: R = 1637.02 ft.
- Chord definition: R = 1637.28 ft.
- The two radii differ by 0.26 ft; the tangent distances differ by 0.06 ft and the long chords by 0.10 ft.
- Under either definition the stationed curve length is 100Δ/D = 692.86 ft; the true arc on the chord-definition circle is 692.96 ft.
Check
Run each radius back through the other definition. R = 1637.02 ft gives a chord-definition degree of 2 arcsin(50/1637.02) = 3°30′02″, and R = 1637.28 ft gives an arc-definition degree of 5729.5780/1637.28 = 3°29′58″. Both land within a few seconds of 3°30′00″, confirming the two radii are the two readings of the same label.
The arc-definition arc length can be obtained twice: R Δ(rad) = 1637.02 × 0.423242 = 692.86 ft, and 100Δ/D = 100 × 24.25/3.5 = 692.86 ft. Agreement is exact, because both are the same identity written differently.
Sanity check on the ordering: R(chord) must exceed R(arc) for any D. It does, by 0.26 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.