3. Curve Equation

A constant grade change rate means the curve does not have a fixed radius - it's continually changing. This provides a smoother transition when vertical travel direction is changed. A circular arc isn't generally used as a vertical curve because it has constant curvature with a fixed radius. While acceptable at lower speeds, at higher speeds there is a tendency for a vehicle to "fly" at the highest point of a crest curve or "bottom out" at the lowest point of a sag curve. While desirable for roller coaster design, upsetting passenger stomachs is generally frowned upon in road design.

Instead, a parabolic arc is used. A parabola tends to flatten at the vertical direction change making for a more comfortable transition.

The general equation for a parabolic curve, Figure B-7, is Equation B-4.

parabola eqn 
 Equation B-4
Figure B-7


The equation consists of two parts, Figure B-8:

  • a straight line, bx+c, which is tangent to the curve at its beginning (b is the slope, c is the y intercept), and,
  • an offset, ax2, which is the vertical distance from the tangent to the curve.
Figure B-8
Tangent and Offset


Figure B-9 shows the parabolic arc in terms of vertical curve nomenclature.

Figure B-9
Curve Terms


The curve begins at the BVC station and elevation.
The tangent line slope is g1, the incoming grade.

At distance d:

  • the tangent elevation is g1d+ElevBVC
  • the tangent offset is ad2, where a is a function of k

The parabolic equation written in curve terms is the Curve Equation, Equation B-5.

CurveEqn       Equation B-5

(For a complete graphic derivation of the Curve Equation see the Family of Curves section.)

di is the distance from the BVC to any point i on the curve. It is computed from Equation B-6 and ranges from 0 to L.

dist eqn       Equation B-6

Recall the previous warning on grade and distance units. In Equation B-5 the combination should be either grades in percent with distances in stations, or grades as a ratio with distances in feet. Remember that curve length, L, is a distance and must play by the same rules.

The grade at any point on the curve is a function of the beginning grade along with k and the distance from the BVC, Equation B-7.

grade eqn       Equation B-7


Using Equations B-5 through B-7 allows us to compute the elevation and grade at any point along the curve.