Computer Aided Engineering Design

96 COMPUTER AIDED ENGINEERING DESIGN

Also

d

d

d

du

du

d

uu

du

du

ji uu ujiu

rv rv

rv

r

r

()

=

()

() = ( – )

()

= ( – ) ( )

v

v

v

v

⇒ v v

The tangents of rv(v) at u = ui and u = uj are
d
d
uu
du
jidu

rv(0) = ( – ) r()i

v

and

d

d

uu

du

jidu

rv(1) r j

= ( – )

()

v

The geometric matrix Gvfor the trimmed segment becomes

Gv = [rv(0) rv(1) rvv(0) rvv(1)]T

= [r(ui) r(uj) (uj – ui)ru(ui) (uj – ui)ru(uj)]T (4.20)

and the equation for the same is

rv(v) = VMGv (4.21)

whereV = [v^3 v^2 v 1]. A similar approach can be used to re-parameterize two Ferguson segments
joined together with C^1 continuity.

4.1.3 Blending of Curve Segments

Curve blending is quite common in design and can be easily accomplished between two Ferguson
segments. Consider two curve segments AB and CD (r(1)(u 1 ) and r(3)(u 3 )) shown in Figure 4.7. The
gapbetweenB and C is filled by a blending curve r(2)(u 2 ) which can be determined as follows:
LetG 1 and G 3 be the geometric matrices of r(1)(u 1 ) and r(3)(u 3 ), respectively. From Figure 4.7
G 1 = [Pi Pi+1 Ti Ti+1]T
G 3 = [Pi+2 Pi+3 Ti+2 Ti+3]T

The geometric matrix G 2 for the blending curve between BandC can be written as

G 2 = [Pi+1 Pi+2 αTi+1 βTi+2]T (4.22)

where parameters α and β can be suitably varied to blend a large variety of curves maintaining C^1
continuity at B and C.

Example 4.4.A planar Hermite-Ferguson curve (1) starts at A (0, 0) and ends at B (4, 2). The tangent
vectors are given as Ti = (7, 7) and Ti+1 = (5, –8). Another curve (3) starts at C (8, 4) and ends at D

C

B

r(2)(u 2 )

r(3)(u 3 )

r(1)(u 1 )

Pi(xi,yi,zi)

Ti(pi,qi,ri)

Pi+1(xi+1,yi+1,zi+1)

Ti+1(pi+1,qi+1,ri+1)

Pi+2(xi+2,yi+2,zi+2)

Ti+2(pi+2,qi+2,ri+2)

Pi+3(xi+3,yi+3,zi+3)

Ti+3(pi+3,qi+3,ri+3)

Figure 4.7 Blending of two Ferguson curve segments

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D

A