DESIGN OF CURVES 105
(c) Symmetry:
Bu Bin( ) = nin–(1 – )u (4.36)
Though suggested in Figure 4.14, the property is shown as follows:
Bu C u u
n
in i
inn( ) = i(1 – )ni i = uuni i
!
!( – )!
––(1 – )
=
!
!( – )!
( ) (1 – )–
n
pn p
ttpnpfort= (1 – u),n–i = p
= nCttBtB up(1 – )np p– = pn( ) = nin–(1 – )
(d) Recursion: The polynomials can be computed by the recursive relationship
Buin( ) = (1 – )uB u uB uin–1( ) + in–1–1( ) (4.37)
This is expected inherently from the de Casteljau’s algorithm. We can show Eq. (4.37) to be true
using the definition of Bernstein polynomials. Considering the right hand side
(1 – ) ( ) + ( ) =
( – 1)!
( )!( – 1 – )!
(1 – ) +
( – 1)!
( – 1)!( – )!
uB–1u uB–1–1u – (1 – )–
n
in i
uu
n
ini
in inniiniiuu
=
( – 1)!
( – 1)!( – 1 – )!
(1 – )^1 +
1
n –
in i
uu
ini
ni i⎛
⎝
⎞
⎠
=
( – 1)!
( – 1)!( – 1 – )!
(1 – )
( – )
n –
in i
uu
n
in i
ni i⎛
⎝
⎞
⎠
=
()!
()!( – )!
(1 – )– = ( )
n
in i
uuBuni i in
(e) Derivative: The derivative with respect to u has a recursive form
dB u
du
i nB u B u
n
i
n
i
() = () – n ()
–1
–1 –1
[]
where BuBu–1n–1() = nn–1() = 0 (4.38)
By definition
Bu
n
in i
innii() = uu
!
!( – )!
(1 – )–
dB u
du
n
in i
i ni u u i u u
n
() =! nii nii
!( – )!
[–( – )(1 – )–1– + (1 – )– –1]
= –
!
!( – 1 – )!
(1 – ) +
!
( – 1)!( – )!
–1– (1 – )– –1
n
in i
uu
n
ini
nii uunii