Graphics Reference
In-Depth Information
Figure 11.31.
A clothoid or Cornu spiral.
for some constant q
0
. Substituting this into equation (11.120) gives us our general
wooden spline F(s). There are well-known special cases.
Definition.
The planar curve
2
2
Ê
Á
Ê
Á
x
ˆ
˜
Ê
Á
x
ˆ
˜
ˆ
˜
s
s
()
=
Ú
cos
Ú
Fs
dx
,
sin
dx
2
2
0
0
is called a
clothoid
or
Cornu spiral
. The curve
n
+
1
n
+
1
Ê
Á
Ê
Á
x
n
ˆ
˜
Ê
Á
x
n
ˆ
˜
ˆ
˜
s
s
()
=
Ú
cos
Ú
Fs
dx
,
sin
dx
+
1
+
1
0
0
is called a
generalized clothoid
or
Cornu spiral
.
Figure 11.31 shows the clothoid. The integrals in the definition are Fresnel inte-
grals and since
2
2
Ê
Á
x
ˆ
˜
Ê
Á
x
ˆ
˜
p
±•
±•
Ú
Ú
cos
dx
=
sin
dx
=±
2
2
2
0
0
we see the the clothoid converges to the two points
pp
22
±
Ê
ˆ
˜
,
.
Á
It is easy to show that the generalized clothoid has signed curvature function
()
=
n
k
s
ss
.
Clothoids play an important role in the construction of freeways and railroad
tracks. As an example for why this might be so, note that the curve corresponding to
an exit ramp for a highway needs to start off with zero curvature and then reach some
