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# Curves and surfaces in computer aided geometric design - Yamaguchi F.

Yamaguchi F. Curves and surfaces in computer aided geometric design - Tokyo, 1988. - 390 p.
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This connection condition expresses the fact that on the No. 2 surface
patch must lie on the tangent plane to a point on the boundary curve of No. 1 surface patch. Equation (5.132) becomes the following:
5. The Bernstein Approximation
[0 0 1 0]MA;1IM?
V V
w2 = A?(w)[3 2 1 0]MBBBlM^ w2
w w
1 1
+ y(w)[l 1 1 ÖÌâÂâëÌ%
'3w
2w
1
0
(5.133)
If we assume that both of the surface patches are bi-cubic, then, for the same reason as before, /i(w) is an arbitrary positive scalar /ë. Also, y(w) can be expressed as the linear form ó(w) = yo + 7iw- In this case, the left-hand side of Eq. (5.133) becomes:
= [0 0 1 0]
-1
i 3 -3 1"
1 -6 3 0
S 3 0 0
0 0 0
3 [w3 w2 w 1]
1 Ml [w3 w2 w l]r
3 -3 Ã 611,00 611,01 611,02 6l,,03
-6 3 0 611,10 611,11 611,12 6„,13
3 0 0 611,20 611,21 6,1,22 611,23
0 V“ ...2 0 0 _ 611,30 611,31 6,1,32 611,33.
611,00 —
3 Q
n,oi + 3 Qu,02 — Qn,îç áíäî + 3 Qll
,n 3flu2+e n,i: — 3 6II>00 + 60II)O1 —3 6ll,02 + 3 611,10 — ^ Qll,11 + 3 Qll, 12 3 611,00 — 3 611,01— 3 611,10 + 3 611,11 — 611,00 + 611,10
The 1-st term on the right-hand side of Eq. (5.133) is:
1-st term on the right-hand side = n(w) [3 2 1 0] ÌâÂâëÌ% [w3 w2 w l]r
= ö[3 2 1 0]
¦-1 3 -3 Ã 61,00 61,01 61,02 61,03
3 -6 3 0 61,10 61,11 61,12 61,13
-3 3 0 0 61,20 61,21 6,22 61,23
1 0 0 0 _6l,30 61,31 6l,32 6,,3 3_
5.2 Surfaces
225
¦-1 3 -3 1" 'w3'
3-630 w2
-3 3 0 0 w
1 0 0 0 1
= 3 [w3 w2 w 1]
^(Ql,20~~^ Ql,21 + 3 01,22 — Ql,23 ~~ Ql,30 + 3 01,31 — 3 01,32 + 01,çç) ^( — 3 Qj 20 + 6 Q121 — 3 0122 + 3 01,30 — 6 Ql z t + 3 0lj32) /^(301,20-301,21 — 3 01,30 + 3 01>3i)
01,20 + 01,3o)
The 2-nd term on the right-hand side of Eq. (5.133) is:
2-nd term on right-hand side = y(w) [1 1 1 ÖÌâÂâ1Ì1 [3w2 2w 1 0]T
-1 3 -3 ã 01,00
3 -6 3 0 01,10 01,20
(Óî + yiW) [1 1 1 1] -3 3 0 0
1 0 0 0 _0I,3O
¦-1 3 -3 ã "0 3 0 O' 'w3'
3 -6 3 0 0 0 2 0 w2
X -3 3 0 0 0 0 0 1 w
1 0 0 0 0 0 0 0 1
01,01 01,02 01,03
01,11 01,12 01,13
01,21 01,22 01,23
01,31 01,32 01,33
= 3 [w3 w2 w 1]
Ó l( —01,30 + 3 01,31 3 01,32 + 01,3ç)
Óî( — 01,çî + 3 0i,3i-3 0!,32 + 0i,33) + 7i (20i,3O — 40i,3i +2 0I>32)
7o(20i,3o— 40i,3i +20I)32) + 7i(—0i,3o + 0i,3i)
7o(— 01,çî + 0i, 31)
Comparing the constant terms on both sides of Eq. (5.133) we obtain:
0ii,io — 0ii,oo = ^(0i,3o“ 0i,2o) + 7o(0i,3i — 0i,çî)- (5.134)
Next, comparing the coefficients of the w terms:
3 0ii,oo — 3 0ii,oi — 3 0h,io + 30H,u (3 0i,2o —3 0i,2i —3 01,çî + 3 0i,3i)
+ Óî (2 0i,çî-40i,3i +20I;32)
+ yi( — 0i, 3o + 0i, 3i)-
226
5. The Bernstein Approximation
Rearranging terms and using Eq. (5.134):
(5.135)
Similarly, when the coefficients of the w2 and w3 terms are compared the following 2 equations are obtained:
In the above 4 equations, (5.134), (5.135), (5.136) and (5.137), setting yo = y1=0 we can confirm that these reduce to Eq. (5.131).
