A simple circular curve has seven named elements and exactly two degrees of freedom. Give any two of radius, central angle, tangent distance, arc length, long chord, middle ordinate and external distance, and the other five follow with no room for judgement. That is the whole subject, and every curve problem you will meet is an exercise in noticing which two you were handed.
The reason curve questions still go wrong is not the trigonometry. It is stationing. The curve is shorter than the two tangents it replaces, so chainage does not run through the point of intersection, and the single most common error in the whole topic is computing the station of the PT by adding the tangent distance to the PI.
Below is one curve carried all the way through: the elements, the stations, the deflection table it is staked from, and the field checks that catch a mistake before it becomes a stake in the wrong place.
The elements, and why they all scale together
Every element of a simple curve is the radius multiplied by a function of the central angle alone. That is worth stating explicitly, because it means the shape of a curve is fixed by Δ and only its size is fixed by R.
T = R · tan(Δ/2) L = R · Δ (Δ in radians)
C = 2R · sin(Δ/2) M = R · (1 − cos(Δ/2))
E = R · (sec(Δ/2) − 1)- Δ, the central angle
- Also the deflection angle between the back tangent and the forward tangent at the PI. The two are the same angle, which is why a plan sheet can label it once and expect you to use it twice.
- Degree of curve
- The central angle subtended by a 100-unit length of the curve. Under the arc definition that length is measured along the arc, giving R = 5,729.578 ⁄ D. Under the chord definition it is measured along a 100-unit chord, giving a slightly different radius. US highway plans still label curves by D rather than R.
The worked curve
Take a highway curve labelled on the plan as D = 5°00′ by the arc definition, with a deflection at the PI of Δ = 24°30′ and the PI at station 45+67.20. The arc definition gives the radius immediately.
| Element | Symbol | Value (ft) |
|---|---|---|
| Radius | R | 1,145.92 |
| Central angle | Δ | 24°30′00″ |
| Tangent distance | T | 248.80 |
| Arc length | L | 490.00 |
| Long chord | C | 486.28 |
| Middle ordinate | M | 26.09 |
| External distance | E | 26.70 |
| Degree of curve (chord definition) | D𝑐 | 5°00′06″ |
Two of those values are worth pausing on. The arc length is exactly 490.00 ft, which is not a coincidence: under the arc definition a curve of D degrees turns one degree every 100 ⁄ D feet, so a 24.5° deflection at D = 5° gives 100 × 24.5 ⁄ 5 = 490 ft of arc exactly. And the chord-definition degree of curve for the same radius is 5°00′06″, not 5°00′ - the two definitions agree to about a tenth of a percent at this radius and diverge as the curve tightens.
Stationing, and the trap
Stationing runs along the alignment. Before the curve that alignment is the back tangent, through the curve it is the arc, and after it the forward tangent. So the PC is found by backing up the tangent distance from the PI, and the PT is found by running the arc length forward from the PC.
PC = PI − T = 4567.20 − 248.80 = 4318.40 → 43+18.40
PT = PC + L = 4318.40 + 490.00 = 4808.40 → 48+08.40The discrepancy is 2T − L, here 7.61 ft, and it is the chainage that disappears because the curve cuts the corner. On an alignment with a dozen curves the accumulated difference is substantial, and it is why an alignment is stationed once, continuously, rather than reconstructed curve by curve.
The deflection table
A curve is staked from the PC by turning deflection angles from the back tangent and measuring chords. The deflection to a point l along the arc is l ⁄ (2R) radians - half the central angle it subtends - and that halving is the entire theory of the method.
| Station | Arc from PC (ft) | Deflection | Chord from PC (ft) | Chord from previous (ft) |
|---|---|---|---|---|
| 44+00.00 | 81.60 | 2°02′24″ | 81.59 | 81.59 |
| 45+00.00 | 181.60 | 4°32′24″ | 181.41 | 99.97 |
| 46+00.00 | 281.60 | 7°02′24″ | 280.89 | 99.97 |
| 47+00.00 | 381.60 | 9°32′24″ | 379.84 | 99.97 |
| 48+00.00 | 481.60 | 12°02′24″ | 478.07 | 99.97 |
| 48+08.40 (PT) | 490.00 | 12°15′00″ | 486.28 | 8.40 |
Two checks fall out of that table and both should be made before anyone leaves the PC. The total deflection at the PT is 12°15′00″, which is exactly Δ ⁄ 2; if it is not, either Δ or the arc length is wrong. And the chord from the PC to the PT is 486.28 ft, which is the long chord C computed independently from the elements. A curve that satisfies both is staked correctly.
Notice also that the sub-chord between full stations is 99.97 ft rather than 100.00 ft. The 100 ft is arc; the 99.97 ft is the straight line across it. Taping 100.00 ft between stations on a 5° curve puts every point progressively further out, and the error is cumulative rather than random.
Solving from any two elements
Plan sheets are inconsistent about which pair they give you. When Δ is known, every other element yields R by a single division. When R is known and one shape element is given, the shape function has to be inverted - sometimes in closed form, sometimes numerically. When two shape elements are given, their ratio depends on Δ alone, which is the fact that makes that case solvable at all.
- R and Δ: everything follows directly. The easiest case and the one most plans give.
- T and Δ, or L and Δ, or C, M or E with Δ: divide by the shape function to get R, then proceed as above.
- R and M: Δ = 2 · arccos(1 − M ⁄ R), which is the one non-obvious inversion worth memorising, because middle ordinate is what you can measure on the ground with a tape.
- T and L, or any other pair of shape elements: their ratio is a function of Δ alone. There is no general closed form, so solve numerically and confirm the answer reproduces both inputs.
- D and one element: convert D to R first, being explicit about arc or chord definition, then treat it as the R case.
Whichever pair you start from, finish by recomputing an element you were not given and checking it against the plan. A curve is over-determined once you have two elements, and that redundancy is free error-checking that costs one extra line of arithmetic.