Graphics Reference
In-Depth Information
2
¢¢
()
+¢
()
fx
b
Ú
dx
.
(11.116)
52
a
(
)
2
1
fx
Here we have used the facts that
¢¢
()
+¢
()
fx
=+¢
()
2
()
=
ds
1
f
x dx
and
k
x
,
s
32
(
)
2
1
fx
where k
s
(x) is the signed curvature function. (They follow from equation (9.2) and
Proposition 9.3.4 in [AgoM05]). Assuming that f¢(x) is very small, one can drop the
denominator in (11.116) and conclude that
b
Ú
fx x
¢¢
()
2
(11.117)
a
is a good approximation to the integral. An easy application of the calculus of varia-
tions shows that the function f(x) that minimizes the integral in (11.117) must satisfy
f
(4)
(x) = 0, so that it is a cubic polynomial, that is, a cubic spline. On the other hand,
without the simplifying hypothesis that f¢(x) is small, the integral in (11.116) is harder
to solve. The interested reader should consult [Mehl74], [Malc77], or [HosL93].
Next, consider a wooden spline F(s) defined by equation (11.115). These curves
can also be approximated by cubic splines because for graphs of functions y = f(x),
the signed curvature k
s
(x) can be approximated by f≤(x) and so, like for mechanical
splines, they are approximated by functions satisfying f
(4)
(x) = 0. Integrating equation
(11.115) shows that
()
=+,
k
s
s sb
(11.118)
for some constants a and b. From this we could already guess at the shape of such a
curve. Its curvature increases with s and hence would have to spiral in on itself like
the spring of a clock. To get an actual formula, let q(s) be the turning angle function
for F(s). We know (see Chapter 9 in [AgoM04]) that
d
ds
q
()
=
k
s
s
(11.119)
and
(
)
s
s
()
=
Ú
()
Ú
()
Fs
cos
q
sdsc
+
,
sin
q
sdsd
+
,
(11.120)
0
0
where c and d are constants. Integrating equation (11.119) and using (11.118) implies
that
1
2
s
s
()
=
Ú
()
Ú
(
)
2
q
s
k
s ds
+=
q
as
+
b ds
+=
q
as
++
bs
q
,
(11.121)
s
0
0
0
0
0
