Civil Engineering Reference
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Y
B
g
e
V
e
g
s
p
y
X
x
A
0
X
/2
X
/2
X
Figure 18.11
Vertical parabola and its local coordinate system.
where:
g
denotes grade
subscripts
s
and
e
denote starting and ending of parabola, respectively
X
is the total horizontal length of parabola
x
is the curve horizontal ordinate
Taking the starting point as the origin of the local coordinate system as
shown in Figure 18.11, by integrating Equation 18.6, the ordinate
y
of point
on curve at
x
can be obtained as
g
−
2
g
X
e
s
y
g x
x
2
=
+
(18.7)
s
In addition to connecting grades,
g
s
and
g
e
, the horizontal length for the
transition
X
is a critical characteristic of a vertical parabola as it controls
the rate of grade change. Considering that the vertical line at
x
1 2
passing through point
V
as shown in Figure 18.11, the relationship between
the external distance
e
and
X
can be written as
=
(
)
X
8
e
X
=
(18.8)
g
−
g
e
s
18.4 curve and surface tessellatIon
When showing a 3D road curve that contains spirals/arcs in plane and
parabolas in vertical curve on screen, curve has to be subdivided into small
straight lines or arc segments as computer graphics technologies cannot
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