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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.
Download (direct link): curvesandsurfacesincomputer1988.djvu
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Q\ = Ql
1
-(4'Qi+O-Ql)
(5.94)
This relation is shown in Fig. 5.26.
In general, when an n-th degree Bezier curve segment is rewritten in an (n + l)-st degree format, the expressions of the curve defining vectors can be predicted from the right-hand side of Eq. (5.94). We will derive these relations
Fig. 5.26. Formal increase of the degree of a cubic Bezier curve segment (to quartic)
=[t [(i-o+(]
The first term of this expression can be expanded as follows: ..........................
= (p) (i-tr+1Q8 + - -+(^)
=(||)(i-t)"+I0eo"+-
...
Similarly, the second term can be expanded as:
z(;)-r'r-e
=^(i-o"ies+-+(^)(i-o0("+1e; =
207
(5.95)
5. The Bernstein Approximation
_l7(l-0+1^oft-1+(v0 1(1 -t)ntlQno+...
((l-ty+'-t'e?.,.
Therefore, Eq. (5.95) becomes:
n+1 /n\ n +1 / n \
pn(t)=i (^J(i-trl",tier+1 -*
(5.96)
"+1 / + 1\
n+i(o= i(^ . er+1. (5.97)
For all t in the range we must have:
Pn(t)=PH+1(t)
which gives:
This equation can be rewritten as:
er1=~ [i-!+(n+1 - er] (5.98)
(i = 0, 1, ..., +1).
Here Qn_x and Qn+1 are undefined, but their coefficients are 0, so they are irrelevant in calculating Q"+l.
Next, in the case of changing from an expression in terms of Q" to one in terms of Q"+k, let us focus on the expression for Q1+k. Using Eq. (5.98) gives:
5.1 Curves
209
=~[es+(-+i)^T (es+ne;)]
n + 2
(2 Qo + nQl).
Continuing this same procedure gives: +=^ <*+)
Of course we have:
Qo+k = Qno-
(5.99)
(5.100)
5.1.9 Partitioning of a Bezier Curve Segment
By a method to that used to divide a cubic Bezier curve segment (refer to Sect. 5.1.2), an n-th degree Bezier curve segment P(t) can be divided at point t = ts into two Bezier curve segments P^u) (O^i^l) and P2{u) (O^i^l). Equation
(5.69) is used for P(t). Performing the parameter transformation u = t/ts in Eq.
(5.69) gives:
[Qol
PM = [t>n tns~lUn~l ... tsu 1] P
= [un un 1 ... 1]
Since:
~t ... 0" Qo
0 trl ts 0 p Qi
0 ... 0 1 Qn_
0 t*~\
0 ... 1
l-*s
(l-O2
(1-0"
... O'
ts 0
... 1
0
ts
2(1-tjfs (i-ts)n~lts
(i -tr2ti
... ...
0
210
we obtain
5. The Bernstein Approximation
1() = [ "'1 ... 1] P
1 0
1 ts ts
(1 -tf 2(1 -ts)ts
(1-0 l'ly)(i-0"-4 ( 2 )(i-0"2^2
0 0 0
= [un un 1 ... 1]
Qo
(l ts)Qo + tsQi (^-ts)2Qo + 2(l-ts)tsQ1 + t2Q2
(i-oOo+( Ja-or^Qi+l 2 )(i-o"'2^2e2+-+^a
= [w" w" 1 ... M 1] /?
Ql00]
Ql0l]
Ql02]
Ql0n]
where:
(5.101)
Qlok]= a-ts)kQo+i1ia--tsritsQ1+i2\(i-ts)k-2ts2Q2+...+tskQb
= ? (;)d-Uf-HlQ,
}=o W /
= tBJ.MQj ().
j =
(5.102)
This could perhaps be anticipated from the n = 3 case discussed in Sect. 5 1.2
5.1 Curves
211
The vertex vectors for the curve defining polygon in the range 0^t^ts agree with the vectors Q[q], Q[01], Ql02], ..., Ql0n] found in the process of graphically determining the point t = ts on the curve.
Next, to find P2(u), perform the parameter transformation u = (t ?s)/(l ts) in equation (5.69) to give:
P2^) [{(1 ts)u + ts}n {(1 ts) + ts}n 1 ... (1 ts)u + ts 1] P
Qo
Qx
(i -t,r

0 o'
1 -ts 0
-1
Since we also have:
212
then**:
5. The Bernstein Approximation
P2(u) = [un un 1... 1]
(1-

L
(1 -0"'4 (1-f.r1


Qn-1 ft,
= [w" u" 1 ... W 1] P
a-ts)nQo+[ '1)a-ts)n-1tsQ1+^2)(i-tsr-2t?Q2+...+fsQn
(l-tsrlQ,+
(i-tsy-2tsQ2 + ... + trlQ
= [w" w"
1]/?
0,
e?] 1 er1
fi1-1!
er
(5.103)
Qrk] = (l-ts)n-kQk +
"f (7')(1-<-^
j = 0 \ J /
(l-ts)n-k-ltsQk+l + ... + trkQn
(5.104)
= 1 Bhn_k(ts)Qk+J (). j =
The vertex vectors for the curve defining polygon in the range ts^t^ 1 agree with the vectors Q\?\ ..., Qln0] found in the process of graphically
determining the point corresponding to t = ts on the curve.
* This can be anticipated from the case n = 3 in Sect 5 1.2
5.1 Curves 213
5.1.10 Connection of Bezier Curve Segments
To the Bezier curve segment:
= I (ostgi)
we wish to connect a second Bezier curve segment:
,= i (ostsi)
j= \J/
with continuity up to the curvature vector (refer to Fig. 5.27).
First, at the connection point, from the condition of continuity of position we have, from (1.49):
Qu,o = Qhm- (5-105)
The condition of continuity of slope at the connection points is expressed by Eq. (1.50). Using (5.81) and (5.82) gives:
(6ll,l Qll,o) {Ql,m~ Ql.m-l) = t- (5.106)
a 2 a!
Here ab a2 are the magnitudes of the tangent vectors Px( 1) and /*(0), respectively. Equation (5.106) requires that the 3 points Qhm-i, Qi,m=Qn,o and Qn i are colinear.
214 5. The Bernstein Approximation
The condition of continuity of curvature vectors at the connection point is expressed by Eq. (1.52). Using Eqs. (5.83) and (5.84), the 2nd derivative vectors at the connection point are, respectively:
(1) = m(m-1) (Qhm-2Qhm_1 + Qhm_2)
i*n(0) = n(n 1) (Qn2 2Qn i + QiU0).
Substituting these into Eq. (1.52) gives: n(n 1) (Qnt2 20 + Quo)
Using Eqs. (5.105) and (5.106) to eliminate 0IIO and Qn i gives:
(- i)/,V
n(n-l) UJ &-2
-^= (~)2+- -+Wl a--*
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