<s>
In	O
3D	O
computer	O
graphics	O
,	O
ray	B-Algorithm
tracing	I-Algorithm
is	O
a	O
technique	O
for	O
modeling	O
light	O
transport	O
for	O
use	O
in	O
a	O
wide	O
variety	O
of	O
rendering	O
algorithms	O
for	O
generating	O
digital	B-Algorithm
images	I-Algorithm
.	O
</s>
<s>
On	O
a	O
spectrum	O
of	O
computational	O
cost	O
and	O
visual	O
fidelity	O
,	O
ray	O
tracing-based	O
rendering	O
techniques	O
,	O
such	O
as	O
ray	B-Algorithm
casting	I-Algorithm
,	O
recursive	O
ray	B-Algorithm
tracing	I-Algorithm
,	O
distribution	B-Algorithm
ray	I-Algorithm
tracing	I-Algorithm
,	O
photon	B-Application
mapping	I-Application
and	O
path	B-Algorithm
tracing	I-Algorithm
,	O
are	O
generally	O
slower	O
and	O
higher	O
fidelity	O
than	O
scanline	B-Algorithm
rendering	I-Algorithm
methods	O
.	O
</s>
<s>
Thus	O
,	O
ray	B-Algorithm
tracing	I-Algorithm
was	O
first	O
deployed	O
in	O
applications	O
where	O
taking	O
a	O
relatively	O
long	O
time	O
to	O
render	O
could	O
be	O
tolerated	O
,	O
such	O
as	O
still	O
computer-generated	O
images	O
,	O
and	O
film	O
and	O
television	O
visual	O
effects	O
(	O
VFX	O
)	O
,	O
but	O
was	O
less	O
suited	O
to	O
real-time	B-General_Concept
applications	O
such	O
as	O
video	O
games	O
,	O
where	O
speed	O
is	O
critical	O
in	O
rendering	O
each	O
frame	O
.	O
</s>
<s>
Since	O
2019	O
,	O
however	O
,	O
hardware	B-Algorithm
acceleration	I-Algorithm
for	O
real-time	B-General_Concept
ray	B-Algorithm
tracing	I-Algorithm
has	O
become	O
standard	O
on	O
new	O
commercial	O
graphics	O
cards	O
,	O
and	O
graphics	O
APIs	O
have	O
followed	O
suit	O
,	O
allowing	O
developers	O
to	O
use	O
hybrid	O
ray	B-Algorithm
tracing	I-Algorithm
and	O
rasterization-based	O
rendering	O
in	O
games	O
and	O
other	O
real-time	B-General_Concept
applications	O
with	O
a	O
lesser	O
hit	O
to	O
frame	O
render	O
times	O
.	O
</s>
<s>
Ray	B-Algorithm
tracing	I-Algorithm
is	O
capable	O
of	O
simulating	O
a	O
variety	O
of	O
optical	B-Algorithm
effects	O
,	O
such	O
as	O
reflection	B-Algorithm
,	O
refraction	O
,	O
soft	O
shadows	O
,	O
scattering	O
,	O
depth	O
of	O
field	O
,	O
motion	O
blur	O
,	O
caustics	O
,	O
ambient	O
occlusion	O
and	O
dispersion	O
phenomena	O
(	O
such	O
as	O
chromatic	O
aberration	O
)	O
.	O
</s>
<s>
It	O
can	O
also	O
be	O
used	O
to	O
trace	O
the	O
path	O
of	O
sound	O
waves	O
in	O
a	O
similar	O
fashion	O
to	O
light	O
waves	O
,	O
making	O
it	O
a	O
viable	O
option	O
for	O
more	O
immersive	O
sound	O
design	O
in	O
video	O
games	O
by	O
rendering	O
realistic	O
reverberation	O
and	O
echoes	B-Protocol
.	O
</s>
<s>
In	O
fact	O
,	O
any	O
physical	O
wave	O
or	O
particle	O
phenomenon	O
with	O
approximately	O
linear	O
motion	O
can	O
be	O
simulated	O
with	O
ray	B-Algorithm
tracing	I-Algorithm
.	O
</s>
<s>
Ray	O
tracing-based	O
rendering	O
techniques	O
that	O
involve	O
sampling	O
light	O
over	O
a	O
domain	O
generate	O
image	B-Algorithm
noise	I-Algorithm
artifacts	O
that	O
can	O
be	O
addressed	O
by	O
tracing	O
a	O
very	O
large	O
number	O
of	O
rays	O
or	O
using	O
denoising	O
techniques	O
.	O
</s>
<s>
The	O
idea	O
of	O
ray	B-Algorithm
tracing	I-Algorithm
comes	O
from	O
as	O
early	O
as	O
the	O
16th	O
century	O
when	O
it	O
was	O
described	O
by	O
Albrecht	O
Dürer	O
,	O
who	O
is	O
credited	O
for	O
its	O
invention	O
.	O
</s>
<s>
Others	O
determine	O
what	O
geometry	O
is	O
visible	O
along	O
a	O
given	O
ray	O
,	O
as	O
is	O
done	O
with	O
ray	B-Algorithm
