Computer Aided Engineering Design

122 COMPUTER AIDED ENGINEERING DESIGN

PPH

i

n

i

n

i

H

i

n

i

n

ii

ii

ii

i

t

Xt

Yt

Zt

Wt

Bt Bt

wx

wy

wz

w

()

()

()

()

()

= ( ) = ( )

=0 =0

≡

⎡

⎣

⎢

⎢

⎢

⎢

⎢

⎤

⎦

⎥

⎥

⎥

⎥

⎥

⎡

⎣

⎢

⎢

⎢

⎢

⎢

⎤

⎦

⎥

⎥

⎥

⎥

⎥

ΣΣ

(4.65)

The corresponding Euclidean coordinates can then be computed as

xt

Xt

Wt

Btwx

Btw

i Br t x

n

i

n ii

i

n

i

n

i

i

n

i

n

() = i

()

()

=

()

()

=0 = ( )

=0

=0

Σ

Σ

Σ

xt

Yt

Wt

Btwy

Btw

i Br t y

n

i

n

ii

i

n

i

n i i

n

i

n

() = i

()

()

=

()

()

=0 = ( )

=0

=0

Σ

Σ

Σ

xt

Zt

Wt

Btwz

Btw

i Br t z

n

i

n

ii

i

n

i

n

i

i

n

i

() = () n i

()

=

()

()

=0 = ( )

=0

=0

Σ

Σ

Σ

(4.66)

where Br t

wB t

wB t

i

n i i

n

i

n

i i

n

() =

()

()

=0

Σ

are the rational Bernstein polynomials in t for which reason P(t) =

Figure 4.21 A C^2 continuous composite Bézier curve

λ = 0.5, μ = 1

λ = 1, μ = 0.5

first segment

2

1.5

1

0.5

0

z

2

0

–2

–4

–6

–8

0

5

10

x

y