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us that the strip will assume a shape that minimizes the strain. The equation for this
minimum energy problem is difficult to solve directly, however, an approximation to
it can be solved and leads to a solution that is a cubic spline. We shall explain this a
little more in Section 11.6. At any rate, it is this ability of cubic splines to model curves
from physical splines, plus the fact that their low degree makes them easy to compute,
that makes them the most popular spline by far.
The spline interpolation problem:
Given an integer k and real numbers x
i
and y
i
, i = 0,
..., n, with x
0
< x
1
< ...< x
n
, find a spline g(x) of order k so that the x
i
are the knots for g
and g(x
i
) = y
i
.
11.2.3.1
Theorem.
The spline interpolation problem has a solution.
Proof.
For a general solution see [BaBB87] or [deBo78]. One can also prove this
using B-splines, which are discussed later. See [RogA90]. Here we shall only show the
existence of a solution in the important special case of a cubic spline. We rephrase
the problem as follows: Given points (x
0
,y
0
), (x
1
,y
1
),..., and (x
n
,y
n
), find cubic poly-
nomials p
i
(x), so that for i = 0,1,..., n - 1,
()
=
(
px
y
,
i
i
i
)
=
px
y
,
i
i
+
1
i
+
1
and for i = 1,2,..., n - 1,
¢
()
=¢
()
¢¢
()
= ¢
()
px
p
x
,
.
i
i
i
-
1
i
px
p
x
i
i
i
-
1
i
In this situation we have 4n degrees of freedom and only 4n - 2 constraints. The two
extra degrees of freedom can be handled in several ways depending on how we choose
to specify m
0
and m
n
, the slope at the beginning and end of the spline, respectively.
We mention four approaches, but there are others.
End condition choices for interpolating splines:
(1) (
Clamped end condition
) We can specify the slopes m
0
and m
n
explicitly.
(2) (
Bessel end condition
) We can let m
0
and m
n
be the end slope of the interpo-
lating parabola for the first, respectively, last three data points.
(3) (
Natural end condition
) We can require that the second derivative of the spline
vanishes at the ends. This amounts to requiring zero curvature of the spline at the
ends and is closer to what happens in the case of a physical spline. The spline will act
like a straight line near its endpoints. This type of spline is called a
natural spline
.
(4) (
Periodic end condition
) We require that the value of the spline and the value
of its first and second derivative are the same at both endpoints. This is of interest
mainly in the context of closed spline curves.
In the first approach we are obligated to specify the end conditions ourselves, whereas
in the other approaches it is done automatically for us. No matter what choice we
