Computer Aided Engineering Design

204 COMPUTER AIDED ENGINEERING DESIGN

Construction of a tensor product surface patch using Hermite blending functions can be similarly
accomplished. We have to consider two parameters u and vand correspondingly, the two Hermite
blending functions Φi(u), (i = 0, 1, 2, 3) and Φj(v), (j = 0, 1, 2, 3). The equation of the surface (or
the position vector of any general point P on the surface) is given by

r

CCCC

CCCC

CCCC

CCCC

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

()

()

()

()

0123

00 01 02 03

10 11 12 13

20 21 22 23

30 31 32 33

0

1

2

3

uuuuuv

v

v

v

v

φφφφ

φ

φ

φ

φ

⎡

⎣

⎢ ⎢ ⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥ ⎥ ⎥

⎡

⎣

⎢ ⎢ ⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥ ⎥ ⎥

(7.7)

EachCij has 3 components and there are 16 of them. Thus there are (16 × 3) 48 unknowns to be
determined for constructing the Hermite tensor product surface. These can be determined from the
following data:

(a) four corner pointsr(0, 0), r(0, 1), r(1, 0) and r(1, 1) of the surface patch,
(b) eight tangents along the boundary curves r(0,v),r(1,v),r(u, 0), r(u, 1) with two at each corner
point. These slopes are given as

d

dv

d

dv

du

du

du

du

d

d

d

d

du

du

du

du

u

u

u

u

u

u

u

r

r

r

r

r

r

r

r

r

r

r

r

r

r

r

(0, )

= (0, 0)

(0, )

= (0, 1)

( , 0)

= (0, 0)

( , 0)

= (1, 0)

(1, )

= (1, 0)

(1, )

= (1, 1)

( , 1)

= (0, 1)

( , 1)

=

=0 =1 =0

=1 = 0 =1

=0 =1

vv

v

v

v

v

v

v

v

v

v

v

v

v

ru(1, 1)

(7.8)

(c) four twist vectors at the corners

∂

∂

∂

∂

∂

∂

∂

∂

2

=0, =0

2

=0, =1

2

=1, = 0

2

=1, =1

(, )

= (0, 0)

(, )

= (0, 1)

(, )

= (1, 0)

(, )

= (1, 1)

r

r

r

r

r

r

r

r

u

ud

u

ud

u

ud

u

ud

u

u

u

u

u

u

u

u

v

v

v

v

v

v

v

v

v

v

v

v

v

v

v

v

(7.9)

At any u = uj, there is a curve r(uj,v) and a tangent ru(uj,v). As we move along r(uj,v)
by varying v, we get different points r(uj,vi) on the surface as well as different tangents ru(uj,vi),
which vary both in direction and magnitude. Twist vectors ruv(uj,vi) represent the rate of change of
the tangent vector ru(uj,v) with respect to vatr(uj,vi). Function r(u,v) is such that the twist vectors
ruv(u,v) = rvu(u,v), that is, the partial mixed derivatives are symmetric with respect to u and v at
every point on the surface.
Expanding the right hand side of Eq. (7.7) and using the Hermite blending functions φ and
derivatives, we can evaluate Cij as