Graphics Reference
In-Depth Information
The polynomials L
i,n
are called the
Lagrange basis functions
with respect to x
0
,
x
1
,...,x
n
. It is easy to check that
()
=d
Lx
,
(11.3)
in
,
j
ij
so that the nth degree polynomial
n
Â
()
=
()
px
yL
x
(11.4)
iin
,
i
=
0
satisfies the interpolatory conditions. This proves the existence.
To prove uniqueness assume that q(x) is another nth degree polynomial satisfy-
ing this condition, then the nth degree polynomial h(x) = p(x) - q(x) has n + 1 roots
x
0
, x
1
,..., x
n
. This is impossible unless h(x) is the zero polynomial, since a nontriv-
ial polynomial of degree n can have at most n roots (Corollary E.5.4 in [AgoM05]).
The theorem is proved.
Note that the Lagrange basis functions satisfy the equation
n
Â
0
,
()
=
Lx
in
1
(11.5)
i
=
for all x. This is so because the equation is trivially true for x = x
j
, and a nontrivial
nth degree polynomial can take on the same value at most n times.
The discussion above can be applied to interpolating points in
R
m
with a
parametric curve because it is simply a case of applying formula (11.4) to the m
components of the points separately. In other words, given distinct real numbers u
0
,
u
1
,..., u
n
and points
p
0
,
p
1
,...,
p
n
in
R
m
, the function
n
Â
()
=
(
)
pu
L
uu u
;
,
,...,
u
p
(11.6)
in
,
01
n
i
i
=
0
is the
unique
polynomial curve of degree n which interpolates the points
p
i
at the
values u
i
. It is called the
Lagrange interpolating polynomial curve
. Because cubic curves
will be of special interest to us throughout this chapter, we look at the formulas for
this case in detail. To begin with
(
)
(
)
(
)
uuuu uu
uuuuuu
-
-
-
1
2
3
()
=
Lu
,
03
,
(
)
(
)
(
)
-
-
-
0
1
0
2
0
3
(
)
(
)
(
)
uu uu uu
uuuuuu
-
-
-
0
2
3
()
=
Lu
,
13
,
(
)
(
)
(
)
-
-
-
1
0
1
2
1
3
(
)
(
)
(
)
u
uuuuuu
uu uuu
-
-
-
0
1
3
()
=
Lu
,
and
23
,
(
)
(
)
(
)
-
-
-
2
0
2
1
2
3
(
)
(
)
(
)
uu uuuu
uuuuuu
-
-
-
0
1
2
()
=
(11.7)
Lu
.
33
,
(
)
(
)
(
)
-
-
-
3
0
3
1
3
2
