Computer Aided Engineering Design

DESIGN OF SURFACES 205

rCrCr Cr C

rCrCrCrC

rCr Cr Cr C

rCrC

(0, 0) = ; (1, 0) = ; (0, 0) = ; (1, 0) =

(0, 1) = ; (1, 1) = ; (0, 1) = ; (1, 1) =

(0, 0) = ; (1, 0) = ; (0, 0) = ; (1, 0) =

(0, 1) = ; (1, 1) =

00 10 20 30

01 11 21 31

02 12 22 32

03

uu

uu

vvvvuu

vv 1313 ; rCrCuuvv(0, 1) = 23 ; (1, 1) = 33

(7.10)

so that Eq. (7.7) can now be written in the form

r

rrr r

rrr r

rrr r

rrr r

( , ) = [ ( ) ( ) ( ) ( )]

(0, 0) (0, 1) (0, 0) (0, 1)

(1, 0) (1, 1) (1, 0) (1, 1)

(0, 0) (0, 1) (0, 0) (0, 1)

(1, 0) (1, 1) (1, 0) (1, 1)

()

()

0123

0

1

uuuuu

uuu u

uuu u

v

v

v

vv

vv

vv

vv

φφφφ

φ

φ

⎡

⎣

⎢

⎢

⎢

⎢

⎢

⎤

⎦

⎥

⎥

⎥

⎥

⎥

φφ

φ

2

3

()

()

v

v

⎡

⎣

⎢

⎢

⎢

⎢

⎢

⎤

⎦

⎥

⎥

⎥

⎥

⎥

(7.11)

Knowing that [() () () ()]φφφφ 0123 uu uu = UMin Eq. (4.7), Eq. (7.11) can be written as

r(u,v) = UMGMTVT (7.12)

withM as the Ferguson’s coefficient matrix in Eqs. ((4.7), (7.6)), V as [v^3 v^2 v 1] and G, the
geometric matrix as

G

rrr r

rrr r

rrr r

rrr r

=

(0, 0) (0, 1) (0, 0) (0, 1)

(1, 0) (1, 1) (1, 0) (1, 1)

(0, 0) (0, 1) (0, 0) (0, 1)

(1, 0) (1, 1) (1, 0) (1, 1)

vv

vv

vv

vv

uuu u

uuu u

⎡

⎣

⎢

⎢

⎢

⎢

⎤

⎦

⎥

⎥

⎥

⎥

(7.13)

It is convenient to express G in the partitioned matrix form given in Eq. (7.14). The top left entries
are corner points, the bottom left are the corner tangents (with respect to u) to the boundary curves
v= 0 and v= 1, and top right are corner tangents to the boundary curves at u= 0 and u= 1. The
bottom right entries indicate the twist vectors at the corners of the surface patch.

rv(0, 1)

ru(0, 1)

Curve (v= 1)

r 11

ru(1, 1)

Curve (u = 1)

ru(1, 0)

r 10

rv(1, 0)

Curve (v = 0)

r 00

rv(0, 0)

ru(0, 0)

r(ui,vj)

Curve (u = 0) r^01

rv(1, 1)

Figure 7.2 Hermite-Ferguson patch