tracing	I-Algorithm
.	O
</s>
<s>
Using	O
a	O
computer	O
for	O
ray	B-Algorithm
tracing	I-Algorithm
to	O
generate	O
shaded	B-General_Concept
pictures	O
was	O
first	O
accomplished	O
by	O
Arthur	O
Appel	O
in	O
1968	O
.	O
</s>
<s>
Appel	O
used	O
ray	B-Algorithm
tracing	I-Algorithm
for	O
primary	O
visibility	O
(	O
determining	O
the	O
closest	O
surface	O
to	O
the	O
camera	O
at	O
each	O
image	O
point	O
)	O
,	O
and	O
traced	O
secondary	O
rays	O
to	O
the	O
light	O
source	O
from	O
each	O
point	O
being	O
shaded	B-General_Concept
to	O
determine	O
whether	O
the	O
point	O
was	O
in	O
shadow	O
or	O
not	O
.	O
</s>
<s>
Later	O
,	O
in	O
1971	O
,	O
Goldstein	O
and	O
Nagel	O
of	O
MAGI	O
(	O
Mathematical	O
Applications	O
Group	O
,	O
Inc	O
.	O
)	O
published	O
“	O
3-D	O
Visual	O
Simulation	O
”	O
,	O
wherein	O
ray	B-Algorithm
tracing	I-Algorithm
is	O
used	O
to	O
make	O
shaded	B-General_Concept
pictures	O
of	O
solids	O
by	O
simulating	O
the	O
photographic	O
process	O
in	O
reverse	O
.	O
</s>
<s>
They	O
cast	O
a	O
ray	O
through	O
each	O
pixel	B-Algorithm
in	O
the	O
screen	O
into	O
the	O
scene	O
to	O
identify	O
the	O
visible	O
surface	O
.	O
</s>
<s>
This	O
non-recursive	O
ray	O
tracing-based	O
rendering	O
algorithm	O
is	O
today	O
called	O
"	O
ray	B-Algorithm
casting	I-Algorithm
"	O
.	O
</s>
<s>
At	O
the	O
ray-surface	O
intersection	B-Algorithm
point	O
found	O
,	O
they	O
computed	O
the	O
surface	O
normal	O
and	O
,	O
knowing	O
the	O
position	O
of	O
the	O
light	O
source	O
,	O
computed	O
the	O
brightness	O
of	O
the	O
pixel	B-Algorithm
on	O
the	O
screen	O
.	O
</s>
<s>
The	O
helicopter	O
was	O
programmed	O
to	O
undergo	O
a	O
series	O
of	O
maneuvers	O
including	O
turns	O
,	O
take-offs	O
,	O
and	O
landings	O
,	O
etc.	O
,	O
until	O
it	O
eventually	O
is	O
shot	O
down	O
and	O
crashed.	O
”	O
A	O
CDC	B-Device
6600	I-Device
computer	O
was	O
used	O
.	O
</s>
<s>
Another	O
early	O
instance	O
of	O
ray	B-Algorithm
casting	I-Algorithm
came	O
in	O
1976	O
,	O
when	O
Scott	O
Roth	O
created	O
a	O
flip	B-Algorithm
book	O
animation	O
in	O
Bob	O
Sproull	O
's	O
computer	O
graphics	O
course	O
at	O
Caltech	O
.	O
</s>
<s>
Roth	O
's	O
computer	O
program	O
noted	O
an	O
edge	O
point	O
at	O
a	O
pixel	B-Algorithm
location	O
if	O
the	O
ray	O
intersected	O
a	O
bounded	O
plane	O
different	O
from	O
that	O
of	O
its	O
neighbors	O
.	O
</s>
<s>
The	O
edges	O
are	O
jagged	O
because	O
only	O
a	O
coarse	O
resolution	O
was	O
practical	O
with	O
the	O
computing	O
power	O
of	O
the	O
time-sharing	O
DEC	B-Device
PDP-10	I-Device
used	O
.	O
</s>
<s>
Roth	O
extended	O
the	O
framework	O
,	O
introduced	O
the	O
term	O
ray	B-Algorithm
casting	I-Algorithm
in	O
the	O
context	O
of	O
computer	O
graphics	O
and	O
solid	O
modeling	O
,	O
and	O
later	O
published	O
his	O
work	O
while	O
at	O
GM	O
Research	O
Labs	O
.	O
</s>
<s>
Turner	O
Whitted	O
was	O
the	O
first	O
to	O
show	O
recursive	O
ray	B-Algorithm
tracing	I-Algorithm
for	O
mirror	O
reflection	B-Algorithm
and	O
for	O
refraction	O
through	O
translucent	O
objects	O
,	O
with	O
an	O
angle	O
determined	O
by	O
the	O
solid	O
's	O
index	O
of	O
refraction	O
,	O
and	O
to	O
use	O
ray	B-Algorithm
tracing	I-Algorithm
for	O
anti-aliasing	O
.	O
</s>
<s>
Whitted	O
's	O
deeply	O
recursive	O
ray	B-Algorithm
tracing	I-Algorithm
algorithm	O
reframed	O
rendering	O
from	O
being	O
primarily	O
a	O
matter	O
of	O
surface	O
visibility	O
determination	O
to	O
being	O
a	O
matter	O
