Computer Aided Engineering Design
DESIGN OF SURFACES 203 rD DD D DD D DD D ( , ) = = [... 1] 1 =0 =0 –1 ( –1) 0 ( –1) ( –1)( –1) ( –1) 0 0( –1) 00 –1 uuuu j n i m ...
204 COMPUTER AIDED ENGINEERING DESIGN Construction of a tensor product surface patch using Hermite blending functions can be sim ...
DESIGN OF SURFACES 205 rCrCr Cr C rCrCrCrC rCr Cr Cr C rCrC (0, 0) = ; (1, 0) = ; (0, 0) = ; (1, 0) = (0, 1) = ; (1, 1) = ; (0, ...
206 COMPUTER AIDED ENGINEERING DESIGN G rr r r rr r r rr r r rr r r = (0, 0) (0, 1) | (0, 0) (0, 1) (1, 0) (1, 1) | (1, 0) (1, 1 ...
DESIGN OF SURFACES 207 n rr rr (, ) = (, ) (, ) | ( , ) ( , ) | u uu uu u u v vv vv v v × × (7.16) n Q P vj ui Z Y X O rv(ui,vj) ...
208 COMPUTER AIDED ENGINEERING DESIGN Suudud u = |u( , ) ( , ) | ,v ∫∫ rrvvv× v (7.19) For Ferguson’s bi-cubic patch, from Eq. ( ...
DESIGN OF SURFACES 209 n ij (0.5, 0.5) = ij 46.5 + 46.5 46.5 + 46.5 =^1 2 ( + ) 22 Now ruu(u,v) = UMuuGMTVT = [(42u– 26) (–42u + ...
210 COMPUTER AIDED ENGINEERING DESIGN Figure 7.5 Bi-cubic surface patch in 16 point form r(0, 1) r(1/3, 1) r(2/3, 1) r(1, 1) r(0 ...
DESIGN OF SURFACES 211 rrrr rrrr rrrr rrrr rr r r rr r r rr r r 33 32 31 30 23 22 21 20 13 12 11 10 03 02 01 00 = (1, 1) 1,^2 3 ...
212 COMPUTER AIDED ENGINEERING DESIGN r 00 r 01 r 02 r 03 r 13 r 23 r 33 r 30 r 10 r^20 r 11 r 21 r 31 r(ui,vj) r 32 r 22 r 12 F ...
DESIGN OF SURFACES 213 = [ 1] 1 32 00 01 02 03 10 11 12 13 20 21 22 23 30 31 32 33 3 2 uuuMB BT rrrr rrrr rrrr rrrr M ⎡ ⎣ ⎢ ⎢ ⎢ ...
214 COMPUTER AIDED ENGINEERING DESIGN Figure 7.7 (a) Example of a quadratic-cubic Bézier surface with control polynet and (b) ch ...
DESIGN OF SURFACES 215 Example 7.4. The control points for a bi-cubic Bézier surface are given by rrrr rrrr rrrr rrrr 00 10 20 3 ...
216 COMPUTER AIDED ENGINEERING DESIGN P 00 = {0, 0, 3}; P 10 = {1, 1, 3};P 20 = {1, 2, 3};P 30 = {0, 3, 3}; P 01 = {0, –1, 2};P ...
DESIGN OF SURFACES 217 i + j + k = n. The value of n is user’s choice. A large n will carry finer details of the patch but at in ...
218 COMPUTER AIDED ENGINEERING DESIGN 7.2 Boundary Interpolation Surfaces Ruled and lofted patches are some examples of boundary ...
DESIGN OF SURFACES 219 7.2.1 Coon’s patches Coon’s patches can use either linear or Hermite blending in surface approximation us ...
220 COMPUTER AIDED ENGINEERING DESIGN From Eqs. (7.35a) and (7.35b), we realize that the two boundary curves for r 3 (u,v),r 3 ( ...
DESIGN OF SURFACES 221 that the conditions in Eq. (7.37) are satisfied. In other words, using Eqs. (7.37) to determine the corre ...
222 COMPUTER AIDED ENGINEERING DESIGN ⇒ r 3 (u, 0) = φ 0 (u)P 00 + φ 1 (u)P 10 + φ 2 (u)s 0 (0) + φ 3 (u)s 1 (0) (7.40a) r(u, 1) ...
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