206 COMPUTER AIDED ENGINEERING DESIGN
Grr r r
rr r rrr r r
rr r 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
vvvv
vvuu u u
uu u u⎡⎣⎢
⎢
⎢
⎢⎤⎦⎥
⎥
⎥
⎥(7.14)The algebraic form in Eq. (7.5) and the geometric form in Eq. (7.12) are equivalent. For m = n = 3,
rDDDDD
DDDD
DDDD
DDDD( , ) = = [ 1]1=03
=03
3233 32 31 30
23 22 21 20
13 12 11 10
03 02 01 003
2
uuuuu
ji ij
vvijv
v
vΣΣ
⎡⎣⎢
⎢
⎢
⎢⎤⎦⎥
⎥
⎥
⎥⎡⎣⎢
⎢
⎢
⎢⎤⎦⎥
⎥
⎥
⎥=UDVT = UMGMTVT ⇒ D = MGMT or G = M–1D(MT)–1implying that the algebraic coefficients Dij and geometric coefficients Gij can be obtained from each
other, each having three components (3-tuple).
A simple Ferguson’s patch can be expressed using the following geometric matrixGrr rr
rr rrrr
rr=(0, 0) (0, 1) | (0, 0) (0, 1)
(1, 0) (1, 1) | (1, 0) (1, 1)
––|––
(0, 0) (0, 1) | 0 0
(1, 0) (1, 1) | 0 0vv
vvuu
uu⎡⎣⎢ ⎢ ⎢ ⎢ ⎢ ⎢⎤⎦⎥ ⎥ ⎥ ⎥ ⎥ ⎥(7.15a)This has been found to be convenient because one can have an intuitive feeling for the corner points
and for the tangents, to a certain extent. It is quite difficult to have any intuitive feel about the twist
vectors.
In general, the tangents and twist vectors in Eq. (7.13) can be expressed as unit vectors (t) in given
directions along with magnitude values. Using short notation such as ru(a,b) = αabtabu, the geometric
matrix can be expressed as follows
Grr tt
rr tt
tttt
tttt=(^000100000101)
(^101110101111)
(^0000010100000101)
(^1010111110101111)
ββ
ββ
ααγ γ
ααγ γ
vv
vv
vv
vv
uuuu
uuuu
⎡
⎣
⎢
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥
⎥
(7.15b)
The 12 coefficients α,β and γ can be selectively changed to get a desired surface (recall the change
in shape of the Hermite PC curve by selecting the values of the tangent magnitudes, in Chapter 4).
7.1.2 Shape Interrogation
Shape interrogation is to extract the differential properties like curvatures, normal and tangents
(discussed in Chapter 6) for a surface patch. The unit normal is given by Eq. (6.4) as