of	O
light	O
transport	O
.	O
</s>
<s>
His	O
paper	O
inspired	O
a	O
series	O
of	O
subsequent	O
work	O
by	O
others	O
that	O
included	O
distribution	B-Algorithm
ray	I-Algorithm
tracing	I-Algorithm
and	O
finally	O
unbiased	B-Application
path	B-Algorithm
tracing	I-Algorithm
,	O
which	O
provides	O
the	O
rendering	O
equation	O
framework	O
that	O
has	O
allowed	O
computer	O
generated	O
imagery	O
to	O
be	O
faithful	O
to	O
reality	O
.	O
</s>
<s>
For	O
decades	O
,	O
global	B-Algorithm
illumination	I-Algorithm
in	O
major	O
films	O
using	O
computer	O
generated	O
imagery	O
was	O
faked	O
with	O
additional	O
lights	O
.	O
</s>
<s>
Early	O
feature	O
films	O
rendered	O
entirely	O
using	O
path	B-Algorithm
tracing	I-Algorithm
include	O
Monster	O
House	O
(	O
2006	O
)	O
,	O
Cloudy	O
with	O
a	O
Chance	O
of	O
Meatballs	O
(	O
2009	O
)	O
,	O
and	O
Monsters	O
University	O
(	O
2013	O
)	O
.	O
</s>
<s>
Optical	B-Algorithm
ray	B-Algorithm
tracing	I-Algorithm
describes	O
a	O
method	O
for	O
producing	O
visual	O
images	O
constructed	O
in	O
3D	O
computer	O
graphics	O
environments	O
,	O
with	O
more	O
photorealism	O
than	O
either	O
ray	B-Algorithm
casting	I-Algorithm
or	O
scanline	B-Algorithm
rendering	I-Algorithm
techniques	O
.	O
</s>
<s>
It	O
works	O
by	O
tracing	O
a	O
path	O
from	O
an	O
imaginary	O
eye	O
through	O
each	O
pixel	B-Algorithm
in	O
a	O
virtual	O
screen	O
,	O
and	O
calculating	O
the	O
color	O
of	O
the	O
object	O
visible	O
through	O
it	O
.	O
</s>
<s>
Scenes	O
in	O
ray	B-Algorithm
tracing	I-Algorithm
are	O
described	O
mathematically	O
by	O
a	O
programmer	O
or	O
by	O
a	O
visual	O
artist	O
(	O
normally	O
using	O
intermediary	O
tools	O
)	O
.	O
</s>
<s>
Typically	O
,	O
each	O
ray	O
must	O
be	O
tested	O
for	O
intersection	B-Algorithm
with	O
some	O
subset	O
of	O
all	O
the	O
objects	O
in	O
the	O
scene	O
.	O
</s>
<s>
Once	O
the	O
nearest	O
object	O
has	O
been	O
identified	O
,	O
the	O
algorithm	O
will	O
estimate	O
the	O
incoming	O
light	O
at	O
the	O
point	O
of	O
intersection	B-Algorithm
,	O
examine	O
the	O
material	O
properties	O
of	O
the	O
object	O
,	O
and	O
combine	O
this	O
information	O
to	O
calculate	O
the	O
final	O
color	O
of	O
the	O
pixel	B-Algorithm
.	O
</s>
<s>
Therefore	O
,	O
the	O
shortcut	O
taken	O
in	O
ray	B-Algorithm
tracing	I-Algorithm
is	O
to	O
presuppose	O
that	O
a	O
given	O
ray	O
intersects	O
the	O
view	O
frame	O
.	O
</s>
<s>
After	O
either	O
a	O
maximum	O
number	O
of	O
reflections	O
or	O
a	O
ray	O
traveling	O
a	O
certain	O
distance	O
without	O
intersection	B-Algorithm
,	O
the	O
ray	O
ceases	O
to	O
travel	O
and	O
the	O
pixel	B-Algorithm
's	O
value	O
is	O
updated	O
.	O
</s>
<s>
The	O
idea	O
is	O
to	O
find	O
the	O
position	O
of	O
each	O
viewport	O
pixel	B-Algorithm
center	O
which	O
allows	O
us	O
to	O
find	O
the	O
line	O
going	O
from	O
eye	O
through	O
that	O
pixel	B-Algorithm
and	O
finally	O
get	O
the	O
ray	O
described	O
by	O
point	O
and	O
vector	O
(	O
or	O
its	O
normalisation	O
)	O
.	O
</s>
<s>
First	O
we	O
need	O
to	O
find	O
the	O
coordinates	O
of	O
the	O
bottom	O
left	O
viewport	O
pixel	B-Algorithm
and	O
find	O
the	O
next	O
pixel	B-Algorithm
by	O
making	O
a	O
shift	O
along	O
directions	O
parallel	O
to	O
viewport	O
(	O
vectors	O
i	O
)	O
multiplied	O
by	O
the	O
size	O
of	O
the	O
pixel	B-Algorithm
.	O
</s>
<s>
One	O
can	O
think	O
of	O
this	O
"	O
ray	O
"	O
as	O
a	O
stream	O
of	O
photons	B-Application
traveling	O
along	O
