Computer Aided Engineering Design

DESIGN OF CURVES 107

′≡

′

′

⎡

⎣

⎢

⎤

⎦

⎥

⎡

⎣

⎢

⎤

⎦

⎥ ′≡

′

′

⎡

⎣

⎢

⎤

⎦

⎥

⎡

⎣

⎢

⎤

⎦

A ⎥

x

y

xp

yq

B

x

y

xp

yq

=

• 
  


• 
  

and =

• 
  


• 
  

1

1

1

1

2

2

2

2

Let ′

′

′

⎡

⎣

⎢

⎤

⎦

⎥

′

′

⎡

⎣

⎢

⎤

⎦

C ⎥

x

y

x

y

* = +^1

1

2

2

λμ

=

• 
  


• 
  

+

• 
  


• 
  

=

+ – ( + )

+ – ( + )

1

1

2

2

12

12

λμ

λμ λμ

λμ λμ

xp

yq

xp

yq

xx p

yy q

⎡

⎣

⎢

⎤

⎦

⎥

⎡

⎣

⎢

⎤

⎦

⎥

⎡

⎣

⎢

⎤

⎦

⎥

=

{ + – } – ( + – 1)

{ + – } – ( + – 1)

12

12

λμ λμ

λμ λμ

xxp p

yyq q

⎡

⎣

⎢

⎤

⎦

⎥

Like points A and B, if C was expressed in the new coordinate system without any change in its
position due to the changed axes, then

′

⎡

⎣

⎢

⎤

⎦

C ⎥

xxp

yyq

=

+ –

+ –

12

12

λμ

λμ

Note that C′ and C′* are not identical, and there is a change in the position of C due to the shift in
the origin O to O′ by (p,q). To make this transformation also affine, the arbitrary scalars λ and μ need
to be constrained as

λ + μ – 1 = 0 or λ + μ = 1 (4.39)
Affine combination is therefore a type of linear combination wherein the respective weights sum
to unity. It is needed to preserve the relative positions of points (describing a geometric entity) during
transformation (change of coordinae axes), which is ensured by Bernstein polynomials in case of
Bézier curves.

4.4 Bézier Segments

Forn + 1 data points Pi,i = 0, ... , n, a Bézier segment is defined as their weighted linear combination
using Bernstein polynomials (Eq. (4.33))

r( ) = uCuu Bu ui=0 (1 – )– = =0 ( ) , 0 1

n

n i ni ii

i

n

i

ΣΣPPn i ≤≤ (4.40a)

y′ y

y′

A C

B

(OC = λOA + μOB)

x′

θ

θ

O x

(a) (b)

y

A C

B

(OC = λOA + μOB)

o′

O x

x′

Figure 4.15 Affine transformations and relation between weights λλλλλ and μ