Graphics Reference
In-Depth Information
Figure 2.1
Geometrical models, like the one pictured, are commonly built from a collection
of polygon meshes.
erroneously ended up inside another object. The additional information mentioned
could include which edges connect to what vertices and what faces connect to a
given face, whether the object forms a closed solid, and whether the object is convex
or concave.
Polygons may be connected to one another at their edges to form a larger polygonal
surface called a
polygon mesh
. Building objects from a collection of polygon meshes is
one of the most common methods for authoring geometrical models (Figure 2.1).
Polygonal objects are defined in terms of their vertices, edges, and faces. When
constructed in this way, objects are said to have an
explicit
representation.
Implicit
objects refer to spheres, cones, cylinders, ellipsoids, tori, and other geometric prim-
itives that are not explicitly defined in such a manner but implicitly through a
mathematical expression. Implicit objects are often described as a function mapping
from 3D space to real numbers,
f
3
:
R
→
R
, where the points given by
f
(
x
,
y
,
z
)
<
0
constitute the interior,
f
(
x
,
y
,
z
)
0 the exterior of the
object (Figure 2.2). An object boundary defined by an implicit function is called an
implicit surface
. Implicit objects can be used as rough approximations of scene objects
for quick rejection culling. The implicit form may allow for fast intersection tests,
especially with lines and rays — a fact utilized in ray tracing applications. Several
examples of implicit tests are provided in Chapter 5.
Convex polygonal objects can also be described as the intersection of a number
of halfspaces. For example, a cube can be expressed as the intersection of six half-
spaces, each halfspace“trimming away”the portion of space that lies outside a face of
=
0 the boundary, and
f
(
x
,
y
,
z
)
>