the	O
same	O
path	O
.	O
</s>
<s>
Any	O
combination	O
of	O
four	O
things	O
might	O
happen	O
with	O
this	O
light	O
ray	O
:	O
absorption	O
,	O
reflection	B-Algorithm
,	O
refraction	O
and	O
fluorescence	O
.	O
</s>
<s>
Between	O
absorption	O
,	O
reflection	B-Algorithm
,	O
refraction	O
and	O
fluorescence	O
,	O
all	O
of	O
the	O
incoming	O
light	O
must	O
be	O
accounted	O
for	O
,	O
and	O
no	O
more	O
.	O
</s>
<s>
The	O
idea	O
behind	O
ray	B-Algorithm
casting	I-Algorithm
,	O
the	O
predecessor	O
to	O
recursive	O
ray	B-Algorithm
tracing	I-Algorithm
,	O
is	O
to	O
trace	O
rays	O
from	O
the	O
eye	O
,	O
one	O
per	O
pixel	B-Algorithm
,	O
and	O
find	O
the	O
closest	O
object	O
blocking	O
the	O
path	O
of	O
that	O
ray	O
.	O
</s>
<s>
Think	O
of	O
an	O
image	O
as	O
a	O
screen-door	O
,	O
with	O
each	O
square	O
in	O
the	O
screen	O
being	O
a	O
pixel	B-Algorithm
.	O
</s>
<s>
This	O
is	O
then	O
the	O
object	O
the	O
eye	O
sees	O
through	O
that	O
pixel	B-Algorithm
.	O
</s>
<s>
Using	O
the	O
material	O
properties	O
and	O
the	O
effect	O
of	O
the	O
lights	O
in	O
the	O
scene	O
,	O
this	O
algorithm	O
can	O
determine	O
the	O
shading	B-General_Concept
of	O
this	O
object	O
.	O
</s>
<s>
The	O
shading	B-General_Concept
of	O
the	O
surface	O
is	O
computed	O
using	O
traditional	O
3D	O
computer	O
graphics	O
shading	B-General_Concept
models	O
.	O
</s>
<s>
One	O
important	O
advantage	O
ray	B-Algorithm
casting	I-Algorithm
offered	O
over	O
older	O
scanline	B-Algorithm
algorithms	I-Algorithm
was	O
its	O
ability	O
to	O
easily	O
deal	O
with	O
non-planar	O
surfaces	O
and	O
solids	O
,	O
such	O
as	O
cones	O
and	O
spheres	O
.	O
</s>
<s>
If	O
a	O
mathematical	O
surface	O
can	O
be	O
intersected	O
by	O
a	O
ray	O
,	O
it	O
can	O
be	O
rendered	O
using	O
ray	B-Algorithm
casting	I-Algorithm
.	O
</s>
<s>
In	O
the	O
method	O
of	O
volume	O
ray	B-Algorithm
casting	I-Algorithm
,	O
each	O
ray	O
is	O
traced	O
so	O
that	O
color	O
and/or	O
density	O
can	O
be	O
sampled	O
along	O
the	O
ray	O
and	O
then	O
be	O
combined	O
into	O
a	O
final	O
pixel	B-Algorithm
color	O
.	O
</s>
<s>
In	O
SDF	O
ray	O
marching	O
,	O
or	O
sphere	O
tracing	O
,	O
each	O
ray	O
is	O
traced	O
in	O
multiple	O
steps	O
to	O
approximate	O
an	O
intersection	B-Algorithm
point	O
between	O
the	O
ray	O
and	O
a	O
surface	O
defined	O
by	O
a	O
signed	O
distance	O
function	O
(	O
SDF	O
)	O
.	O
</s>
<s>
Recursive	O
ray	B-Algorithm
tracing	I-Algorithm
continues	O
the	O
process	O
.	O
</s>
<s>
When	O
a	O
ray	O
hits	O
a	O
surface	O
,	O
additional	O
rays	O
may	O
be	O
cast	O
because	O
of	O
reflection	B-Algorithm
,	O
refraction	O
,	O
and	O
shadow	O
.	O
</s>
<s>
A	O
reflection	B-Algorithm
ray	O
is	O
traced	O
in	O
the	O
mirror-reflection	O
direction	O
.	O
</s>
<s>
The	O
closest	O
object	O
it	O
intersects	O
is	O
what	O
will	O
be	O
seen	O
in	O
the	O
reflection	B-Algorithm
.	O
</s>
<s>
Effects	O
such	O
as	O
reflections	O
and	O
shadows	O
,	O
which	O
are	O
difficult	O
to	O
simulate	O
using	O
other	O
algorithms	O
,	O
are	O
a	O
natural	O
result	O
of	O
the	O
ray	B-Algorithm
tracing	I-Algorithm
algorithm	O
.	O
</s>
<s>
The	O
computational	O
independence	O
of	O
each	O
ray	O
makes	O
ray	B-Algorithm
tracing	I-Algorithm
amenable	O
to	O
a	O
basic	O
level	O
of	O
parallelization	B-Operating_System
,	O
but	O
the	O
divergence	O
of	O
ray	O
paths	O
makes	O
high	O
utilization	O
under	O
parallelism	B-Operating_System
quite	O
difficult	O
to	O
