Computer Aided Engineering Design

DESIGN OF CURVES 125

|OD | = | OP 0 | sin θ

and so

| | = | | – | | = | |

1

sin

DP 11 OP OD OP 0 – sin

θ

⎛ θ

⎝

⎞

⎠

Also,

|DP | = | OP | – | OP | = | OP 0 | – | OD | = |OP 0 | (1 – sin θ)

Thus

||

||

=

(1 – sin ) sin

(1 – sin )

= sin

1 2 (1 + sin )

DP

DP

θθ

θ

θ

θ (4.69)

Comparing Eqs. (4.68) and (4.68), we have w = sin θ.

Example 4.11.For given data points P 0 = (1, 0), P 1 = (a,a) and P 2 = (0, 1), determine the circular
arcs using rational quadratic Bézier curves for different values of a. Also, draw the corresponding
circles of which the arcs are a part.
The included angle is given by

2 = cos

( – ) ( – )

| – | | – |

= cos

• 2 (1 – )
  
  
  • (1 – )
  
  
  
  

–1 21 01

2121

–1

22

θ

PP PP

PPPP

⎧ ⋅

⎨

⎩

⎫ ⎬ ⎭ ⎧ ⎨ ⎩

⎫

⎬

⎭

aa

aa

using which θ can be computed and the weight w = sin θ can be assigned to P 1. The center O of the
circle lies on the lines perpendicular to P 0 P 1 and P 2 P 1 with P 0 and P 2 as two points on the circle. The
equations of the lines containing the center are

y a

a

x

y a

a

x

• 1 =
  1 –

= 1 – ( – 1)

solving which gives the coordinates of the center as

1 –

1 – 2 ,

1 –

1 – 2

a

a

a

a

⎛

⎝

⎞

⎠. The radius r of the circle is

|OP 0 | = | OP 2 | =

(1 – ) +

(1 – 2 )

22

2

aa

a

Figure 4.24 depicts the circular arcs (thick lines) and the corresponding circles (dashed lines) for
different positions of P 1 on the line y = x. Note Figure 4.24 (e) for a =^12 when the three points are
collinear and the circular arc degenerates to a straight line.