… or – How to convert a curve into an angle?

In the track design handbook we have a too complex formula that refers to “degree of curvature”.

Before discussing about the equivalent gradient, let’s see first what this is – what is the degree of curvature?

This measure is used in United States in road and railway design engineering as an alternative measure to radius and curvature.

The degree of curvature is the angle at the centre D° that corresponds to a conventional arc length, A. This length is 100 ft.(30.48m).

Bigger the radius, smaller the degree of curvature:

To calculate D° we consider the full circle. D° will fit n times into 360° and A also n times in the circumference, 2πR. This n, obviously, is not necessarily an integer.

From this we can write the three equations below:

And, if we divide (1) by (3), we will get (4):

From it we get the equation for D°, the degree of curvature:

In feet, that becomes:

and in metres, is this:

These are a few values of D° and R, relevant for a future post on how to convert a curve into a gradient …