## Background:

Vertical curves for roads are often designed under certain
conditions which lead to the curves having a vertical, parabolic
cross-section. In laying out these curves surveyors use four
design parameters to determine the coefficients a,b and c.
Looking at the diagram these four values can be explained. The
first one is the elevation above the datum line at BVC, the
beginning of the vertical curve; the second is the grade (G1) of
the road at BVC i.e. the slope of the tangent to the parabola at
BVC; the third is the grade (G2) of the road at EVC, the end of
the vertical curve; and the last is the horizontal length (L) of
the curve from BVC to EVC. G1 and G2 are usually expressed as %
grades. These grades are often posted at the roadside at the
beginning of steep hills. The conditions under which the curves
are designed are:

- The rate of change of the slope of the curve over the
length of the curve must be constant.
- The slope at the beginning of the curve must be the same as
the slope of the road section joining the curve.
- The elevation of the road at the beginning of the curve
must be the same as the elevation of the road section to which
it is joined. For ease of notation the elevation will be named
BVC.

## Problem:

Find the equation of the curve given these three
conditions.

## Solution:

Consider the first condition. The rate of change of slope of a
curve is given by

In this case the slopes at two points are known and the
distance between the points is also known. The slope of the curve
at BVC is G1 and the slope of the
curve at EVC is G2. The change in
the slope between these points is G2 - G1 and the
average rate of change of slope is found by dividing the change
by the distance between the points. But this rate of change is
constant by the design criteria so we can set up the
relation:

Remember that grades are % grades. Integrating gives

=

The second condition tells us that at x = 0 the slope of the
curve is the same as the slope of the section of road joining the
curve. So, at x = 0

Therefore:

which gives

So,

Integrating gives

=

The third condition shows that, at x = 0, y = BVC.
Substituting in the equation gives K2 = BVC.

So the equation of the vertical curve is a parabola of the
form:

Actually this result is the starting point of another application
in our table of examples, namely, how linear algebra is used in
surveying