Computer Aided Engineering Design

DESIGN OF SURFACES 217

i + j + k = n. The value of n is user’s choice. A large n will carry finer details of the patch but at
increased computational cost. The number of control points used are^12 (n + 1)(n + 2).
Indexi = 0 corresponds to the left side of the triangle, j = 0 to the base and k = 0 to the right side
of the triangular table of control points. There are n + 1 points on each side of the triangle. The
surface patch is defined by

rr( , , ) =!

++= !!!

uw n

ijk

uw

ijknijk

vvΣ ij k,u + v + w = 1, i + j + k = n (7.29)

The three boundary curves are given by {u = 0, v,w = (1 –v)}, {u = (1 –w),v = 0, w} and
{u,v = (1 –u),w = 0}. Thus,

rr() =! r

!!

(1 – ) =!

!( – )!

(1 – )

+= 0, , =0 0, , –

vvvvvΣΣ –

ik n jk

jk

j

n

jn j jnj

n

jk

n

jn j

rr() =! r

!!

(1 – ) =!

!( – )!

(1 – )

+= ,,0 =0 ,( – ),0

u n –

ij

uu n

in i

uu

ijnij

ij

i

n

ΣΣini ini (7.30)

rr() =! r

!!

(1 – ) =!

!( – )!

(1 – )

+= ,0, =0 ( – ),0,

w n –

ik

ww n

kn k

ww

kinik

ki

k

n

ΣΣnk k knk

Example 7.6. Generate a triangular Bézier patch with n = 2 and the following 6 control points:

r 020 = (1, 3, 1)

r 011 = (0.5, 1, 0); r 110 = (1.5, 1, 0)

r 002 = (0, 0, 0); r 101 = (1, 0, –1);r 200 = (2, 0, 0)

The patch generated using Eq. (7.30) is shown in Figure 7.11.

Figure 7.11 Triangular surface patch.