Graphics Reference
In-Depth Information
(
)
=
(
)
=
(
)
=
(
)
=
pu
,
v
p
,
pu
,
v
p
,
pu
,
v
p
,
and
pu
,
v
p
.
k
-
1
h
-
1
00
mh
+
1
-
1
m
0
k
-
1
n
+
1
0
n
mn
+
1
+
1
n
(5) If m = k - 1, n = h - 1, (u
i
) = (0,...,0,1,...1), and (v
j
) = (0,...,0,1,...,1), then
p(u,v) defines a Bézier surface.
In addition, we have
12.12.1
Theorem.
B-spline surfaces are affinely invariant.
Proof.
This follows from Theorem 11.2.2.3.
B-spline surfaces p(u,v) do not satisfy any known variation diminishing property
(see [PraG92]). Section 12.12.5 will discuss algorithms for evaluating p(u,v) and its
derivatives.
12.12.2
Polynomial Surfaces and Multiaffine Maps
The multiaffine approach to polynomial curves described in Section 11.5.2 can be
extended to polynomial surfaces. What we need to do is extend the notion of polar
form or blossom for a polynomial function of one parameter to a polynomial
function
m
:
RR
2
Æ
p
(12.47)
of two parameters. There are two ways that one can define a blossom P for such p.
One leads to tensor product surfaces, the other, to surfaces based on triangular
patches. We only sketch the approaches. For much more detail see [Gall00] or
[Fari97]. A reader who understands the material in Section 11.5.2 should find what
we do here straightforward.
The Tensor Product Surface Blossom.
Here we treat each variable u and v for a
point (u,v) in
R
2
separately and construct a blossom for each in the same manner as
in Section 11.5.2, that is, if the degree of p in u and v is d1 and d2, respectively, then
the blossom has the form
d
1
d
2
m
P
:
R
¥
R
Æ
R
.
(12.48)
The function P(u
1
,u
2
,...,u
d1
,v
1
,v
2
,...,v
d2
), called the
tensor product polar form
or
tensor product blossom
for p(u,v), is only symmetric and multiaffine in the variables
u
i
and v
j
separately. It follows that keeping one set of variables fixed and thinking of
P as a function of the other set means that all the algorithms and properties of the
blossoms in Section 11.5.2 are valid here. The efficient evaluation of tensor product
Bézier and B-spline surfaces is based on this notion of blossom.
To find the tensor product blossom P of p(u,v) = (u + v - 3,uv,u
2
12.12.2.1
Example.
+ v
2
).
Solution.
For fixed v, the blossom with respect to u is
