Graphics Reference
In-Depth Information
Section 12.7
12.7.1
Find the formula for the Coons surface p(u,v) defined by boundary curves p(u,0) =
(2u,u
2
,0), p(u,1) = (2u,-4u
2
+ 8u + 1,0), p(0,v) = (0,v,0), and p(1,v) = (-4v
2
+ 4v + 2,4v +
1,0). Also find ∂p/∂u(0.5,0.5) and ∂p/∂v(0.5,0.5).
12.7.2
Prove Theorem 12.7.1.
Section 12.9
12.9.1
Prove that the matrix
Q
in equation (12.36a) is just the matrix
B
in equation
(12.35).
12.9.2
Consider the bicubic patches p(u,v) with geometric matrices
(
)
(
)
(
)
(
)
110
,,
3 2 0
, ,
210
,,
210
,,
Ê
ˆ
Á
Á
(
)
(
)
(
)
(
)
˜
˜
200
, ,
510
,,
310
,,
310
,,
(a)
B
=
(
)
(
)
(
)
(
)
1 10
,,
-
2 10
,,
-
100
,,
100
,,
(
)
(
)
(
)
(
,,
)
Ë
1 10
,,
-
2 10
,,
-
100
,,
100
¯
(
)
(
)
(
)
(
)
Ê
100
,,
023
,,
-
123
,,
-
123
,,
ˆ
Á
Á
Á
(
)
(
)
(
)
(
)
˜
˜
˜
300
,,
323
,,
023
,,
023
,,
(b)
B
=
(
)
(
)
(
)
(
)
200
,,
300
,,
100
,,
100
,,
(
)
(
)
(
)
(
)
Ë
200
,,
300
,,
100
,,
100
,,
¯
p
p
Ê
Ê
Ë
ˆ
¯
Ê
Ë
ˆ
¯
ˆ
(
110
,,
)
(
101
,,
)
00
2
,,
0
,
-
,
0
Á
Á
Á
Á
2
˜
˜
˜
˜
p
p
Ê
Ë
ˆ
¯
Ê
Ë
ˆ
¯
(
)
(
)
220
,
,
202
,,
00
2
,,
0
,
-
,
0
2
(c)
B
=
1
1
2
1
2
p
p
Ê
Ë
ˆ
¯
Ê
Ë
ˆ
¯
Ê
Ë
ˆ
¯
Ê
Ë
ˆ
¯
, ,
0
1 0
,,
00
4
,,
0
,
-
,
0
4
1
010
1
22
p
p
Ê
Ë
ˆ
¯
Ê
Ë
ˆ
¯
Ê
Ë
ˆ
¯
Ê
Ë
ˆ
¯
1
,
,
, ,
00
42
, ,
0
,
-
42
0
,
Ë
¯
2
2
Describe the surfaces defined by these patches geometrically like we did in Example
12.9.1.
12.9.3
(a)
Let p(u,v) be the bicubic patch defined by Exercise 12.9.2(b). Consider the rec-
tangle
A
= [1/4,3/4] ¥ [1/3,2/3]. Let q(u,v) be the bicubic patch that is the subpatch
pΩ
A
reparameterized to [0,1] ¥ [0,1]. Find the geometric matrix for q(u,v).
(b)
Generalize (a) to the case of an arbitrary subrectangle
A
= [a,b] ¥ [c,d].
Section 12.12.3
12.12.3.1
The blossom for a triangular Bézier surface p(u,v) can be obtained from its control
points
b
ijk
using the triangular de Casteljau algorithm similar to what was done in
Exercise 11.5.2.1 for curves. To learn about it, see, for example, [Fari97].
