Graphics Reference
In-Depth Information
FIGURE 12.21
Example from video using L-systems to animate plantlike figures.
(Image courtesy of Vita Berezine-Blackburn, ACCAD.)
changes of plant growth can be successfully captured by L-systems. By adding parameters, time
variables, and communication modules, one can model other aspects of plant growth. Most recently,
Deussen et al. [ 5 ] have used open L-systems to model plant ecosystems.
12.3 Subdivision surfaces
Subdivision surfaces are useful in animation for designing objects in a top-down fashion [ 4 ]. Starting
from a coarse polyhedron, the geometry is refined a step at a time. Each step introduces more com-
plexity to the object, usually by rounding corners and edges. There are various methods proposed
in the literature for conducting the subdivision and much research has been performed in determining
what the limit surfaces look like and what their mathematical properties are.
As a simple subdivision example, consider the sequence in Figure 12.22 in which each vertex is
replaced by a face made from vertices one-third along the way down each edge emanating from the
vertex ( Figure 12.23 ) and each face is redefined to include the new vertices on the original edges
( Figure 12.24 ) .
One of the more popular subdivision schemes is due to Charles Loop [ 9 ] . For a closed, two-
dimensional, triangulated manifold, the Loop subdivision method creates a new vertex at the midpoint
of each edge and is repositioned. In addition, each original vertex is repositioned, and each triangle
is divided into four new triangles ( Figure 12.25 ) . The new edge midpoints are positioned according
to V M ¼
(3 V 1 þ
3 V 2 þ V A þ V B )/8 where V 1 and V 2 are the end vertices of the original edge and
 
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