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a binary value that is 1 if the ray intersects the primitive between the image plane
and infinity, and 0 otherwise. This is equivalent to the visibility function applied
to the primary ray origin and a point infinitely far away along the primary ray.
B
For a scene containing multiple primitives, primitives may occlude each other
in the camera's view. The
depth complexity
of a ray is the number of times it
intersects the scene. At any given intersection point
P
,the
quantitative invisibil-
ity
[App67] is the number of other primitive intersections that lie between the ray
origin and
P
. Figure 36.6 shows examples of each of these concepts.
For some applications, backfaces—surfaces where the dot product of the geo-
metric normal and the ray direction is positive—are ignored when computing
depth complexity and quantitative invisibility. The case where the intersection of
the ray and a surface is a segment (e.g., Figure 36.7) instead of a finite number
of points is tricky. We've already discussed that the existence of such an intersec-
tion is already suspect due to limited precision, and in fact for it to occur at all
with nonzero probability requires us to have explicitly placed geometry in just the
right configuration to make such an unlikely intersection representable. Of course,
humans are very good at constructing exactly those cases, for example, by plac-
ing edges at perfectly representable integer coordinates and aligning surfaces with
axes in ways that could never occur in data measured from the real world. We
note that for a closed polygonal model, it is common to ignore these line-segment
intersections, but the definition of depth complexity and quantitative invisibility
for this case varies throughout the literature.
A
Figure 36.6: The quantitative
invisibility of two points is the
number of surface intersections
on the segment between them.
The quantitative invisibility of B
with respect to A is 2 in this fig-
ure. The depth complexity of a
ray is the total number of surface
intersections along the ray. The
ray from A through B has depth
complexity 3 in this figure.
In the case of multiple primitive intersections, we say that the coverage is 1
at the first intersection and 0 at all later ones to match the visibility function
definition.
At the end of this chapter, in Section 36.9, we extend binary coverage to
par-
tial coverage
by considering multiple light paths per pixel.
Figure 36.7: A ray tangent to an
intersected surface.
Most rendering today is on triangles. The triangles may be explicitly created, or
they may be automatically generated from other shapes. Some common modeling
primitives that are reducible to triangles are subdivision surfaces, implicit sur-
faces, point clouds, lines, font glyphs, quadrilaterals, and height field models.
Ray-tracing renderers solve exact visibility determination by
ray casting
(Sec-
tion 36.2): intersecting the model with a ray to produce a sample. Data structures
optimized for ray-triangle intersection queries are therefore important for efficient
evaluation of the visibility function. Chapter 37 describes several of these data
structures.
Backface culling
(Section 36.6) is implicitly part of ray casting.
Hardware rasterization renderers today tend to use
frustum culling
(Sec-
tion 36.5),
frustum clipping
(Section 36.5),
backface culling
(Section 36.6),
and a
depth buffer
(Section 36.3) for per-sample visible surface determination.
Those methods provide correctness, but they require time linear in the number of
primitives. So relying on them exclusively would not scale to large scenes. For
efficiency it is therefore necessary to supplement those with conservative meth-
ods for determining occlusion and hierarchical methods for eliminating geometry
outside the view frustum in sublinear time.
A handful of applications rely on the
painter's algorithm
(Section 36.4.1)
of simply drawing everything in the scene in back-to-front order and letting the
