This curve calculator solves the two curves that shape every road, railway and pipe alignment. The horizontal curve calculator returns the tangent, arc length, long chord, mid-ordinate, external distance and degree of curve, plus the PC and PT stations. The vertical curve tab returns the algebraic grade change, the K value, the PVC and PVT stations and elevations, and the location of the high or low point.
The deflection angle Δ is the angle between the two tangents, which equals twice the deflection angle measured from the PC. Decimal degrees and DMS both work.
A horizontal curve is described by a small set of interlocking elements. The deflection angle Δ is the total change of direction between the two tangents, and the radius R fixes the size of the circle. From those two the rest follow: the tangent distance T = R tan(Δ/2) is how far back from the intersection point the curve begins, the arc length L = R Δ (with Δ in radians) is the distance along the curve, the long chord LC = 2R sin(Δ/2) is the straight line from PC to PT, the mid-ordinate M = R(1 − cos(Δ/2)) is how far the curve bulges away from that chord, and the external distance E = R(sec(Δ/2) − 1) is the gap between the curve and the intersection point.
The degree of curve is an alternative to the radius. Under the arc definition it is the central angle subtended by a 100 m or 100 ft arc, so D = 5729.577951 / R. Under the chord definition, used historically in North American practice, it is the angle subtended by a 100 ft chord, so D = 2 asin(50 / R). The two give different radii for the same D, which is a common source of confusion when a drawing does not state which definition applies.
A vertical curve is a parabola that joins two grades smoothly. The algebraic grade change A = g2 − g1 sets the shape: a positive A gives a sag curve with a low point, a negative A a crest with a high point. The K value, K = L / |A|, is the horizontal distance needed to change grade by one percent, and it is the number highway design standards are written against because it also governs sight distance. The turning point sits at x = −g1 L / A measured from the PVC, inside the curve whenever that value falls between zero and L.
A circular curve cannot be joined straight to a tangent without a kink in the alignment, so road and rail design inserts a transition between the two. The classic form is the clothoid, better known as the Euler spiral or simply a spiral curve, in which curvature grows linearly with distance travelled so that lateral acceleration builds at a constant rate. A spiral curve calculator normally works from the design speed and a permissible rate of change of centripetal acceleration, or from the superelevation runoff length, and returns the minimum transition length together with the spiral angle and the shift needed to set the curve out.
The spiral transition has three practical effects. It shifts the circular arc inwards by the shift distance, so a slightly larger radius is needed to hold the same clearance; it shortens the tangent at each end by the spiral contribution; and it moves the start and end of curvature outwards from the PC and PT to the TS (tangent to spiral) and ST (spiral to tangent) stations. The horizontal curve elements above describe the central circular portion only, so if a drawing gives TS and ST stations rather than PC and PT the alignment is spiralled, and the tangent distance from the PI is the circular value plus the spiral contribution.
Superelevation is the crossfall built into the carriageway through a horizontal curve so that
part of the centripetal force is carried by the road surface instead of entirely by the tyres. The
design relation is e + f = V² / (127 R) in metric units, where e is the
superelevation rate, f the assumed side friction factor, V the design speed in km/h and
R the radius in metres; the US customary form uses 15 in place of 127 with V in mph and R in feet. A
superelevation calculator solves that relation for whichever quantity is unknown — the
crossfall needed for a given radius and design speed, the maximum speed a sharp curve can carry, or
the smallest radius that is acceptable at a chosen e and f.
Where the split between e and f falls is a matter of policy rather than arithmetic, but the values used in most standards are a maximum superelevation of 6 to 8 percent with a side friction factor between 0.10 and 0.15 depending on design speed. The crossfall must also be introduced gradually over a runoff length long enough that the outer edge does not rise faster than drivers can follow, and it is often that runoff requirement, rather than comfort alone, that fixes the minimum spiral length. Because a crossfall is simply a slope expressed as a percentage, the slope percentage calculator is a quick way to move between a crossfall, a rise and run, and a grade in degrees.
A station equation is a deliberate break in the stationing, used when a design changes length
without the physical alignment moving — a realignment, a re-survey or a change of origin. It is
written as back = ahead: the station measured from the start of the earlier survey is the
back station, the value in the new chainage is the ahead station, and when the ahead
value is the larger of the two the stationing gains length at that point. A stationing
calculator exists for exactly this bookkeeping, because a drawing that omits the equation is one
of the commonest sources of setting-out blunders.
Once an equation is in play a station number on its own no longer identifies a unique point, so the station must always be quoted together with the side of the equation it falls on. The curve tab above computes PC and PT stations by subtracting the tangent distance from the PI station and adding the arc length back, which assumes continuous stationing; where a project carries an equation, apply the offset first and only then compare the result with the drawing. Turning a chainage and an alignment geometry into coordinates is COGO work rather than curve work, which is what the traverse calculator covers.
Curve elements are needed at every stage of an alignment, from concept through to setting out:
The tangent distance is T equals R times the tangent of half the deflection angle, where the angle is in degrees and the tangent is taken of the angle in radians. For a radius of 300 m and a deflection angle of 60 degrees, T is 300 times tan(30 degrees), which is 173.205 m.
Under the arc definition the degree of curve is the central angle subtended by a 100 m or 100 ft arc, so D equals 5729.577951 divided by the radius. Under the chord definition it is the angle subtended by a 100 ft chord, so D equals two times the arc sine of 50 divided by the radius. For the same D the chord definition gives a slightly different radius, so always check which definition a drawing uses.
Measure from the PVC along the curve a distance of minus g1 times L divided by A, where g1 is the entry grade in percent, L is the curve length and A is the algebraic grade change g2 minus g1. If that distance falls between 0 and L the turning point is inside the curve; otherwise the curve never turns and the elevation is monotonic. For example with g1 of minus 2 percent, g2 of 3 percent and L of 300 m, the low point is 120 m from the PVC.
K is the curve length divided by the algebraic grade change, so it expresses how many metres of curve are needed to change grade by one percent. A larger K means a flatter, more comfortable curve. Design standards such as those for stopping sight distance are usually written as a minimum K for each design speed, so K is the value to compare against the standard.
No. At a deflection angle of 180 degrees the two tangents are parallel and opposite, the tangent length becomes infinite and no circular curve can join them. Practical curve design keeps the deflection angle well below 180 degrees.
Yes. Enter a station as a full station-format value such as 12+34.500, or as a plain number in the same units as the curve, and the results are returned in the same station format.
Long enough to satisfy two separate limits: the superelevation must be run off at a rate the driver can follow comfortably, and the rate of change of centripetal acceleration must stay within the value the standard allows for the design speed. The longer of the two governs. Rates around 0.3 to 0.6 m per second cubed are typical, and a transition is often also quoted as a length per unit of speed, such as 0.5 to 1.0 metres of spiral for each km/h of design speed.
Between 6 and 8 percent in most standards for roads where snow and ice are expected, rising to 10 percent or occasionally 12 percent in warm, dry regions where a slippery surface is less of a concern. The limit exists because at a high crossfall a slow vehicle can slide down the curve, so the design speed assumed for the curve sets how much of the centripetal force may be met by crossfall rather than by side friction.
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