Graphics Reference
In-Depth Information
Using the rational basis functions, equation (12.53) can be rewritten as
m
n
Â
Â
(
)
=
(
)
puv
,
R uv
ij
,
p
.
(12.55)
ij
i
=
0
j
=
0
The most commonly used type of rational B-spline surface is the following:
Definition.
If the B-splines N
i,k
(u) and N
j,h
(v) (defined by equations (11.69)) are
defined with respect to knot vectors
Uaauu
=
( ,..., ,
,
,...,
ubb dVc
, ,..., )
=
( ,..., ,
c v
,
v
,...,
v
, ,..., ),
dd
kk
+
1
m
hh
+
1
n
144
144
144
144
k
k
h
h
(12.56a)
respectively, then the rational B-spline surface
m
n
Â
Â
()
()
NuNvw
p
i k
,
j h
,
ij
ij
i
=
0
j
=
0
(
)
=
puv
,
.
(12.56b)
m
n
Â
Â
()
()
NuNvw
ik
,
jh
,
ij
i
=
0
j
=
0
has domain [a,b] ¥ [c,d] and is called a
n
onuniform
r
ational
B
-
s
pline (NURBS)
surface
.
Note that if the weights of a NURBS surface are all 1, then formula (12.56b)
reduces to (12.46) and we just have an ordinary B-spline surface. This follows from
Theorem 11.5.1.5. For NURBS surfaces one usually has k = h and the domain is
assumed to be [0,1] ¥ [0,1]. Algorithms for evaluating NURBS surfaces will be dis-
cussed in the next section.
Compare the properties of NURBS surfaces listed in the next theorem to the cor-
responding ones in Theorem 11.5.3.2 for NURBS curves.
12.12.4.1 Theorem.
Let p(u,v) be a NURBS surface of order (k,h) with domain
[0,1] ¥ [0,1], u-knots u
i
, v-knots v
j
, control points
p
ij
, and weights w
ij
.
(1) The rational basis functions R
ij
(u) for p(u,v) satisfy
m
n
Â
Â
(
)
=
Ruv
,
1
ij
i
=
0
j
=
0
and R
ij
(u,v) ≥ 0 if all the weights are nonnegative.
(2) The surface p(u,v) interpolates the four corner points. More precisely, p(0,0)
=
p
00
, p(1,0) =
p
m,0
, p(0,1) =
p
0n
, and p(1,1) =
p
mn
.
