DESIGN OF SURFACES 245
r
P
(, ) =
() ()
() ()
=0=0
,+ ,+
=0 =0 ,+ ,+
u
wN uN
wN uN
i
m
j
n
ij ppi qq j ij
i
m
j
n
ij ppi qq j
v
v
v
ΣΣ
ΣΣ
(7.62)
with user chosen weights wij.
Exercises
- A bi-linear surface r(u,v) is defined by the points r(0, 0) = {0, 0, 1}, r(0, 1) = {1, 1, 1}, r(1, 0) = {1,
0, 0} and r(1, 1) = {0, 1, 0}. Show the plot of the surface. Determine the unit normal to the surface at
(u= 0.5, v= 0.5). - A bi-cubic Ferguson patch is defined by the following:
Corner points r(0, 0) = {–100, 0, 100}, r(0, 1) = {100, –100, 100}, r(1, 1) = {–100, 0, –100}, r(1, 0)
= { –100, –100, –10}, u-tangent vectors ru(0, 0) = {10, 10, 0}, ru(0, 1) = { –1, –1, 0}, ru(1, 1) = { –1,
1, 0}, ru(1, 0) = {1, –1, 0}; v-tangent vectors rv(0, 0) = {0, –10, –10}, rv(0, 1) = {0, 1, –1}, rv(1, 1) =
{0, 1, 1}, rv(1, 0) = {0, 1, 1}; twist vectors ruv(0, 0) = {0, 0, 0}, ruv(0, 1) = {0.1, 0.1, 0.1}, ruv(1, 1) =
{ 0, 0, 0}, ruv(1, 0) = {–0.1, –0.1, –0.1}.
Generate the surface and find tangents, normal and curvatures for the surface at (0.5, 0.5). - A Coon’s patch is generated using quadratic Bézier curves φ 0 (u),φ 1 (u) and ψ 0 (v),ψ 1 (v) having control
points [{0, 0, 0}, {1, 0, 3},{3, 0, 2}]; [{0, 3, 0},{1, 3, 3},{3, 3, 2}] and [{0, 0, 0},{0, 1, 3},{0, 3, 2}];
[{3, 0, 2},{3, 2, 3},{3, 3, 2}]. Work out the complete analysis of individual patches and the final Coon’s
patch.
Figure 7.31 Closed B-spline surfaces
Z axis
X axis
Y axis
Z axis
X axis Y axis
3
2
1
–5
0
50
5
10
0
5
–10 10
0
10
3 2 1 3 2 1 0
5
10
–2 0
(^24)
(a)
(b)
(c)