Graphics Reference
In-Depth Information
12.12.4
Rational B-spline Surfaces
Rational B-spline surfaces are the surface analogs of rational B-spline curves and their
use is motivated by similar reasons, namely, that, although B-spline surfaces are very
general, they do not include the quadric surfaces and can only approximate them. By
using rational B-spline curves and surfaces a modeling system needs to support only
one uniform representation for its geometric objects. Some general references for
rational Bézier and B-spline surfaces are [PieT95], [Pieg91], [Fari95], [Roge01], or
[RogA90].
Proceeding just like we did for curves in Section 11.5.3, consider the tensor
product surfaces defined by equation (12.31). These were surfaces in affine space.
Using homogeneous coordinates, their projective space analogs would have the form
m
n
Â
Â
(
)
=
() ()
Puv
,
a ub v
i
P
,
(12.52)
j
ij
i
=
0
j
=
0
where the
P
ij
are points described with homogeneous coordinates. Expressing the
points
P
ij
in the form (w
ij
x
ij
,w
ij
y
ij
,w
ij
z
ij
,w
ij
), the projective space surface defined by
P(u,v) will project to the surface
m
n
Â
Â
() ()
aubvw
p
i
j
ij
ij
i
=
0
j
=
0
(
)
=
puv
,
.
(12.53)
m
n
Â
Â
() ()
aubvw
i
j
ij
i
=
0
j
=
0
where
p
ij
= (x
ij
,y
ij
,z
ij
).
Definition.
The parametric surface p(u,v) defined by equation (12.53) is called a
rational tensor product surface
. It is called a
rational Bézier surface
if its domain is [0,1]
¥ [0,1] and a
i
(u) = B
i,m
(u) and b
j
(v) = B
j,n
(v). (The B
s,t
(u) are the functions defined by
equation (11.50).) The surface p(u,v) is called a
rational B-spline surface
of order (k,h)
and degree (k - 1,h - 1) if the a
i
(u) and b
j
(v) are B-splines of order k and h, respec-
tively. The knots of a
i
(u) are called the
u-knots
of the surface and those of b
j
(v), the
v-knots
. In both the Bézier and B-spline case, the points
p
ij
are called the
control points
of the surface and the numbers w
ij
are called its
weights
.
Definition.
The functions
() ()
aubvw
i
j
ij
(
)
=
Ruv
,
.
(12.54)
ij
m
n
Â
Â
() ()
aubvw
s
t
st
s
=
0
t
=
0
are called
rational basis functions
for the surface defined by equation (12.53).
