Parabolic Curves

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Parabolic curves find there usage when we have to connect sections of highways or railroads of differing gradient. The use of a parabolic curve provides a gradual change in direction along the curve.

where
PVC= point of vertical curvature, beginning of curve
PVI= point of vertical intersection of grades on either side of curve
PVT= point of vertical tangency, end of curve
G₁= grade at beginning of curve, ft/ft (m/m)
G₂= grade at end of curve, ft/ft (m/m)
L=length of curve, ft (m)
R =rate of change of grade, ft/ft2 (m/m2)
V =elevation of PVI, ft (m)

E_o= elevation of PVC, ft (m)
E_t= elevation of PVT, ft (m)
x =distance of any point on the curve from the
PVC, ft (m)
E_x= elevation of point x distant from PVC, ft (m)
x_s= distance from PVC to lowest point on a
sag curve or highest point on a summit curve, ft (m)
E_s= elevation of lowest point on a sag curve or highest point on a summit curve, ft (m)

Equations of Parabolic Curves

Upward grades are positive and downward grades are negative.

R=(G₂-G₁)/L

E_o=V-LG₁/2

E_x=E_o+G₁x +Rx²/2

x_s1/R

_Eso-G1²/2R

_{For xs>L:The curve has no high or low point}

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