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2. Catmull-Clark Subdivision Surfaces 37

S

left

4

0

1

5

6

9

10

15141312

11109

Boundary

patch

8

7654

3210

S

diagonal

S

top

V

boundary

V

right

V

down

V

diagonal

Figure 2.17. Extrapolated supporting control points for a convex corner boundary

vertex.

• Non-corner boundary vertices require one extrapolated support:

S

edge

=2V

boundary

− V

opposite

. (2.8)

• Convex corner boundary vertices require three extrapolated supports:

S

left

=2V

boundary

− V

right

,

S

diagonal

=4V

boundary

− 2V

right

− 2V

down

+ V

diagonal

,

S

top

=2V

boundary

− V

down

.

(2.9)

• Concave corner boundary vertices may not be extrapolated and require

recursive subdivision.

If all vertices of a control mesh face are regular or are borders that ...