Graphics Programs Reference
In-Depth Information
y
Curve fitting
Interpolation
Figure 3.1.
Interpolation and curve fitting of data.
Data points
x
One meansofobtaining this polynomial is the
formula of Lagrange
n
P
n
−
1
(
x
)
=
y
i
i
(
x
)
(3.1a)
=
i
1
where
x
−
x
1
x
−
x
2
x
−
x
i
−
1
x
−
x
i
+
1
x
−
x
n
i
(
x
)
=
x
1
·
x
2
···
x
i
−
1
·
x
i
+
1
···
x
i
−
x
i
−
x
i
−
x
i
−
x
i
−
x
n
n
x
−
x
j
=
x
j
,
i
=
1
,
2
,...,
n
(3.1b)
x
i
−
j
=
1
j
=
i
arecalled the
cardinal functions
.
For example, if
n
=
2, the interpolant is the straight line
P
1
(
x
)
=
y
1
1
(
x
)
+
y
2
2
(
x
),
where
x
−
x
2
x
−
x
1
1
(
x
)
=
2
(
x
)
=
x
1
−
x
2
x
2
−
x
1
=
=
y
1
1
(
x
)
+
y
2
2
(
x
)
+
y
3
3
(
x
), where now
With
n
3, interpolationis parabolic:
P
2
(
x
)
(
x
−
x
2
)(
x
−
x
3
)
1
(
x
)
=
(
x
1
−
x
2
)(
x
1
−
x
3
)
(
x
−
x
1
)(
x
−
x
3
)
2
(
x
)
=
(
x
2
−
x
1
)(
x
2
−
x
3
)
−
−
(
x
x
1
)(
x
x
2
)
3
(
x
)
=
(
x
3
−
x
1
)(
x
3
−
x
2
)
The cardinalfunctions are polynomials of degree
n
−
1 and have the property
0
if
i
=
j
i
(
x
j
)
=
=
δ
i j
(3.2)
=
1 f
i
j
δ
i j
is the Kroneckerdelta. This propertyis illustrated in Fig. 3.2 for three-point
interpolation (
n
where
=
3) with
x
1
=
0,
x
2
=
2 and
x
3
=
3
.
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