Graphics Reference
In-Depth Information
Chapter 24
of Shape
We introduced implicit functions in Chapters 7 and 14 as a means for defining
shapes. Implicitly defined shapes, like the circle defined by
x
2
+
y
2
=
1, or the
sphere defined by
x
2
+
y
2
+
z
2
=
1, or far more general shapes defined by equations
of the form
F
(
P
)=
c
for some complicated function
F
, serve several roles in
graphics. First, for a wide class of functions, computing ray-surface intersections
with such shapes is fairly easy. Second, it's sometimes convenient to represent
surfaces like “the boundary between water and air” in a simulation implicitly,
because it's very easy to change the topology of an implicitly defined surface (by
changing either
F
or
c
), while it's generally difficult to do so for parametrically
defined surfaces. Third, in many applications we find ourselves with data defined
on a grid of points (the temperature at each point in a nuclear reactor, for instance,
or the material density at each point in a CAT scan of a brain) and we wish to
visualize this data; often, seeing the level surface (the set of points where the
function has a particular value) for a function that's consistent with the observed
data can help us understand the data. In this chapter, we introduce implicit curves
and surfaces and discuss how they are used to model shapes, how they can be used
in ray tracing and animation, and how they can be converted to polyhedral meshes.
The main advantages of implicit representations are the general smoothness
of the shapes defined this way, the simplicity of creating quite general shapes,
the ease of defining shapes whose topology changes over time, and the ability to
exactly compute surface normals and other geometric properties (many of which
are difficult to estimate for polyhedral surfaces). The disadvantages are that con-
verting an implicit representation to a polygon mesh suitable for most renderers
can be very expensive, and that the ability of implicits to represent multiple topolo-
gies can also make it difficult to
control
the topology of an implicitly defined
shape.
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