Computer Aided Engineering Design

DESIGN OF CURVES 89

From Eq. (4.9), T(u) = ru(u) = UM 1 G with Ti = citi and Ti+1 = ci+1ti+1. Therefore

T( ) = [ 1]

0000

6–63 3

–6 6 – 4 –2

0010

–1 2

85

83

6–14

uuuu^32

⎡

⎣

⎢

⎢

⎢

⎢

⎤

⎦

⎥

⎥

⎥

⎥

⎡

⎣

⎢

⎢

⎢

⎢

⎤

⎦

⎥

⎥

⎥

⎥

= (–12uu^22 + 10 + 8) + (– 51ijuu + 34 + 3) = r ̇

d

du

uu uu u

T

= rij( ) = (– 24 + 10) + (–102 + 34) = ̇ ̇r

Thus,

T( = 0.5) = (0.5) = (–3 + 5 + 8) + (–12.75 + 17 + 3) = 10 + 7.25u ̇ri jij

|T | = √ (10^2 + 7.25^2 ) = 12.35

̇ ̇rij( = 0.5) = – 2 – 17u

The curvature is given by

κ = | | / | | = 155.5

(12.35)

3

r ̇× ̇ ̇r r ̇ 3. So ρ = 1/κ = 1/0.083 = 12.11

The bi-normal B is
̇r / | | = –1×× ̇ ̇r r ̇ ̇ ̇r.k k55 5 / 155.5 = –

The principal normal

N = B× T

|T|

= –k× (0.81i + 0.59j) = –0.81j + 0.59i

It may be left as an exercise to show that the torsion τ =

( )

| |^2

̇r ̇ ̇r ̇ ̇ ̇r

r ̇ ̇ ̇r

×⋅

×

at u = 0.5 is zero.

Effect of the tangent magnitudes ciandci+1
Keeping the end points fixed, Figure 4.3 shows the change in curve shape when the end tangent
magnitudes are altered. For increase in ci, the curve leans towards Pi+1 and eventually forms a cusp.
At higher values, a loop is formed. For a two-dimensional cubic segment, there are 8 unknowns (a 0 x,
a 1 x,a 2 x,a 3 x,a 0 y,a 1 y,a 2 y and a 3 y) needing eight conditions for evaluation. Four conditions are available
from the given end locations (x(0),y(0);x(1),y(1). The direction cosines of unit tangents ti = (tix,tiy)
andti+1 = (ti+1x,ti+1y) provide only two of the other four conditions. This is because ttix^22 + = 1iy and

ttix+1^2 + iy+1^2 = 1 implying that only two among the parameters (tix,tiy,ti+1x,ti+1y) may be supplied
while normalization constraints the other two. The remaining two conditions are supplied as the
magnitudes with vectors ti and ti+1, that is, ci and ci+1. The tangent magnitudes, therefore, play a vital
role in shaping a Hermite-Ferguson curve while preserving the location of the end points and the
direction of tangent vectors.

4.1.1 Composite Ferguson Curves

Following on the notion that the overall curve is a piecewise fit of individual cubic Ferguson
segments, consider any two neighboring segments of a composite curve, r(1)(u 1 ) and r(2)(u 2 ) with
0 ≤u 1 ,u 2 ≤ 1. Here, the superscripts (1) and (2) refer to the curve segments and u 1 and u 2 are the