Computer Aided Engineering Design

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

1. 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).


2. 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).


3. 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)