Graphics Reference
In-Depth Information
()
=
(
)
Œ
[]
ft
t
,, ,
00
t
01
, ,
(
)
Œ
[]
=
10
,,
t
-
1
,
t
12
, ,
(
)
Œ
[]
=-
301
t
,, ,
t
23
, ,
(
)
Œ
[
]
=
004
,,
-
t
,
t
34
, .
We shall rotate the square as it is swept along g(s). This can be accomplished by
defining
(
)
=
(
)
T xyz
, ,
x
cos , , sin
syz
s
.
s
The functions g
i
(s,t) are easily computed. For example, for t Π[1,2],
()
=
()
=
()
=-
(
)
g
s t
,
cos ,
s
g
s t
,
0
,
and
g
s t
,
t
1
sin .
s
1
2
3
Figure 12.9(b) shows the resulting screw sweep surface.
12.6
Bilinear Surfaces
Let four points
p
00
,
p
01
,
p
10
, and
p
11
be given. The easiest way to define a surface
p(u,v) that interpolates these points is by means of a double linear interpolation.
Define
()
=-
(
)
[
(
)
]
+-
[
(
)
]
puv
,
11
v
-
u
pp
+
u
v
1
u
pp
+
u
,
(12.17a)
(12.17b)
00
10
01
11
[
]
+-
[
]
=-
(
11
uv
)
(
-
)
pp
+
v
uv
(
1
)
pp
+
v
,
00
01
10
11
(
)
(
)
(
)
(
)
=-
11
uv
-
p
+-
1
uv uv
p
+-
1
p
+
v
p
,
00
01
10
11
where 0 £ u, v £ 1. Equation (12.17a) says that to find p(u,v) we first find p(u,0) and
p(u,1) by a linear interpolation in the u-direction and then do a linear interpolation
of those points in the v-direction. Equation (12.17b) says that we get the same answer
if we first interpolate in the v-direction and then in the u-direction. See Figure
12.10(a).
Definition.
The parametric surface defined by equations (12.17) is called the
bilinear surface
determined by the four points or the
four-point interpolating surface
.
Figure 12.10.
Bilinear surfaces.
