Graphics Reference
In-Depth Information
CHAPTER
12
Special Models for Animation
Often, how an object is defined is intimately woven with how it is animated. We have already seen
examples of this, such as hierarchical linkages, special models for fluids, and so forth. This chapter
contains a collection of such special models that often have a role to play in animation. These models
are implicit surfaces, L-systems, and subdivision surfaces. Some of these modeling techniques have
already been mentioned in previous chapters (e.g., implicit surfaces in
Chapter 8
and subdivision
surfaces in the
Chapter 10
). This chapter is intended to give the interested reader some additional
information on these modeling techniques.
12.1
Implicit surfaces
Implicitly defined surfaces are surfaces defined by all those points that satisfy an equation of the form,
f
(
P
)
0;
f
(
P
) is called the
implicit function
. A common approach to using implicit surfaces to define
objects useful for animation is to construct a compound implicit function as a summation of implicitly
defined primitive functions. Interesting animations can be produced by animating the relative position
and orientation of the primitive functions or by animating parameters used to define the functions them-
selves. Implicit surfaces lend themselves to shapes that are smooth and organic looking. Animated
implicit surfaces are useful for shapes that change their topology over time. They are often used for
modeling liquids, clouds, and animals.
An extensive presentation of implicit surface formulations is not appropriate material for this topic,
but it can be found in several sources, in particular in Bloomenthal's edited volume [
3
]. A brief over-
view of commonly used implicit surfaces and their use in animation is presented here.
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12.1.1
Basic implicit surface formulation
Typically, an
implicit surface
is defined by the collection of points that are the zero points of some
implicit function,
f
(
P
)
0. The surface is referred to as implicit because it is only implicitly defined,
not explicitly represented. As a result, when an explicit definition of the surface is needed, as in a
graphics display procedure, the surface has to be searched for by inspecting points in space in some
organized way.
Implicit surfaces can be directly ray traced. Rays are constructed according to the camera param-
eters and display resolution, as is typical ray tracers. Points along the ray are then sampled to locate a
surface point within some error tolerance.
An explicit polygonal representation of an implicitly defined surface can be constructed by sam-
pling the implicit function at the vertices of a three-dimensional mesh that is constructed so that its
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