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In-Depth Information
make it will definitely influence the curve. As an example, consider the first three
approaches in the context of a uniformly spaced spline curve that interpolates points
on an arc of a circle (we shall show how our results about spline
functions
applies
to spline
curves
shortly). One would like the curvature of the curve to be approxi-
mately constant since that holds for the circle. What one finds is the following (see
[Fari97]): With the correct choice of start and end tangents, the clamped end condi-
tion approach has the best curvatures, the Bessel end condition approach is the next
best, and the natural end condition approach is the worst because it forces the biggest
deviations from constant curvature near the endpoints.
For more on spline interpolation see [Beac91], [Fari97], or [RogA90]. Here we
only sketch the solution to the spline interpolation problem for clamped end
conditions.
Let us assume for the moment that we know the slopes m
i
= p
i
¢(x
i
) for i = 1,2,
..., n - 1. Assume further that we also know the slopes m
0
and m
n
at x
0
and x
n
, respec-
tively. Because the slopes are known, equation (11.21) defines the polynomials p
i
(x).
A simple computation using equation (11.21) shows that
(
)
-
(
)
D
xm m
+
2
D
y
2
3
DD
D
yxmm
x
-
2
+
i
i
i
+
1
i
i
i
i
i
+
1
¢¢
()
=
(
)
+
px
6
xx
-
,
i
i
3
2
D
x
i
i
where Dx
i
= x
i+1
- x
i
, and Dy
i
= y
i+1
- y
i
. Setting p
i
≤(x
i
) equal to p
i-1
≤(x
i
) leads to the
equation
(
)
(
)
2
2
d
m
+
2
d
+
d
m
+
d
m
=
3
d
D
y
+
d
D
y
,
i
-
1
i
-
1
i
-
1
i
i
i
i
+
1
i
-
1
i
i
-
1
i
for i = 0,1,..., n - 1, where d
i
= 1/Dx
i
. The matrix form of this system of equations is
(
)
-
2
2
m
m
3
d
D
y
+
d
D
y
d
m
Ê
ˆ
Ê
ˆ
1
0
1
0
0
0
1
Á
Á
Á
Á
Á
Á
(
)
˜
˜
˜
˜
˜
˜
Á
Á
Á
Á
Á
Á
˜
˜
˜
˜
˜
˜
2
2
3
d
D
y
+
d
D
y
2
1
2
1
2
.
.
.
.
S
=
,
(11.30)
n
(
)
2
2
m
m
3
d
D
y
+
d
D
y
n
-
2
n
-
3
n
-
2
n
-
3
n
-
2
(
)
-
Ë
¯
Ë
2
2
¯
3
d
D
y
+
d
D
n
y
d
m
n
-
1
n
-
2
-
1
n
-
1
n
n
-
2
n
-
1
where S
n
is the (n - 1) ¥ (n - 1) matrix
2
(
dd
+
)
d
0
...
...
...
0
0
Ê
ˆ
0
1
1
Á
Á
Á
Á
Á
Á
Á
˜
˜
˜
˜
˜
˜
˜
˜
˜
d
2
(
d
+
d
)
d
0
0
1
1
2
2
0
d
2
(
d
+
d
)
0
0
2
2
3
.
.
.
...
.
.
S
n
=
.
(11.31)
.
.
.
...
.
.
0
0
0
...
...
...
d
0
Á
Á
n
-
3
0
0
0
2
(
d
+
d
)
d
n
-
3
n
-
2
n
-
2
Ë
¯
0
0
0
d
2
(
d
+
d
)
n
-
2
n
-
2
n
-
1
