Graphics Reference
In-Depth Information
With this true generalization of a blossom, everything we did in Section 11.5.2
carries over to here (as long as we translate results correctly). We need to remember,
however, that barycentric coordinates are now triples of real numbers, not pairs, so
that, as a consequence, arrays of control points are replaced by triangular arrays of
control points as we shall see shortly. To simplify the notation, define the triangular
set of integer triples I
d
by
{
}
d
=
(
)
Σ
3
I
i jk
,,
Z
0
i jk and i
,,
+
j k
+
=
d
.
d
Definition.
If (i,j,k) ΠI
d
, then the
Bernstein polynomial
B
i,j,k
(u,v,w), u + v + w = 1, is
defined by
d
ijk
uvw
!
!! !
d
(
)
=
i
j
k
Buvw
,,
.
ijk
,,
Like their cousins in Section 11.4, these Bernstein polynomials also satisfy a
recurrence relation (see Exercise 11.4.1).
12.12.2.4
Theorem.
The Bernstein polynomials satisfy the recurrence relation
d
(
)
=
d
(
)
+
d
(
)
+
d
(
)
(
)
Œ
B
u v w
,,
uB
u v w
,,
vB
u v w
,,
wB
u v w
,, , ,,
i j k
I
.
d
ijk
,,
i
-
1
,,
j k
ij
,
-
1
,
k
ijk
,,
-
1
Proof.
This follows easily from the binomial coefficient identity
(
)
(
)
(
)
d
ijk
!
!! !
d
-
!
!! !
1
d
ij
-
1
1
!
d
ij k
-
1
!
=
+
+
!
.
(
)
(
)
(
)
i
-
1
j k
!
-
! !
k
!!
-
1
The next theorem states the main facts about triangular blossoms.
Let
r
,
s
, and
t
be three linearly independent points in
R
3
. Let
12.12.2.5
Theorem.
{
(
)
Œ
}
b
ijk
ijk
,,
I
,,
d
be any set of (d + 1)(d + 2)/2 points in
R
3
for some integer d
≥
1.
(1) There is a unique polynomial surface p(u,v) of total degree d whose triangular
blossom P satisfies
P
( ,..., , ,..., , ,..., )
r
r s
s t
t
=
b
144144123
ijk
,,
i
j
k
for all (i,j,k) ΠI
d
.
(2) If (u,v) = a
r
+ b
s
+ c
t
, a + b + c = 1, then
