Graphics Reference
In-Depth Information
12.8
Tensor Product Surfaces
Tensor product surfaces are one of the most common surfaces encountered in CAGD.
Some simple versions can be computed with matrices. This section only gives an
overview, leaving the details with regard to some important special cases for subse-
quent sections.
Consider a curve
m
Â
0
()
=
()
pu
f up
,
i
i
i
=
where the f
i
(u) are basis functions, and we treat the p
i
as a 1-parameter family of
vector-valued functions
n
Â
()
=
()
pv
gv
p
i
j
ij
j
=
0
with respect to some other basis functions g
j
(v) and points
p
ij
.
Definition.
The parametric surface p(u,v) defined by
m
n
Â
Â
(
)
=
() ()
puv
,
f ug v
p
(12.31)
i
j
ij
i
=
0
j
=
0
is called a
tensor product
or
Cartesian product
surface with basis functions f
i
(u)g
j
(v).
In matrix form, equation (12.31) becomes
()
()
◊
()
p
◊
◊
p
gv
gv
Ê
ˆ
Ê
ˆ
00
0
n
0
Á
Á
Á
˜
˜
˜
Á
Á
Á
˜
˜
˜
◊
◊
◊
1
(
)
=
(
() ()
◊◊◊
()
)
puv
,
f u f u
f
u
(12.32)
.
0
1
m
◊
◊
◊
Ë
¯
Ë
¯
p
◊
◊
p
gv
m
0
mn
n
Note how the partial derivatives of tensor product surfaces are easily obtained
from the derivatives for the curves:
n
m
È
Í
˘
˙
∂
∂
p
u
∂
∂
ÂÂ
0
(
)
=
()
()
uv
,
g v
fu
p
(12.33a)
j
i
ij
u
j
=
i
=
0
m
n
È
Í
˘
˙
∂
∂
p
v
∂
∂
ÂÂ
0
(
)
=
()
()
uv
,
f u
gv
p
(12.33b)
i
j
ij
v
i
=
j
=
0
