100 COMPUTER AIDED ENGINEERING DESIGN

Note that for α =^12 , the coefficient of u^3 is zero and r(u) is of degree 2. In fact, we can show that

r(u) = (16u^2 – 16u + 4)i+ (16u – 8)j=x(u)i + y(u)j

Fory(u) = 16u – 8 or u
y

• 8
  16
  , substituting into x(u), the above results in

x

yy

= 16

+ 8

16

• 16

+ 8

16

+ 4

2

⎛

⎝

⎞

⎠

⎛

⎝

⎞

⎠

or 16x = y^2 which is a parabola. Without proof, for α = 0.3 and 0.8, respectively, the ellipse and
hyperbola are also shown in Figure 4.10.

–4 –2 0 2 4

x(u)

8

6

4

2

0

–2

–4

–6

–8

y(u) Hyperbola Parabola

Ellipse

Figure 4.10 Conics with Ferguson’s segments

4.1.5 Need for Other Geometric Models for the Curve

Relocating the end points or altering the magnitudes and/or direction of the end tangents results in
shape change of a Ferguson segment. However, (a) it is not as intuitive to specify the tangent
information, and the designer is more comfortable in specifying the data points. (b) For C^1 continuous
composite Ferguson curves, modifying a data point or its slope would result in local shape change
of the curve as discussed in Section 4.1.1. For C^2 continuous composite Ferguson curves, however,
modifying any data point would result in re-computation of slopes (Eq. 4.17) resulting in an overall
shape change of the composite curve. A user would therefore seek a design method that allows
specifying only data points while maintaining local shape control properties for the entire curve.
Pierre Etienne Bézier , who worked with Renault, a French car manufacturer in 1970s, developed
a method to mathematically describe the curves and surfaces of an automobile body using data points
(henceforth referred to as control points). By shifting these points, the shape of the curve contained
within some local region could be changed predictably. Bézier created the UNISURF CAD system
for designing car bodies which utilized his curve theories. Paul de Faget de Casteljau’s work with car
manufacturer Citroen had similar results earlier than Bézier (1960s). Both works on Bézier curves are
based on Bernstein polynomials developed much earlier by the Russian mathematician Sergei Natanovich
Bernstein in 1912 as his research on approximation theory. The geometric construction of a parabola
using the three tangent theorem is first discussed. The construction is generalized to generate a curve
of any degree, attributed to de Casteljau. The resultant Bernstein polynomials have certain properties
useful in predicting shape change in Bézier curves.