achieve	O
in	O
practice	O
.	O
</s>
<s>
A	O
serious	O
disadvantage	O
of	O
ray	B-Algorithm
tracing	I-Algorithm
is	O
performance	O
(	O
though	O
it	O
can	O
in	O
theory	O
be	O
faster	O
than	O
traditional	O
scanline	B-Algorithm
rendering	I-Algorithm
depending	O
on	O
scene	O
complexity	O
vs.	O
number	O
of	O
pixels	B-Algorithm
on-screen	O
)	O
.	O
</s>
<s>
Until	O
the	O
late	O
2010s	O
,	O
ray	B-Algorithm
tracing	I-Algorithm
in	O
real	O
time	O
was	O
usually	O
considered	O
impossible	O
on	O
consumer	O
hardware	O
for	O
nontrivial	O
tasks	O
.	O
</s>
<s>
Scanline	B-Algorithm
algorithms	I-Algorithm
and	O
other	O
algorithms	O
use	O
data	O
coherence	O
to	O
share	O
computations	O
between	O
pixels	B-Algorithm
,	O
while	O
ray	B-Algorithm
tracing	I-Algorithm
normally	O
starts	O
the	O
process	O
anew	O
,	O
treating	O
each	O
eye	O
ray	O
separately	O
.	O
</s>
<s>
However	O
,	O
this	O
separation	O
offers	O
other	O
advantages	O
,	O
such	O
as	O
the	O
ability	O
to	O
shoot	O
more	O
rays	O
as	O
needed	O
to	O
perform	O
spatial	B-Algorithm
anti-aliasing	I-Algorithm
and	O
improve	O
image	O
quality	O
where	O
needed	O
.	O
</s>
<s>
Although	O
it	O
does	O
handle	O
interreflection	O
and	O
optical	B-Algorithm
effects	O
such	O
as	O
refraction	O
accurately	O
,	O
traditional	O
ray	B-Algorithm
tracing	I-Algorithm
is	O
also	O
not	O
necessarily	O
photorealistic	B-Application
.	O
</s>
<s>
Ray	B-Algorithm
tracing	I-Algorithm
,	O
if	O
it	O
is	O
limited	O
to	O
Whitted	O
's	O
algorithm	O
,	O
is	O
not	O
necessarily	O
the	O
most	O
realistic	O
.	O
</s>
<s>
Methods	O
that	O
trace	O
rays	O
,	O
but	O
include	O
additional	O
techniques	O
(	O
photon	B-Application
mapping	I-Application
,	O
path	B-Algorithm
tracing	I-Algorithm
)	O
,	O
give	O
a	O
far	O
more	O
accurate	O
simulation	O
of	O
real-world	O
lighting	O
.	O
</s>
<s>
The	O
process	O
of	O
shooting	O
rays	O
from	O
the	O
eye	O
to	O
the	O
light	O
source	O
to	O
render	O
an	O
image	O
is	O
sometimes	O
called	O
backwards	B-Algorithm
ray	I-Algorithm
tracing	I-Algorithm
,	O
since	O
it	O
is	O
the	O
opposite	O
direction	O
photons	B-Application
actually	O
travel	O
.	O
</s>
<s>
Early	O
ray	B-Algorithm
tracing	I-Algorithm
was	O
always	O
done	O
from	O
the	O
eye	O
,	O
and	O
early	O
researchers	O
such	O
as	O
James	O
Arvo	O
used	O
the	O
term	O
backwards	B-Algorithm
ray	I-Algorithm
tracing	I-Algorithm
to	O
mean	O
shooting	O
rays	O
from	O
the	O
lights	O
and	O
gathering	O
the	O
results	O
.	O
</s>
<s>
Therefore	O
,	O
it	O
is	O
clearer	O
to	O
distinguish	O
eye-based	O
versus	O
light-based	O
ray	B-Algorithm
tracing	I-Algorithm
.	O
</s>
<s>
While	O
the	O
direct	O
illumination	O
is	O
generally	O
best	O
sampled	O
using	O
eye-based	O
ray	B-Algorithm
tracing	I-Algorithm
,	O
certain	O
indirect	O
effects	O
can	O
benefit	O
from	O
rays	O
generated	O
from	O
the	O
lights	O
.	O
</s>
<s>
This	O
integration	O
of	O
eye-based	O
and	O
light-based	O
rays	O
is	O
often	O
expressed	O
as	O
bidirectional	B-Algorithm
path	I-Algorithm
tracing	I-Algorithm
,	O
in	O
which	O
paths	O
are	O
traced	O
from	O
both	O
the	O
eye	O
and	O
lights	O
,	O
and	O
the	O
paths	O
subsequently	O
joined	O
by	O
a	O
connecting	O
ray	O
after	O
some	O
length	O
.	O
</s>
<s>
Photon	B-Application
mapping	I-Application
is	O
another	O
method	O
that	O
uses	O
both	O
light-based	O
and	O
eye-based	O
ray	B-Algorithm
tracing	I-Algorithm
;	O
in	O
an	O
initial	O
pass	O
,	O
energetic	O
