124 COMPUTER AIDED ENGINEERING DESIGN

P

PPP

() =

(1 – ) + 2 (1 – ) +

(1 – ) + 2 (1 – ) +

(^20122)
t 22
twttt
twttt
(4.67)
wherew 0 = w 1 = 1 and w 2 = w to retain symmetry in the rational polynomials like in Bernstein
polynomials. Referring to Figure 4.9, the midpoint P of the curve is given by
P
PPP

• 2 +
  2(1 + )
  012 w
  w
  Now,
  DP P D
  PPP
  PP
  PP P
  = – =


• 2 +
  2(1 + )

-^1
2
( + ) =

(2 – – )

2(1 + )

012

02

w 10 2

w

w

w

while

DP P D P P P

PP P

11 1 0 2

= – = –^1 10 2

2

( + ) =

(2 – – )

2

and so

||

||

= =

1 1 +

DP

DP

α w

w (4.68)

We can achieve different conic sections as follows. For α < 0.5 (or w < 1), one gets an elliptic
segment. For α = 0.5 (or w = 1), the segment is parabolic while for α > 0.5 (or w > 1), the segment
is hyperbolic.
To draw a circular arc with an included angle 2θ (Figure 4.23) using the rational quadratic Bézier
segment, we have

|| =

||

(^1) sin
OP OP^0
θ
P 2
P 1
P 0
θ
θ
D
P
O
Figure 4.23 Design of a circular arc with rational Bézier curve