The geometrical significance of Eq. (5.134) is that 611,10 °f the No. 2 surface patch lies on the plane formed by the vectors 61,30 — 61,20 and 61,31— 61,30 °f the No.l surface patch (refer to Fig. 5.35). Similarly, the significance of Eq. (5.137) is that 611,13 °f the No. 2 surface patch lies on the plane formed by
6l,33 — 6l,23 an<i 6l,33 — 6l,32-
The geometrical interpretation of Eq. (5.135) and (5.136) is not as simple as the above 2 cases.
5.2.4 Triangular Patches Formed by Degeneration
When a triangular surface patch is to be expressed by an oridnary quadrangular Bezier surface patch, let us consider the conditions for the unit normal
(5.136)
611,13 611,03 — /^(61,33 61,23) + (Óî + Ói) (61,33 61,32)- (5.137)
Fig. 5.35. Connection of cubic Bezier surface patches (general method)
5.2 Surfaces
227
vector at the degenerate point to be uniquely determined (refer to Sect. 1.3.8). Assume that we are dealing with a bi-cubic Bezier surface patch. Then we have:
P(u,w)=UMBBBMlWT
Pu(0, w) in Eq. (1.128) can be found to be:
PB(0,w) = [0 0 1 0~]MBBBMl[_w3 w2 w 1]T
= ÇÁ0?3(èÎ (áþ~áîî) + 3Blj3(w) (6ii —601)
+ 3#2,ç0^) {Q\2~ Qoi) + Ç^ç,3(w) (É1ç-Ñîç)-
In Fig. 5.36, at the degenerate point D we have Q0o = Qoi==Qo2 = Qo3^ so:
^(^^-ÇÂî^Ì (Cio~Coo) + 3Blj3(w) (fin-Coo)
+ 3B2j3{w) (Q12 — Coo)+ 3 #3,3 (w) (Ci3-Coo)- (5.138)
We also have, for PMW(0,w):
ÐèË0,W) = [0 0 1 0] MBBBMl [3w2 2w 1 or
~32?0?3(w) (áþ —Coo) + 32?13(w) (fin-fioo)
+ 3B2f3(w) (Ñ12-å00) + ÇÂ3.çÈ (fiia-fioo). (5.139)
As can be seen from Eqs. (5.138) and (5.139), P„(0,w) and Puw(0,w) are both
expressed as linear combinations of the four vectors Ci0 — fi0o> Qn~Qoo^
Ci2 — Coo and C13-O00. From the properties of a Bezier surface, the direction of the tangent of curve P(u, 0) at point D coincides with the direction of 610 “600? also> the direction of the tangent to curve P(u, 1) at point D coincides with the direction of fii3 — Q00, so the tangent plane at point D is formed by these 2 vectors. Therefore, if fiu and fi12 lie on this plane, then
228
5. The Bernstein Approximation
Fig. 5.37. Application of a degenerate Bezier surface patch (rounded convex corner). D, degenerate point of patch; •, surface defining point Qtj
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