72 COMPUTER AIDED ENGINEERING DESIGN
ii=0ΣΣ Σxxi i i iΣyi
3
=03
=03
2
=03
1 = 4, = 8.5, = 39.25, = 5ΣΣΣ Σ
i i i ii i i i i i
xxyx xy
=03
3
=03
=03
4
=03
= 220.37, = 2, = 1302.06, 2 = – 25and the linear system is
4 8.5 39.25
8.5 39.25 220.37
39.25 220.37 1302.06=5
2
–25, or =- 0.16
3.35 - 0.58
0
1
20
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a01 23 45 6
x5 4 3 2 1 0–1–2yFigure 3.3 Results of curve fitting using a quadratic polynomial. Curve shape is
changed globally when a data point is relocatedThe resultant quadratic y = – 0.16 + 3.35x – 0.58x^2 is plotted (dashed lines) and compared with the
previous curve to note that the shape change is global.
The method described above is known as least square fitting. It is not mandatory always to employ
a polynomial for the purpose. Instead, any non-polynomial (trigonometric or exponential) function
may be chosen to best fit the given data. With polynomials of a chosen degree, although curve fitting
may resolve unwarranted fluctuations as is the case with curve interpolation, it would still not impart
local shape control to the designer (e.g. Example (3.2)) and changing a data point would require re-
computing the entire curve.
Both curve interpolation and fitting methods discussed above have some limitations with regard
to curve design. Before alternative methods are explored, it is first imperative to understand and
choose the best possible mathematical representation for curves to particularly suit their design in
three-dimensions. Based on the forgoing discussion, we may surmize that the use of low degree
polynomials (usually cubic) is preferable over the high degree ones to avoid unwarranted fluctuations.
Further, to allow local control on curve shape and close proximity to data points, piecewise fitting of
the consecutive subsets of data points could be considered. For instance, a cubic polynomial can
interpolate four data points, we may choose to interpolate four consecutive points at a time from a