Computer Aided Engineering Design

234 COMPUTER AIDED ENGINEERING DESIGN

To match the degree in v,λ(v) = λ, a constant, while μ(v) = μ 0 + μ 1 v, a linear function in v. With this
condition, cross boundary tangents are discontinuous across patch boundaries. Composite patch
boundaries would have positional continuity but gradient discontinuity at all patch corners. However,
the tangent directions of all four patch boundaries meeting at the intersection are coplanar. Resulting
conditions impose the constraints shown in Figure 7.25 on the two adjacent polyhedra. For polyhedron
II, for instance, the number of data points to be freely chosen becomes 10 as opposed to 8 in Case I.
For patch C (Figure 7.24) using this scheme, the number of freely chosen data points is 10 while for
patch D, they are 8. Case II is, therefore, less restrictive than Case I in terms of freely specifying the
control points.

Example 7.11.To blend two bi-cubic Bézier patches, the control points for a bi-cubic Bézier patch
rI(u,v) are given as

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

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

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

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

⎡

⎣

⎢

⎢

⎢

⎢

⎤

⎦

⎥

⎥

⎥

⎥

while for an adjacent patch are given as

rrrr

rrrr

00 01 02 03

10 11 12 13

II

(0, 6, 0) (1, 6, 5) (2, 6, 5) (3, 6, 0)

(0, 7, 0) (1, 8, 0) (2, 8, 0) (3, 7, 0)

⎡

⎣

⎢

⎢

⎢

⎢

⎤

⎦

⎥

⎥

⎥

⎥

where [r 00 ,r 01 ,r 02 ,r 03 ,r 10 ,r 11 ,r 12 ,r 13 ]II are to be determined to achieve position and slope
continuity at the common boundary. Determine the unknown control points and show the composite
surfaces.
The position continuity requires that the boundary polygon must be common between the two
patches. Thus

Figure 7.25 Arrangement of polyhedral edges for gradient continuity (Case II)

Coplanar edges

(shown using

dark edges)

Polyhedron I Polyhedron II