Vertical curves are used by engineers in road design. According to The University Transportation Center for Alabama (UTCA), second order equations are used to describe vertical curves: these equations can be solved for both station and elevation for either sag vertical curves (which look similar to a concave lens) or crest vertical curves (which look similar to a convex lens). The procedure for calculating both is the same, UCTA states, except that each has a different minimum length that needs to be taken into account when computing the curve.

Calculate A, the percentage difference in grades, by subtracting the smaller grade from the largest. For example, if grade one is 5 per cent and grade two is 15 per cent, then 15 per cent - 5 per cent =10 per cent .

Compute the required vertical length. According to King Fahd University (KFU), this is obtained by multiplying the absolute value of A (from step 1) by K, the rate of vertical curvature. K is usually obtained from pre-defined tables, according to the type of curve and speed of the road. If K is 50, then vertical length = 50*10=500m.

Calculate the station of VPC (the vertical point of curvature) and VPT (the vertical point of tangency) using the following equations:

VPC=VPI-1/2L VPT=VPI+1/2L

The University of Idaho states that the VPI is the vertical point of intersection--the place where the two roads would have intersected if they had been allowed to meet instead of being diverted with the vertical curve.

Calculate the distance from VPC to the maximum and minimum elevations using the following equation: x=-(LG1)/A

Calculate how high VPC and VPT high. KFU states that G1, G2, and L should be used in this calculation.

Use the equation y=a(xsquared)+bx+c to calculate the curve elevation at each mandatory station. That's according to KFU, who states that:

y=curve elevation a=(G2-G1)/200L b=g1/100 c="elevation of VPC" x="horizontal distance from VPC to station of interest."