Graphics Reference
In-Depth Information
Figure 11.23.
The geometry of the
Bézier polynomial con-
struction.
() =
(
)
pu
Puu u
aPsuuuaPt uuu
a a Psuu
, ,...,
, , ,...
(
) +
(
)
=
, , ,...
0
1
(
(
) +
(
)
)
=
, , ,...,
u
a Ps t u
, , ,...,
u
00
1
(
(
) +
(
)
)
+
a a Ptsu
, , ,...,
u
aPt tu
, , ,.
..,
u
10
1
2
(
)
(
) +
(
) (
) +
2
(
)
=-
1
aPs s uua aPs t uuaPt t uu
, , ,...,
2
1
-
, , ,...,
, , ,...,
.
(11.93)
1
1
1
1
Figure 11.23 shows the geometry in all this for a cubic polynomial curve in the plane.
Recall that we can permute parameters, so that
(
) =
(
)
(
) =
(
)
Putt
, , ,...
Pttu
, , ,...
and
Psut
, , ,...
Pstu
, , ,....
Again, the reader needs to stare at Figure 11.23 and match its geometry with equa-
tions (11.93) until the relationship is clear. The mathematics in this section is basi-
cally simple, but it may look complicated because it is easy to get lost in the subscripts
and parameters.
The recursive formula for the b r (u) in (11.89) leads to the recursive algorithm for
computing p(u) from its control points b i shown in Algorithm 11.5.2.1. Note how this
algorithm generalizes the corresponding Algorithm 11.4.1 in Section 11.4. In par-
ticular, letting d = n, s = 0, and t = 1, equations (11.51) and (11.92) define the same
curve. See also Exercise 11.5.2.1.
There is also a formula for the derivatives of p(u) using forward differences.
Forward difference methods lead to fast evaluation of polynomials at uniformly
spaced intervals. They are discussed in most topics on numerical analysis. Wallis
([Wall90]) has a brief tutorial.
Definition.
For any sequence of points b i , define the rth forward difference operator
D r as follows:
0
D
DD
bb
b
=
=
i
i
r
r
-
1
r
-
1
b
-
D
b
i r
,
>
0
.
i
i
+
1
The operator D 1 is usually abbreviated to D.
Search WWH ::




Custom Search