photons	B-Application
are	O
traced	O
along	O
rays	O
from	O
the	O
light	O
source	O
so	O
as	O
to	O
compute	O
an	O
estimate	O
of	O
radiant	O
flux	O
as	O
a	O
function	O
of	O
3-dimensional	O
space	O
(	O
the	O
eponymous	O
photon	B-Application
map	O
itself	O
)	O
.	O
</s>
<s>
In	O
a	O
subsequent	O
pass	O
,	O
rays	O
are	O
traced	O
from	O
the	O
eye	O
into	O
the	O
scene	O
to	O
determine	O
the	O
visible	O
surfaces	O
,	O
and	O
the	O
photon	B-Application
map	O
is	O
used	O
to	O
estimate	O
the	O
illumination	O
at	O
the	O
visible	O
surface	O
points	O
.	O
</s>
<s>
The	O
advantage	O
of	O
photon	B-Application
mapping	I-Application
versus	O
bidirectional	B-Algorithm
path	I-Algorithm
tracing	I-Algorithm
is	O
the	O
ability	O
to	O
achieve	O
significant	O
reuse	O
of	O
photons	B-Application
,	O
reducing	O
computation	O
,	O
at	O
the	O
cost	O
of	O
statistical	O
bias	O
.	O
</s>
<s>
An	O
additional	O
problem	O
occurs	O
when	O
light	O
must	O
pass	O
through	O
a	O
very	O
narrow	O
aperture	B-General_Concept
to	O
illuminate	O
the	O
scene	O
(	O
consider	O
a	O
darkened	O
room	O
,	O
with	O
a	O
door	O
slightly	O
ajar	O
leading	O
to	O
a	O
brightly	O
lit	O
room	O
)	O
,	O
or	O
a	O
scene	O
in	O
which	O
most	O
points	O
do	O
not	O
have	O
direct	O
line-of-sight	O
to	O
any	O
light	O
source	O
(	O
such	O
as	O
with	O
ceiling-directed	O
light	O
fixtures	O
or	O
torchieres	O
)	O
.	O
</s>
<s>
In	O
such	O
cases	O
,	O
only	O
a	O
very	O
small	O
subset	O
of	O
paths	O
will	O
transport	O
energy	O
;	O
Metropolis	B-Algorithm
light	I-Algorithm
transport	I-Algorithm
is	O
a	O
method	O
which	O
begins	O
with	O
a	O
random	O
search	O
of	O
the	O
path	O
space	O
,	O
and	O
when	O
energetic	O
paths	O
are	O
found	O
,	O
reuses	O
this	O
information	O
by	O
exploring	O
the	O
nearby	O
space	O
of	O
rays	O
.	O
</s>
<s>
First	O
,	O
a	O
ray	O
is	O
created	O
at	O
an	O
eyepoint	O
and	O
traced	O
through	O
a	O
pixel	B-Algorithm
and	O
into	O
the	O
scene	O
,	O
where	O
it	O
hits	O
a	O
diffuse	O
surface	O
.	O
</s>
<s>
From	O
that	O
surface	O
the	O
algorithm	O
recursively	O
generates	O
a	O
reflection	B-Algorithm
ray	O
,	O
which	O
is	O
traced	O
through	O
the	O
scene	O
,	O
where	O
it	O
hits	O
another	O
diffuse	O
surface	O
.	O
</s>
<s>
Finally	O
,	O
another	O
reflection	B-Algorithm
ray	O
is	O
generated	O
and	O
traced	O
through	O
the	O
scene	O
,	O
where	O
it	O
hits	O
the	O
light	O
source	O
and	O
is	O
absorbed	O
.	O
</s>
<s>
The	O
color	O
of	O
the	O
pixel	B-Algorithm
now	O
depends	O
on	O
the	O
colors	O
of	O
the	O
first	O
and	O
second	O
diffuse	O
surface	O
and	O
the	O
color	O
of	O
the	O
light	O
emitted	O
from	O
the	O
light	O
source	O
.	O
</s>
<s>
For	O
example	O
,	O
if	O
the	O
light	O
source	O
emitted	O
white	O
light	O
and	O
the	O
two	O
diffuse	O
surfaces	O
were	O
blue	O
,	O
then	O
the	O
resulting	O
color	O
of	O
the	O
pixel	B-Algorithm
is	O
blue	O
.	O
</s>
<s>
As	O
a	O
demonstration	O
of	O
the	O
principles	O
involved	O
in	O
ray	B-Algorithm
tracing	I-Algorithm
,	O
consider	O
how	O
one	O
would	O
find	O
the	O
intersection	B-Algorithm
between	O
a	O
ray	O
and	O
a	O
sphere	O
.	O
</s>
<s>
This	O
is	O
merely	O
the	O
math	O
behind	O
the	O
line	B-Algorithm
–	I-Algorithm
sphere	I-Algorithm
intersection	I-Algorithm
and	O
the	O
subsequent	O
determination	O
of	O
the	O
colour	O
of	O
the	O
pixel	B-Algorithm
being	O
calculated	O
.	O
</s>
<s>
There	O
is	O
,	O
of	O
course	O
,	O
far	O
more	O
to	O
the	O
general	O
process	O
