Horizontal Curve Layout Calculator

What a simple circular curve is made of

A highway alignment is a sequence of straight tangents. Where two tangents meet, at the point of intersection or PI, a circular arc is inserted so that vehicles can change direction at speed. The arc leaves the back tangent at the PC (point of curvature) and rejoins the forward tangent at the PT (point of tangency), touching each tangent without a kink.

Because the radii to the PC and the PT are perpendicular to their tangents, the central angle of the arc is exactly the angle between the tangents. That single fact gives you everything else. The figure PI–PC–centre–PT is a kite with two right angles, so the two tangent distances are equal, and the whole curve is symmetric about the line from the PI to the centre.

Six numbers describe the curve. T tells you where to set the PC and PT relative to the PI. L is what goes in the stationing, because alignment stations follow the arc, not the tangents. LC is the straight-line distance between the two ends. M is the offset from the middle of that chord to the curve, and it is the number that governs sight distance past an obstruction on the inside of the curve. E is the offset from the PI to the curve, which tells you how far outside the intersection the pavement actually runs.

Where each formula comes from

Drop a line from the PI to the centre of the circle. It bisects the angle Δ, so it forms a right triangle with the radius and one tangent, containing the angle Δ/2 at the centre.

T = R·tan(Δ/2) is the opposite over adjacent in that triangle. Notice that the tangent function grows without limit as Δ approaches 180°, which is why a near-reversal cannot be made with a single curve of any usable radius.

L = R·Δ·π/180 is nothing but arc length. Δ must be converted to radians, which is where the π/180 comes from. If you are working in degree of curve on the arc definition, there is a much easier form: L = 100·Δ/D, because D is defined as the angle subtended by exactly 100 ft of arc.

LC = 2R·sin(Δ/2) is the chord of a central angle Δ, split into two right triangles of half-angle each.

M = R(1 − cos(Δ/2)) is the radius minus its projection onto the bisector. Because the cosine is nearly 1 for small angles, M is tiny on flat curves and grows quickly on sharp ones.

E = R(sec(Δ/2) − 1) is the hypotenuse from the centre to the PI minus the radius. E is always larger than M, and both approach infinity with T as Δ approaches 180°.

The degree of curve is an older way of stating sharpness that survives on highway and railroad plans. Under the arc definition, D is the central angle subtended by a 100 ft arc, so R = 100 × (360/2π) / D = 5729.578/D. Under the chord definition, used by railroads, D subtends a 100 ft chord, giving R = 50/sin(D/2). The two agree to within a foot on flat curves and diverge on sharp ones — 1,432.39 ft against 1,432.69 ft at D = 4°, but 286.48 ft against 287.94 ft at D = 20°.

Reading the numbers as a designer

Middle ordinate is the sight-distance number. A barrier, cut slope or building on the inside of a curve blocks the driver's line of sight. The relationship M = R(1 − cos(Δ/2)) applied to the sight-distance sub-arc rather than the whole curve gives the horizontal sightline offset, which AASHTO uses to set the clear zone inside a curve. If M for the sight sub-arc exceeds the available offset to the obstruction, the design speed cannot be met.

External distance tells you whether the curve fits. On a constrained corner — an urban intersection, a bridge approach, a corner of a parcel — E is the amount by which the pavement centreline runs outside the PI. Compare it to the available right of way before anything else.

Tangent distance decides whether adjacent curves collide. Two successive PIs must be far enough apart that the PT of the first curve precedes the PC of the second, with tangent between them for superelevation runoff. When the tangent lengths overlap, you need a compound or reverse curve rather than two simple ones, or a larger radius.

Curve length drives the stationing of everything downstream. Every station after the PT shifts by the tangent deficiency 2T − L. Set the PC and PT correctly and the rest of the alignment follows; get them wrong and every culvert, drive and utility crossing beyond the curve is misplaced by the same amount.

Radius has to satisfy the speed, not the other way round. The minimum radius for a design speed comes from R = V² / (15(e + f)) with V in mph, e the superelevation rate and f the side-friction factor. That relationship, not the geometry on this page, is what sets the smallest curve you are allowed to use.

Degree of curve and radius, both definitions

The two definitions differ by less than a tenth of a foot below D = 1° and by nearly a foot and a half at D = 20°. Always state which definition a plan uses before converting.

Errors that put a curve in the wrong place

• Computing the PT as PI + T. Stationing runs along the arc, so PT = PC + L. The difference 2T − L is 12.30 ft on the worked example and grows rapidly with Δ.
• Mixing the arc and chord definitions. A 20° curve differs by 1.46 ft in radius between the two, which is more than the tolerance on any staking.
• Forgetting that Δ is the tangent deflection, not the interior angle at the PI. If the field book records the interior angle I, then Δ = 180° − I.
• Using the long chord where the arc is required. Pavement quantities, guardrail lengths and stationing all follow L; only the straight-line tie between PC and PT is LC.
• Staking long chords on a sharp curve. The mid-ordinate of a sub-chord grows with the square of its length. On a 200 ft radius, a 100 ft sub-chord departs from the arc by about 3.1 ft at its midpoint, so tighten the interval to 25 ft.
• Ignoring the spiral. Most modern high-speed alignments insert a spiral transition between tangent and circular arc. The circular portion still uses these formulas, but T, PC and PT all shift, and the spiral has its own set of elements.
• Treating a right deflection and a left deflection the same in the field. The geometry is identical, but the deflection angles are turned in the opposite rotational sense from the back tangent.

Simple curves among the alternatives

A simple curve — one arc of one radius — is the base case and covers most low-speed and rural work. A compound curve joins two arcs of different radii turning the same way, used where terrain or right of way makes a single radius impossible. A reverse curve joins two arcs turning opposite ways; without a tangent between them, superelevation cannot be run off and drivers get a sudden steering reversal, so agencies restrict them. A spiral (clothoid) has a radius that decreases linearly with length, matching the path a driver actually steers and allowing superelevation to be introduced gradually.

Before any of these can be laid out you need the alignment directions, which arrive as bearings or azimuths from the traverse — the bearing to azimuth converter handles the notation, and the deflection angle Δ is simply the difference between the back-tangent and forward-tangent azimuths. Once the curve is staked, the same alignment drives earthwork quantities through a cut and fill computation, the utility trenches through a trench volume estimate, and the surfacing through an asphalt tonnage estimate once the paved area is known.

Vertical curves are the other half of alignment geometry and use a completely different shape: a parabola, not a circle, because it produces a constant rate of grade change. Do not carry the formulas on this page over to a crest or sag curve.

Key terms

PI — point of intersection

Where the two tangents would meet if extended. The curve does not pass through it; the external distance E measures how far short of it the curve runs.

PC and PT

Point of curvature and point of tangency: where the curve leaves the back tangent and rejoins the forward tangent. Also written BC/EC or TS/ST on spiralled alignments.

Station

Distance along the alignment, written in hundreds of feet: station 102+55.65 is 10,255.65 ft from the alignment origin. Metric alignments station in kilometres with the same plus notation.

Deflection angle (staking)

Half the central angle subtended by an arc, turned from the back tangent at the PC. It is what the instrument reads; the chord is what the tape or EDM measures.

Tangent deficiency

2T − L, the amount of stationing lost by cutting the corner. Every station beyond the PT shifts by this quantity.

Superelevation

The cross-slope banking of the roadway through a curve, expressed as a rate e. Together with the side-friction factor it sets the minimum radius for a design speed.