106 COMPUTER AIDED ENGINEERING DESIGN

=

( – 1)!

( – 1)!( – )!

(1 – ) –

( – 1)!

!( – 1 – )!

n – –1 (1 – )–1–

n

ini

uu

n

in i

⎡ ni i uun i i

⎣⎢

⎤

⎦⎥

=n [n–1Ci–1(1 – u)n–iui–1 – n–1Ci(1 – u)n–1–iui]

= [nBin–1–1( ) – u Bin–1( )]u

4.3 Barycentric Coordinates and Affine Transformation

In addition to constraining a Bézier curve to lie within the convex hull of the control polyline,
Bernstein polynomials also allow to describe the curve in space independent of the coordinate frame.
The shape of a given curve, surface, or solid should not depend on the choice of the coordinate
system. In other words, the relative positions of points describing a curve, surface, or solid should
remain unaltered during rotation or translation of the chosen axes. Consider for instance, two points
A(x 1 ,y 1 ) and B(x 2 ,y 2 ) in a two-dimensional space defined by an origin O and a set of axes Ox-Oy with
unit vectors (i,j). Let point C be defined as a linear combination of position vectors OA and OB, that
is,OC = λOA + μOB, where λ and μ are scalars. In terms of the ordered pair, C is then (λx 1 + μx 2 ,
λy 1 , + μy 2 ).
The axes Ox-Oy are rotated through an angle θ about the z-axis to form a new set of axes Ox′-Oy′
with unit vectors (i′,j′). Let A and B be described by (, )xy 11 ′′ and (, )xy 22 ′′ under the new coordinate
system for which the new definition of CisCx x y y′′ ′ ′ ′(λμλ μ 1212 + , + ). From Chapter 2, the rotation
matrix transforming (i,j) to (i′,j′) is given by

Rz =

cos – sin

sin cos

θθ

θθ

⎡

⎣

⎢

⎤

⎦

⎥

A and B ae placed at the same location in space. However, their new coordinates are now

′≡

′

′

⎡

⎣

⎢

⎤

⎦

⎥

⎡

⎣

⎢

⎤

⎦

⎥

⎡

⎣

⎢

⎤

⎦

⎥ ′≡

′

′

⎡

⎣

⎢

⎤

⎦

⎥

⎡

⎣

⎢

⎤

⎦

⎥

⎡

⎣

⎢

⎤

⎦

A ⎥

x

y

x

y

B

x

y

x

y

=

cos – sin

sin cos

and =

cos – sin

sin cos

(^1)
1
1
1
2
2
2
2
θθ
θθ
θθ
θθ
Now, let us define
′ ≡ ′′′′
′
′
⎡
⎣
⎢
⎤
⎦
⎥
′
′
⎡
⎣
⎢
⎤
⎦
Cxxyy ⎥
x
y
x
y

• 
  

  ( 121 + , + ) = +
  1
  1
  2
  2
  λμλμ λ μ

  
  

  cos – sin
  sin cos

  
  

• 
  

  cos – sin
  sin cos
  1
  1
  2
  2
  λ
  θθ
  θθ
  μ
  θθ
  θθ
  ⎡
  ⎣
  ⎢
  ⎤
  ⎦
  ⎥
  ⎡
  ⎣
  ⎢
  ⎤
  ⎦
  ⎥
  ⎡
  ⎣
  ⎢
  ⎤
  ⎦
  ⎥
  ⎡
  ⎣
  ⎢
  ⎤
  ⎦
  ⎥
  x
  y
  x
  y

  
  

  cos – sin
  sin cos

  
  


• 
  


• 
  

  =
  cos – sin
  sin cos

  
  

  12
  12
  θθ
  θθ
  λμ
  λμ
  θθ
  θθ
  ⎡
  ⎣
  ⎢
  ⎤
  ⎦
  ⎥
  ⎡
  ⎣
  ⎢
  ⎤
  ⎦
  ⎥
  ⎡
  ⎣
  ⎢
  ⎤
  ⎦
  ⎥ ′
  xx
  yy
  CC
  This implies that the relative positions of A,B and C remain unaltered after rotation and thus rotation
  transformation is affine.
  Next, consider a new set of axes O′x′-O′y′ formed by shifting the origin O to O′ by a vector (p,q)
  as in Fig. 4.15(b). The set O′x′-O′y′ is parallel to Ox-Oy and thus the unit vectors stay the same, i.e.,
  (i′,j′) = (i,j). The coordinates of points A and B in the transformed system is given by