Graphics Reference
In-Depth Information
Note that
=
(
)
()
=
¢
()
=¢
2
p u
UA
and
p u
U A
3210
u
u
A
.
Furthermore, in analogy with equation (11.9),
0001
1111
0010
3210
Ê
ˆ
Á
Á
Á
˜
˜
˜
AB
h
.
Ë
¯
Recall from (11.10) that the Hermite matrix
M
h
, which is the inverse of the 4 ¥ 4
matrix on the left, is given by
2211
3321
0010
1000
-
Ê
ˆ
Á
Á
Á
˜
˜
˜
-
-
-
M
h
=
.
(11.36)
Ë
¯
By definition,
A
=
M
h
B
h
. We have the following matrix equations:
()
=
=
=
pu
UA
UM B
FB
,
hh
(11.37)
h
where
(
()
()
()
()
)
FUM
=
=
FuF uFuF u
.
h
1
2
3
4
The functions F
i
(u) are just the Hermite basis functions defined earlier in (11.14). They
will now also be called
blending functions
because they blend the geometric coeffi-
cients into p. Equations (11.37) show that a curve can be manipulated by changing
either its algebraic or geometric coefficients.
Derivatives of p can also be computed in matrix form:
¢
()
=
(
)
¢
u
pu
UM
B
=
UM B
=
F B
,
¢
(11.38)
bh
h
h
where
0000
6633
6642
0010
Ê
ˆ
Á
Á
Á
˜
˜
˜
-
M
u
=
,
(11.39)
-
-
-
Ë
¯
