Computer Aided Engineering Design
DESIGN OF SURFACES 223 or in matrix form r 3 (u,v) = [ ( ) ( ) ( ) ( )] (0) (0) (1) (1) (0) (1) (0) (1) () () () () 0123 00 10 0 ...
224 COMPUTER AIDED ENGINEERING DESIGN + [ ( ) ( ) ( ) ( )] () () () () 0101 0 1 2 3 aassvvvv ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎤ ...
DESIGN OF SURFACES 225 To construct a bi-linear Coon’s patch, the four boundary curves are given by a r r r r a r r r r 0 32 00 ...
226 COMPUTER AIDED ENGINEERING DESIGN 7.3 Composite Surfaces Surface patches, in small units, need to be joined (stitched) toget ...
DESIGN OF SURFACES 227 t PP ij i PP ij ij ij ij = D | – | +1 –1 +1 –1 , where Di = min (| Pij – Pij–1 |, | Pij+1 – Pij |) (7.4 ...
228 COMPUTER AIDED ENGINEERING DESIGN For slopes tij t 01 = [min (| P 01 – P 00 |, | P 02 – P 01 |)] PP PP 02 00 02 00 | – | = ...
DESIGN OF SURFACES 229 Solving yields, four relations which can be summarized to the following two (A) sij + 4si+ 1 j + si+ 2 j= ...
230 COMPUTER AIDED ENGINEERING DESIGN G rr rr rr rr rr rr rr rr rrrr rrrr r F = 3( – ) 3( – ) 3( – ) 3( – ) 3( – ) 3( – ) 9( – – ...
DESIGN OF SURFACES 231 qqq qqq qqq 00 01 02 10 11 12 20 21 22 = {0, 2, 2} = {0, 3, 2} = {0, 4, 4} = {1, 3, 3} = {1, 4, 3} = {1, ...
232 COMPUTER AIDED ENGINEERING DESIGN Further, for Sp remaining unaltered while Sq changed to Sq1 having different control point ...
DESIGN OF SURFACES 233 Once the 16 data points for a bi-cubic Bézier patch are chosen, in choosing the data points for adjacent ...
234 COMPUTER AIDED ENGINEERING DESIGN To match the degree in v,λ(v) = λ, a constant, while μ(v) = μ 0 + μ 1 v, a linear function ...
DESIGN OF SURFACES 235 [r 00 r 01 r 02 r 03 ]II = [(0, 3, 0) (1, 3, 3) (2, 3, 3) (3, 3, 0)] For slope continuity, Case I yields ...
236 COMPUTER AIDED ENGINEERING DESIGN or r 10 II = (0, 3, 0) +^1 3 (3, 3, 9) = (1, 4, 3) r 11 II = (1, 3, 3) +^1 3 (4, 3, 9) =^7 ...
DESIGN OF SURFACES 237 Considering the triple scalar product for the vectors on the left gives pp p12 3 ( ) = 113 103 0–10 ⋅× = ...
238 COMPUTER AIDED ENGINEERING DESIGN rB( , 1) = [uuuu 1] 2–2 1 1 –3 3 –2 –1 00 10 10 00 (1, 0, 0) (1, 0.5, 1) (0, 0, 1) (0, 1, ...
DESIGN OF SURFACES 239 Equation of the ruled surface B using two boundary curves is given by rB(u,v) = (1 – v)rB(u, 0) + vrB(u, ...
240 COMPUTER AIDED ENGINEERING DESIGN Equation of the Coon’s patch C (incorporating the correction surface) is given by rC(u,v) ...
DESIGN OF SURFACES 241 Example 7.13 (Closed Bézier Surface). Closing the polyhedron formed by the control points can create clos ...
242 COMPUTER AIDED ENGINEERING DESIGN Here, rrr rrr rrr 00 01 02 10 11 12 20 21 22 = {0, 0, 0} {0, 1, 0} (0, 2, 0} {1, 0, 0} {1, ...
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