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I'm Daniel Finley with the Institute for transportation research and education at NC State University I'm going to discuss the use of the parabolic equation for vertical curves on a roadway I'll cover the basics of vertical curves how the parabolic equation fits into the design of vertical curves and then we'll go over an example to illustrate these concepts vertical curves are used to provide a smooth transition from one grade to another the tangents represent grades which can either be flat uphill or downhill the typical vertical curve is a symmetric parabolic curve whose shape is defined by the parabolic equation the information required to fully define a vertical curve is the elevation of the beginning of the curve the grades of the two tangents that are connected and the length of the curve the entrance grade is known as G 1 and the exit grade is known as G 2 a vertical curve is depicted in a diagram such as this a section through the center line of the road shown in the profile view the horizontal stationing axis remains on a plane that is station points do not follow the elevated or depressed centerline and distances are measured in horizontal distances this graphic shows the profile view of a vertical alignment it illustrates the path that a vehicle would take when driving along this alignment there are two general types of curves a crest curve that has a concave down shape and a SAG curve which has a concave up shape for a symmetric parabolic curve the length of the curve is evenly divided on each side of the PVI in other words the distance from the PVC to the PVI is half of the length of the curve and the distance from the PVI to the PBT is also half of the length of the curve the term a represents the algebraic difference in grades of G 1 and G 2 the P VI is the point of vertical intersection the point where the two tangents meet the PVC is the point of vertical curvature the point at which the curve begins the Pvt is the point of vertical tangency the point at which the curve ends the term ex is used to denote any horizontal distance along the curve we will use X to calculate the elevation of the curve at a point X is bounded by zero and the length of the curve the letter y is used to note the elevation of a point along the curve at the distance X an important note is that the grades G 1 and G 2 are measured in percent and the distances X and L are measured in stations where one station equals 100 feet this terminology will become meaningful when we start the calculations mathematically any point on a parabolic curve can be described using the equation y equals ax squared plus BX plus C in the general parabolic equation a and B...