Computer Aided Engineering Design

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