Computer Aided Engineering Design

DESIGN OF CURVES 87

4.1 Ferguson’s or Hermite Cubic Segments

A cubic Ferguson’s segment is designed like an arch with two end points and two respective slopes
known (Figure 4.2). Let the end points be Pi(xi,yi,zi) at u = 0 and Pi+1 (xi+1,yi+1,zi+1) at u = 1. Also,
let the respective slopes be Ti(pi,qi,ri) and Ti+1(pi+1,qi+1,ri+1).Ti and Ti+1need not be of unit
magnitude and may be written in terms of respective unit vectors ti and ti+1 as Ti = citi and Ti+1 = ci+1
ti+1 for some scalars ci and ci+1. Consider the parametric variation only along the x coordinate, that is

x(u) = a 0 x + a 1 xu + a 2 xu^2 + a 3 xu^3 (4.3)

withx(0) = xi,x(1) = xi+1, and also dx(0)/dt = pi and dx(1)/dt = pi+1. We get

xi =a 0 x

xi+1 =a 0 x + a 1 x + a 2 x + a 3 x

pi =a 1 x

pi+1 =a 1 x + 2a 2 x + 3a 3 x

solving which gives

a 0 x=xi

a 1 x=pi

a 2 x= 3Δxi – 3pi – Δpi

a 3 x=Δpi – 2Δxi + 2pi (4.4)

whereΔxi = xi+1 – xi and Δpi = pi+1 – pi. The polynomial in Eq. (4.3) becomes

x(u) = xi + piu + (3Δxi – 3pi – Δpi)u^2 + (Δpi – 2Δxi + 2pi)u^3

= (1 – 3u^2 + 2u^3 )xi + (3u^2 – 2u^3 )xi+1 + (u – 2u^2 + u^3 )pi + (–u^2 + u^3 )pi+1

= Hux Hux 03 () + i 13 () + i+1 Hup Hup 23 () + i 33 ()i+ 1 (4.5)

Huii^3 ( ), = 0,... , 3 are functions of parameter uand are termed as Hermite polynomials. They serve
asblending functions orbasis functions orweightsto combine the end point and slope information
to generate the shape. At u = (0 and 1), Hu 03 ()= (1 and 0) while Hu 13 () = (0 and 1), and Hu 23 () and
Hu 33 () are both (0 and 0). This implies that at the end points, the slope information is not used. We
can further compute the first derivatives of Hermite basis functions to find that both

dH u

du

dH u

du

0

3

3

()^3

=

()

= 0 at u = 0 and 1. This decouples the data points with slopes and allows their

selective modification to change the shape of the cubic segment.
In matrix form, Eq. (4.5) can be expressed as

xu u u u

x

x

p

P

i

i

i

i

( ) = [ 1]

2–21 1

–3 3 –2 –1

0010

1000

(^32)
+1
+1
⎡
⎣
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥
⎡
⎣
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥
(4.6)
Figure 4.2 A cubic Ferguson segment
Ti+1
Ti Pi+1
Pi