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

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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+1«°Q8 + - -+(^)

=(||)(i-t)"+I«0eo"+-

...

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 ö-*ã

(5.96)

"+1 /è + 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 - î er] (5.98)

(i = 0, 1, ..., è+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 ... è 1]

Since:

~t” î ... 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

... î ... î

î 0

210

we obtain

5. The Bernstein Approximation

Ð1(ì) = [ìè è"'1 ... è 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

î 0 î 0 î 0

= [un un 1 ... è 1]

Qo

(l — ts)Qo + tsQi (^-ts)2Qo + 2(l-ts)tsQ1 + t2Q2

(i-o”Oo+( 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 (Îéêéï).

j = î

(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^Ö) — [{(1— ts)u + ts}n {(1 — ts)è + ts}n 1 ... (1 — ts)u + ts 1] P

Qo

Qx

(i -t,r

î

0 o'

1 -ts 0

ü«-1

Since we also have:

212

then**:

5. The Bernstein Approximation

P2(u) = [un un 1... è 1]

(1-Îè

î

L î

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

î

î

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"

è 1]/?

0,

e?] 1 er1

fi1-1!

er

(5.103)

Qrk] = (l-ts)n-kQk +

"f (”7'‘)(1-<»ã‘-^é

j = 0 \ J /

(l-ts)n-k-ltsQk+l + ... + trkQn

(5.104)

= 1 Bhn_k(ts)Qk+J (Îéêéï). j = î

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:

ëì= I (ostgi)

we wish to connect a second Bezier curve segment:

ë,«= i (ostsi)

j=î \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 /*ï(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:

Ë(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 — 20öä + Quo)

Using Eqs. (5.105) and (5.106) to eliminate 0IIO and Qn i gives:

«(«- i)/»,V

n(n-l) UJ &-2

-Ã^=ã (~)2+- -+Wl a--*

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