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 B7, is Equation B6.


Figure B7 Parabola 
The equation consists of two parts, Figure B8:
 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, ax^{2}, which is the vertical distance from the tangent to the curve.
Figure B8 Tangent and Offset 
Figure B9 shows the parabolic arc in terms of vertical curve nomenclature.
Figure B9 Curve Terms 
The curve begins at the BVC station and elevation.
The tangent line slope is g_{1}, the incoming grade.
At distance d:
 the tangent elevation is g_{1}d+Elev_{BVC}
 the tangent offset is ad^{2}, where a is a function of k
The parabolic equation written in curve terms is the Curve Equation, Equation B7.
Equation B7 
(For a complete graphic derivation of the Curve Equation see the Family of Curves section.)
d_{i} is the distance from the BVC to any point i on the curve. It is computed from Equation B8 and ranges from 0 to L.
Equation B8 
Recall the previous warning on grade and distance units. In Equation B5 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 B9.
Equation B9 
Using Equations B7 through B9 allows us to compute the elevation and grade at any point along the curve. To compute the elevation of the curve's midpoint, L/2 would be used for d_{i} in Equation B7.
A geometric characteristic of a parabolic vertical curve is at its midpoint the vertical offsets from the PVI to the curve and from the curve to the chord, o in Figure B10, are the same.
Figure B10 Midcurve Offsets 
Besides using Equation B7, another way to compute the midpoint elevation is from the following:
 The chord's midpoint elevation is the average elevation of the BVC and EVC.
 The curve's midpoint elevation is the average elevation of the chord midpoint and PVI.
The curve midpoint generally isn't any more or less important than other curve points. Being able to compute its elevation two ways, however, can be used as a partial math check