Graphics Reference
In-Depth Information
(4) Editing disk radii: This allows a user to round, thicken, or thin parts of the
object in uniform or nonuniform ways.
The bending operation in particular shows why the medial axis representation has an
advantage over a b-rep. With a b-rep such an operation can produce tears if one is
not careful. Although bending the medial axis may produce tears or intersections, the
refleshing operation removes all that.
Medial axis computations have many applications. Just to list a few topics and
references, they are used in finite element mesh generation ([STGLS97]), shape opti-
mization and robot path planning ([GelD95]), and pattern analysis and shape recog-
nition ([FarR98]). See [Nack82] for relationships between the curvature of a surface
and curvature functions associated to its medial axis representation.
Finally, related to the medial axis are the
level sets
of [LazV99] and the
Reeb
graph
of [ShKK91] and [ShiK91]. With level sets the goal was to describe both the
topology and geometry of the object, whereas with the Reeb graph the goal was to
encode the topology. Both of these approaches are based on the handle decomposi-
tion of manifolds that is central to the classification of manifolds. See Chapter 8 in
[AgoM04]. Reeb graphs have also been useful for volume data mining ([FTAT00]).
5.4
Modeling Natural Phenomena
Except for the pixel- and voxel-based types, the representation schemes we have dis-
cussed so far are not very useful for modeling natural phenomena. Objects such as
trees, mountains, grass, or various terrain cannot easily be modeled by linear poly-
hedra or smooth surface patches. Using very small pieces in the representation would
overwhelm one with massive amounts of data. Even if this were not a problem, it
would not be a satisfactory solution. The picture might look all right at the start, but
what if one were to zoom in? One would have to adjust the fineness of the subdivi-
sion dynamically to prevent things from eventually looking flat. Modeling and ren-
dering natural phenomena is a digression from the main thrust of this topic. For that
reason, we shall only take a brief look at this subject. The four topics we consider are
fractals, iterated function systems, grammar based models, and particle systems.
Fractals.
One of the most important applications of fractals to graphics is in the
representation of natural phenomena. For a definition of a fractal, see Section 22.3.
They enable one to represent such phenomena in a realistic way with a small amount
of data. The zooming problem also is no problem here. There is one caveat however.
Fractals are typically used to represent “generic” trees, mountains, or whatever. They
do not lend themselves easily to represent a specific tree or mountain. This is usually
not an issue though.
Why are fractals so great for modeling certain natural phenomena? To begin with
let us show how fractal curves and surfaces can be generated. The basic construction
generalizes that of the well-known Koch curve (see Section 22.3).
In the one-dimensional case, the algorithm starts with a given initial polygonal
curve and then generates a sequence of new curves, each of which adds more detail
to the previous one. In every iteration we replace each segment of the old curve with
