Computer Aided Engineering Design

226 COMPUTER AIDED ENGINEERING DESIGN

7.3 Composite Surfaces

Surface patches, in small units, need to be joined (stitched) together to form a larger surface. We can
observe this in surfaces such as car roof-tops, doors, side panals, engine-hood and also in aircraft
fuselage and wing-panels. In general, common boundary curves should match exactly (without any
gap) and the joint should not leave any wrinkles.
Similar treatment is performed when attempting to stitch two patches together at their common
boundaries as is the case with composite curves. Care is taken to maintain position (C^0 ), slope (C^1 )
and/or curvature (C^2 ) continuity at the boundary curves. Position continuity is obtained only when
the boundary curves of two adjoining patches coincide in which case, the slope along the boundary
curves is also continuous. A step further is to ensure a unique normal at any point on the common
boundary. This is accomplished by coinciding the tangent planes of the two adjacent patches at that
point. This section, discusses composite surfaces with Ferguson, Bézier and Coon’s patches.

7.3.1 Composite Ferguson’s Surface

An advantage with Ferguson’s bi-cubic patch is that at least the position (C^0 ) continuity is ensured
across patch boundaries because the corner points and slopes (and thus the boundary curves) are the
same for two adjacent patches. Consider the common boundary for patches I and II, for instance, in
Figure 7.17 which is a cubic curve in parameter v(u = 1 for patch I and u = 0 for patch II). It is
apparent that the slope rv is continuous along this common boundary. For patches I and III, the same
can be stated about the continuity of the slope ru along their common boundary. In addition to
position continuity, therefore, the slopes along the patch boundaries are also continuous for Ferguson’s
patches.

Note, however, that Eqs. (7.12) and (7.13) seem demanding from the user’s viewpoint as they
require higher order input (slopes and twist vectors) as a part of geometric information to be specified.
One way to avoid is: Given a set of data points Pij,i = 0,... , m and j = 0,... , n over which it is
required to fit a composite Ferguson surface (Figure 7.15), intermediate slopes sij (along u) and tij
(alongv) can be estimated as

s

PP

ij i PP

ij ij

ij ij

= C

• | – |

+1 –1

+1 –1

, where Ci = min (| Pij – Pi–1j |, | Pi+1j – Pij |)

Pij+2

Pi+1j+2

III

ti+1j+1

si+1j+1

Pi+1j+1

II Pi+2j+1

s Pi+2j

i+1j

I ti+1j

Sij

u

Pij

tij

Pi+1j

v

tij+1

Sij+1

Pij+1

Figure 7.17 Position and slope continuity across Ferguson’s patch boundaries