of	O
ray	B-Algorithm
tracing	I-Algorithm
,	O
but	O
this	O
demonstrates	O
an	O
example	O
of	O
the	O
algorithms	O
used	O
.	O
</s>
<s>
The	O
laws	O
of	O
reflection	B-Algorithm
state	O
that	O
the	O
angle	O
of	O
reflection	B-Algorithm
is	O
equal	O
and	O
opposite	O
to	O
the	O
angle	O
of	O
incidence	O
between	O
the	O
incident	O
ray	O
and	O
the	O
normal	O
to	O
the	O
sphere	O
.	O
</s>
<s>
where	O
is	O
the	O
intersection	B-Algorithm
point	O
found	O
before	O
.	O
</s>
<s>
Now	O
we	O
only	O
need	O
to	O
compute	O
the	O
intersection	B-Algorithm
of	O
the	O
latter	O
ray	O
with	O
our	O
field	O
of	O
view	O
,	O
to	O
get	O
the	O
pixel	B-Algorithm
which	O
our	O
reflected	O
light	O
ray	O
will	O
hit	O
.	O
</s>
<s>
Lastly	O
,	O
this	O
pixel	B-Algorithm
is	O
set	O
to	O
an	O
appropriate	O
color	O
,	O
taking	O
into	O
account	O
how	O
the	O
color	O
of	O
the	O
original	O
light	O
source	O
and	O
the	O
one	O
of	O
the	O
sphere	O
are	O
combined	O
by	O
the	O
reflection	B-Algorithm
.	O
</s>
<s>
To	O
test	O
for	O
this	O
the	O
ray	B-Algorithm
tracer	I-Algorithm
must	O
compute	O
and	O
keep	O
the	O
product	O
of	O
the	O
global	O
and	O
reflection	B-Algorithm
coefficients	O
as	O
the	O
rays	O
are	O
traced	O
.	O
</s>
<s>
Then	O
from	O
the	O
first	O
surface	O
the	O
maximum	O
contribution	O
is	O
0.5	O
,	O
for	O
the	O
reflection	B-Algorithm
from	O
the	O
second	O
:	O
0.5	O
×	O
0.5	O
=	O
0.25	O
,	O
the	O
third	O
:	O
0.25	O
×	O
0.5	O
=	O
0.125	O
,	O
the	O
fourth	O
:	O
0.125	O
×	O
0.5	O
=	O
0.0625	O
,	O
the	O
fifth	O
:	O
0.0625	O
×	O
0.5	O
=	O
0.03125	O
,	O
etc	O
.	O
</s>
<s>
Enclosing	O
groups	O
of	O
objects	O
in	O
sets	O
of	O
hierarchical	B-Data_Structure
bounding	I-Data_Structure
volumes	I-Data_Structure
decreases	O
the	O
amount	O
of	O
computations	O
required	O
for	O
ray	B-Algorithm
tracing	I-Algorithm
.	O
</s>
<s>
A	O
cast	O
ray	O
is	O
first	O
tested	O
for	O
an	O
intersection	B-Algorithm
with	O
the	O
bounding	B-Algorithm
volume	I-Algorithm
,	O
and	O
then	O
if	O
there	O
is	O
an	O
intersection	B-Algorithm
,	O
the	O
volume	O
is	O
recursively	O
divided	O
until	O
the	O
ray	O
hits	O
the	O
object	O
.	O
</s>
<s>
The	O
best	O
type	O
of	O
bounding	B-Algorithm
volume	I-Algorithm
will	O
be	O
determined	O
by	O
the	O
shape	O
of	O
the	O
underlying	O
object	O
or	O
objects	O
.	O
</s>
<s>
Boxes	O
are	O
also	O
easier	O
to	O
generate	O
hierarchical	B-Data_Structure
bounding	I-Data_Structure
volumes	I-Data_Structure
.	O
</s>
<s>
Note	O
that	O
using	O
a	O
hierarchical	O
system	O
like	O
this	O
(	O
assuming	O
it	O
is	O
done	O
carefully	O
)	O
changes	O
the	O
intersection	B-Algorithm
computational	O
time	O
from	O
a	O
linear	O
dependence	O
on	O
the	O
number	O
of	O
objects	O
to	O
something	O
between	O
linear	O
and	O
a	O
logarithmic	O
dependence	O
.	O
</s>
<s>
This	O
is	O
because	O
,	O
for	O
a	O
perfect	O
case	O
,	O
each	O
intersection	B-Algorithm
test	O
would	O
divide	O
the	O
possibilities	O
by	O
two	O
,	O
and	O
result	O
in	O
a	O
binary	O
tree	O
type	O
structure	O
.	O
</s>
<s>
Kay	O
&	O
Kajiya	O
give	O
a	O
list	O
of	O
desired	O
properties	O
for	O
hierarchical	B-Data_Structure
bounding	I-Data_Structure
volumes	I-Data_Structure
:	O
</s>
<s>
The	O
sum	O
of	O
the	O
volumes	O
of	O
all	O
bounding	B-Algorithm
volumes	I-Algorithm
should	O
be	O
minimal	O
.	O
</s>
<s>
The	O
first	O
implementation	O
of	O
an	O
interactive	O
ray	B-Algorithm
tracer	I-Algorithm
was	O
the	O
LINKS-1	B-Device
