Computer Aided Engineering Design

70 COMPUTER AIDED ENGINEERING DESIGN

α 0 = y 0 = 0

α

α

1

10

10

=

• 
  
  
  
  • =
  
  
  
  

2 – 0

1 – 0 = 2

y

xx

α

αα

2

20 120

2021

=

( – ) – ( – )

( – )( – )

=

(4 – 0) – 2(1.5 – 0)

(1.5 – 0)(1.5 – 1)

=^4

3

yxx

xxxx

α

αα α

3

30 130 23031

303132

=

( – ) – ( – ) – ( – )( – )

( – )( – )( – )

=

(–1) – 2 (6) –^4

3

(6)(5)

(6)(5)(4.5)

= –^53

135

yxxxxxx

xxxxxx

Thus, the new polynomial becomes

yn = 2 + x^4 xx xx x

3

( – 1) –^53

135

( – 1) –

3

2

⎛

⎝

⎞

⎠

Comparative plots of y and yn are provided in Figure 3.2 which shows the change in curve shape.
Moving a data point results in the shape change of the entire interpolated curve.
Curve interpolation provides a simple tool for curve design with data points governing the curve
shape. The degree of the interpolating polynomial is dependent on the number of data points specified.
Note that an n–1 degree polynomial has at most n–1 roots and thus crosses the xaxis at most n− 1
times. In cases where the number of data points is large, the number of real roots of a high degree
polynomial would be large. Such polynomials would then exhibit many oscillations or fluctuations
undesirable from the view point of curve design. It is required, therefore, to choose a polynomial of
a lower order (a polynomial of order m is of degree m–1) known a priori and determine its unknown
coefficients for which the polynomial best fits the given design points.

01 234 56

x

14

12

10

8

6

4

2

0

–2

y

Figure 3.2 Original curve (solid line) changed (dashed line) when a data

point is moved. Change is global with curve interpolation

3.2 Curve Fitting

Consider a set of n data points (xi,yi),i = 0, ..., n– 1 which are to be best fitted, say, by a quadratic
polynomial