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