What this calculator does
Versines are how curves get measured on the ground. Stretch a chord between two points on the rail, measure the offset from the chord to the rail at the midpoint, and you have the versine — the single number a Hallade survey records at every station along a curve. This calculator turns that reading into a curve radius, and converts it to the versine the same curve would read on a different chord length, which is what you need whenever survey records, design drawings and standards disagree about chord conventions.
The formula
For a circular arc, the exact mid-ordinate of a chord C on radius R is v = R − √(R² − C²/4). On railway curves the chord is tiny next to the radius, and that collapses to the relationship every permanent way engineer carries around:
v = C² / (8R) → R = C² / (8v)
Working in the units the field actually uses — chord in metres, versine in millimetres — the constant folds in:
R(m) = 125 × C(m)² / v(mm)
The approximation is not a meaningful source of error here. On a 20 m chord at 5000 m radius it differs from the exact mid-ordinate by about one part in ten million; even at 150 m radius it is inside 0.1%. Your tape and your rail wear will dominate long before the algebra does.
Chord length is not a detail
The single most common versine mistake is comparing readings taken on different chords. Versine goes with the square of the chord, so the same curve reads four times larger on a 20 m chord than on a 10 m one. A 5000 m radius shows 10 mm on a 20 m chord and only 2.5 mm on a 10 m chord — and 2.5 mm is close enough to the noise floor of a hand measurement to be worthless.
Chord conventions vary by region and by purpose: 20 m is the common Hallade chord in metric practice, 10 m chords appear in tight yard and siding work, and North American track charts are built around a 62 ft chord because on that chord the versine in inches is numerically close to the degree of curve. Whenever you move a number between a survey record, a design drawing and a standard, check which chord it belongs to. That is what the target chord field on this tool is for.
Note also the difference between overlapping and end-to-end chords. A Hallade survey normally uses overlapping chords — each station’s chord spans the stations either side of it — so the versines form a continuous string that can be integrated to give the alignment. Readings taken on discrete, non-overlapping chords describe individual arcs but cannot be strung together the same way.
Degree of curve
The tool also reports the degree of curve on the arc definition — the central angle subtended by a 100 ft arc, which is the North American way of naming a curve:
D = 5729.578 / R(ft) = 1746.375 / R(m)
There is also a chord definition (the angle subtended by a 100 ft chord), used mainly in older highway practice. The two agree closely on flat curves and diverge on sharp ones; if you are reconciling with a historic track chart, confirm which definition it uses.
Worked example
A track inspector measures a 10 mm versine on a 20 m chord.
R = 125 × 20² / 10 = 125 × 400 / 10 = 5000 m
The curve is 5000 m radius. On a 10 m chord the same curve would read 125 × 10² / 5000 = 2.50 mm — which is why nobody surveys a 5000 m curve on a 10 m chord. As a degree of curve it is 1746.375 / 5000 ≈ 0.349°, a very flat curve by North American reckoning.
As a field sanity check, the offsets at the quarter points of that same 20 m chord should read about 0.75 × 10 = 7.50 mm. If they do not, the chord is not lying against a uniform circular arc — you are probably measuring across a transition, or across a geometry defect.
FAQ
Which rail do I measure against? Convention is the gauge face of the high (outer) rail, at a consistent height below the rail head, and consistently the same rail for the whole survey. Consistency matters more than the choice.
Can I use this on a transition? Only to spot one. Through a transition the radius is changing continuously, so a single versine describes an average over the chord rather than a real radius. In a Hallade survey the transition shows up as versines ramping steadily between the straight (zero) and the circular curve value — that ramp is the signature you look for.
My versines vary station to station on a “circular” curve. Is the curve wrong? Some scatter is normal — measurement error, rail wear, and genuine geometry variation all contribute. What matters is whether the trend is flat. A consistent drift means the radius really is changing; a single outlier usually means a local defect or a measurement error at that station.
Does this account for cant? No. Versine is a horizontal alignment measurement. If you are measuring against a canted curve, keep the measurement in the plane of the rails and consistent between stations; cant affects the crosslevel record, not the versine string.
This tool provides indicative figures for preliminary design and checking. Track geometry acceptance must be assessed against the standards applicable to your network.