Length selection

The usual distance used to compute degree of curvature in North American road work is 100 feet (30.5 m) of arc.^[2] Conversely, North American railroad work traditionally used 100 feet of chord, which is used in other places for road work. Other lengths may be used—such as 100 metres (330 ft) where SI is favoured or a shorter length for sharper curves. Where degree of curvature is based on 100 units of arc length, the conversion between degree of curvature and radius is Dr = 18000/π ≈ 5729.57795, where D is degree and r is radius.

Since rail routes have very large radii, they are laid out in chords, as the difference to the arc is inconsequential; this made work easier before electronic calculators became available.

The 100 feet (30.48 m) is called a station, used to define length along a road or other alignment, annotated as stations plus feet 1+00, 2+00, etc. Metric work may use similar notation, such as kilometers plus meters 1+000.

Formula from arc length

${\displaystyle r={\frac {180^{\circ }A}{\pi D_{\text{C}}}}}$

where ${\displaystyle A}$ is arc length, ${\displaystyle r}$ is radius of curvature, and ${\displaystyle D_{\text{C}}}$ is degree of curvature, arc definition

Substitute deflection angle for degree of curvature or make arc length equal to 100 feet.

Formula from chord length

${\displaystyle r={\frac {C}{2\sin \left({\frac {D_{\text{C}}}{2}}\right)}}}$

where ${\displaystyle C}$ is chord length, ${\displaystyle r}$ is radius of curvature and ${\displaystyle D_{\text{C}}}$ is degree of curvature, chord definition

Formula from radius

${\displaystyle D_{\text{C}}=5729.58/r}$

Example

As an example, a curve with an arc length of 600 units that has an overall sweep of 6 degrees is a 1-degree curve: For every 100 feet of arc, the bearing changes by 1 degree. The radius of such a curve is 5729.57795. If the chord definition is used, each 100-unit chord length will sweep 1 degree with a radius of 5729.651 units, and the chord of the whole curve will be slightly shorter than 600 units.