Computer Aided Engineering Design

DESIGN OF SURFACES 213

= [ 1]

1

32

00 01 02 03

10 11 12 13

20 21 22 23

30 31 32 33

3

2

uuuMB BT

rrrr

rrrr

rrrr

rrrr

M

⎡

⎣

⎢

⎢

⎢

⎢

⎢

⎤

⎦

⎥

⎥

⎥

⎥

⎥

⎡

⎣

⎢

⎢

⎢

⎢

⎢

⎤

⎦

⎥

⎥

⎥

⎥

⎥

v

v

v (7.28)

whereMB is the Bézier coefficient matrix defined in Eq. (4.40).
The properties of Bézier curves are inherited by Bézier patches, some notable properties being: (a)
the four corner points of the patch are the respective corner points in the control polyhedron, (b)
boundary curves are tangent to the polyhedron edges at corner points and (c) the patch is contained
within the convex hull of the polyhedron. Most solid modeling packages use bi-quintic (m = n = 5)
or bi-septic (m = n = 7) patches to provide more flexibility to a user when designing a composite
surface. We know from Chapter 4 that when designing composite Bézier curves, 3 control points are
constrained to be collinear when requiring C^1 continuity at the junction point and, in addition, 2 more
control points (a total of five) are required to be coplanar for curvature continuity.

Example 7.3. The control points for a quadratic-cubic Bézier patch are given as

rrrr

rrrr

rrrr

00 01 02 03

10 11 12 13

20 21 22 23

=

(0, 0, 0) (1, 0, 1) (2, 0, 1) (3, 0, 0)

(0, 1, 0) (1, 1, 1) (2, 1, 1) (3, 1, 0)

(0, 2, 0) (1, 2, 1) (2, 2, 1) (3, 2, 0)

⎡

⎣

⎢

⎢

⎢

⎤

⎦

⎥

⎥

⎥

⎡

⎣

⎢

⎢

⎢

⎤

⎦

⎥

⎥

⎥

Plot the Bézier’s patch. Determine the unit normal, equation of the tangent plane and curvature at (u
= 0.5, v = 0.5) on the surface.
Equation (7.27) defines the surface with m = 2 and n = 3, two opposite boundaries are quadratic Bézier
curves and the remaining is a pair of cubic Bézier curves. The expression for the surface patch is

r

rrrr

rrrr

rrrr

( , ) = [(1 – ) 2 (1 – ) ]

(1 – )

3 (1 – )

3 (1 – )

22

00 01 02 03

10 11 12 13

20 21 22 23

3

2

2

3

uuuuuv

v

vv

vv

v

⎡

⎣

⎢

⎢

⎢

⎤

⎦

⎥

⎥

⎥

⎡

⎣

⎢

⎢

⎢

⎢

⎢

⎤

⎦

⎥

⎥

⎥

⎥

⎥

= [(1 – ) 2 (1 – ) ]

(0, 0, 0) (1, 0, 1) (2, 0, 1) (3, 0, 0)

(0, 1, 0) (1, 1, 1) (2, 1, 1) (3, 1, 0)

(0, 2, 0) (1, 2, 1) (2, 2, 1) (3, 2, 0)

(1 – )

3 (1 – )

3 (1 – )

22

3

2

2

3

uuuu

⎡

⎣

⎢

⎢

⎢

⎤

⎦

⎥

⎥

⎥

⎡

⎣

⎢

⎢

⎢

⎢

⎢

⎤

⎦

⎥

⎥

⎥

⎥

⎥

v

vv

vv

v

= [3v, 2u, 3v(1 –v)]

which is plotted in Figure 7.7 (a).
Unit normal to the surface is given by

n

rr

rr

(, ) =

(, ) (, )

| ( , ) ( , ) |

=

{6 – 12 , 0, – 6}

(6 – 12 ) + 36^2

u

uu

uu

u

u

v

vv

vv

v

v

v

v

×

× ⇒ n (0.5, 0.5) = {0, 0, –1}

Equation of the tangent plane at r(0.5, 0.5) ≡ {1.5, 1, 0.75} is