cubic spiral

This spiral is a also a simple polynomial cubic.

The spiral forms the first order approximation of the spiral for which the curvature is linearly related to the path, the clothoid.
That's why the curve has been used in designing railways and some modern highways.
Railways are primarily constructed with straight lines and circular arcs, a transition curve makes a smooth transition between two curves with different curvature.

This use gave the curve the name of cubic spiral.

The spiral of the The American Railway Engineering Association, the A.R.E.A. spiral is almost the same as the cubic spiral. Instead of measuring the length along the spiral itself, the A.R.E.A. spiral is measured by 10 equal chords. The two curves concide up to 15 degrees.