Overview

Vertical curves are used to provide a smooth transition between grade lines of a highway or railroad. An equal-tangent parabolic curve is illustrated in Fig. 2-A. It has the property that the vertex (V) is midway between the beginning of the vertical curve (BVC) and its end (EVC) measured horizontally. Parabolic curves can be calculated by two different procedures: Tangent-offset method or Chord-gradient method.

Figure 2-A

Fig. 2-A shows: the center of a parabola is midway between the vertex and the long chord. Parabolic curve has the property that offsets from a tangent to a parabola are proportional to the squares of the horizontal distances from the point of tangency (See Eq. 2-1).

1. Tangent offset

   

   Calculate the tangent offset of Table 2-1.

   

   Table 2-1: Equal-Tangent Vertical Curve

   Station	Point	Tangent Elevation	Tangent Offset	Elevation of Curve
   11+00	BVC	1264	0.00	1264.00
   11+50		1266	0.19	1265.81
   12+00		1268	0.75	1267.25
   12+50		1270	1.69	1268.31
   13+00	V	1272	3.00	1269.00
   13+50		1271	1.69	1269.31
   14+00		1270	0.75	1269.25
   14+50		1269	0.19	1268.81
   15+00	EVC	1268	0.00	1268.00

2. 

   From Fig. 2-A,

3. External distance

   From Fig. 2-A,