Graphics Reference
In-Depth Information
Equation (11.21) in the proof of Theorem 11.2.2.2 shows us that the interpolat-
ing polynomials p
i
(x) can be expanded in the following way:
()
=
()
+
()
()
()
px
fxy fxy
+
fxm f xm
+
1
,
(11.22)
i
1
i
2
i
+
1
3
i
4
i
+
where the functions f
j
(x) (which depend on i) are defined by
(
)
3
2
(
() () () ()
)
=-
(
)
(
)
(
)
(
)
fxfxfxfx
xx
xx
-
xx
-
1
M
D
x
.
(11.23)
1
2
3
4
i
i
i
h
i
The interesting property that we want to record here is that
()
+
()
=
fx f x
1
(11.24)
1
2
for all x. Compare this with equation (11.17). The proof is left as Exercise 11.2.2.2.
Just like in the Lagrange case we can apply the results above to interpolating
points in
R
m
with a parametric curve. In other words, given distinct real numbers u
0
,
u
1
,..., u
n
, points
p
0
,
p
1
,...,
p
n
and tangent vectors
v
0
,
v
1
,...,
v
n
in
R
m
, there is a
unique curve p : [u
0
,u
n
] Æ
R
m
, called the
piecewise Hermite interpolating curve
,
satisfying
(1) p (u
i
) =
p
i
,
(2) p¢ (u
i
) =
v
i
, and
(3) p
i
= p | [u
i
,u
i+1
] is a cubic polynomial.
In fact,
p
p
v
v
Ê
ˆ
i
Á
Á
Á
˜
˜
˜
(
)
3
2
i
+
1
()
=-
(
)
(
)
(
)
(
)
pu
uu
uu
-
uu
-
1
M
D
u
,
(11.25)
i
i
i
i
h
i
i
Ë
¯
i
+
1
where Du
i
= u
i+1
- u
i
. Using equation (11.23) we can also express p
i
(u) in the form
()
=
()
+
()
()
+
()
pu
fu
p
fu
p
+
fu
v
f u
v
,
(11.26)
i
1
i
2
i
+
1
3
i
4
i
+
1
where the functions f
j
(u) are defined as in equation (11.22).
The function p(u) is clearly differentiable by construction. However, this does not
completely solve the problem with which we began, because although we now have
an interpolating curve of low degree without “corners,” we assumed that the tangent
vectors
v
i
were given to us. Such an assumption may not be convenient and it is in
fact unnecessary as we shall see in the next section.
Finally, the answers to the Lagrange and Hermite interpolation problems above
show a pattern that ought to be noted. In each case, the problem was solved in an
elegant way by finding
basis functions
a
i
(x) such that a
i
(x
j
) =d
ij
and similar functions
for derivatives. These functions constructively isolated the effect of each “control
datum” so that it occurred only once as an explicit parameter in the solution. This is