Computer	I-Device
Graphics	I-Device
System	I-Device
built	O
in	O
1982	O
at	O
Osaka	O
University	O
's	O
School	O
of	O
Engineering	O
,	O
by	O
professors	O
Ohmura	O
Kouichi	O
,	O
Shirakawa	O
Isao	O
and	O
Kawata	O
Toru	O
with	O
50	O
students	O
.	O
</s>
<s>
It	O
was	O
a	O
massively	B-Operating_System
parallel	I-Operating_System
processing	I-Operating_System
computer	O
system	O
with	O
514	O
microprocessors	B-Architecture
(	O
257	O
Zilog	B-Device
Z8001s	I-Device
and	O
257	O
iAPX	B-General_Concept
86s	I-General_Concept
)	O
,	O
used	O
for	O
rendering	O
realistic	O
3D	O
computer	O
graphics	O
with	O
high-speed	O
ray	B-Algorithm
tracing	I-Algorithm
.	O
</s>
<s>
According	O
to	O
the	O
Information	O
Processing	O
Society	O
of	O
Japan	O
:	O
"	O
The	O
core	O
of	O
3D	O
image	O
rendering	O
is	O
calculating	O
the	O
luminance	O
of	O
each	O
pixel	B-Algorithm
making	O
up	O
a	O
rendered	O
surface	O
from	O
the	O
given	O
viewpoint	O
,	O
light	O
source	O
,	O
and	O
object	O
position	O
.	O
</s>
<s>
The	O
LINKS-1	O
system	O
was	O
developed	O
to	O
realize	O
an	O
image	O
rendering	O
methodology	O
in	O
which	O
each	O
pixel	B-Algorithm
could	O
be	O
parallel	O
processed	O
independently	O
using	O
ray	B-Algorithm
tracing	I-Algorithm
.	O
</s>
<s>
The	O
earliest	O
public	O
record	O
of	O
"	O
real-time	B-General_Concept
"	O
ray	B-Algorithm
tracing	I-Algorithm
with	O
interactive	O
rendering	O
(	O
i.e.	O
,	O
updates	O
greater	O
than	O
a	O
frame	O
per	O
second	O
)	O
was	O
credited	O
at	O
the	O
2005	O
SIGGRAPH	O
computer	O
graphics	O
conference	O
as	O
being	O
the	O
REMRT/RT	O
tools	O
developed	O
in	O
1986	O
by	O
Mike	O
Muuss	O
for	O
the	O
BRL-CAD	B-Language
solid	O
modeling	O
system	O
.	O
</s>
<s>
Initially	O
published	O
in	O
1987	O
at	O
USENIX	B-Operating_System
,	O
the	O
BRL-CAD	B-Language
ray	B-Algorithm
tracer	I-Algorithm
was	O
an	O
early	O
implementation	O
of	O
a	O
parallel	O
network	O
distributed	B-Algorithm
ray	I-Algorithm
tracing	I-Algorithm
system	O
that	O
achieved	O
several	O
frames	O
per	O
second	O
in	O
rendering	O
performance	O
.	O
</s>
<s>
This	O
performance	O
was	O
attained	O
by	O
means	O
of	O
the	O
highly	O
optimized	O
yet	O
platform	O
independent	O
LIBRT	O
ray	B-Algorithm
tracing	I-Algorithm
engine	O
in	O
BRL-CAD	B-Language
and	O
by	O
using	O
solid	O
implicit	O
CSG	O
geometry	O
on	O
several	O
shared	O
memory	O
parallel	B-Operating_System
machines	I-Operating_System
over	O
a	O
commodity	O
network	O
.	O
</s>
<s>
BRL-CAD	B-Language
'	O
s	O
ray	B-Algorithm
tracer	I-Algorithm
,	O
including	O
the	O
REMRT/RT	O
tools	O
,	O
continue	O
to	O
be	O
available	O
and	O
developed	O
today	O
as	O
open	B-Application
source	I-Application
software	I-Application
.	O
</s>
<s>
Since	O
then	O
,	O
there	O
have	O
been	O
considerable	O
efforts	O
and	O
research	O
towards	O
implementing	O
ray	B-Algorithm
tracing	I-Algorithm
at	O
real-time	B-General_Concept
speeds	O
for	O
a	O
variety	O
of	O
purposes	O
on	O
stand-alone	O
desktop	O
configurations	O
.	O
</s>
<s>
Some	O
real-time	B-General_Concept
software	O
3D	O
engines	O
based	O
on	O
ray	B-Algorithm
tracing	I-Algorithm
have	O
been	O
developed	O
by	O
hobbyist	O
demo	O
programmers	O
since	O
the	O
late	O
1990s	O
.	O
</s>
<s>
In	O
1999	O
a	O
team	O
from	O
the	O
University	O
of	O
Utah	O
,	O
led	O
by	O
Steven	O
Parker	O
,	O
demonstrated	O
interactive	O
ray	B-Algorithm
tracing	I-Algorithm
live	O
at	O
the	O
1999	O
Symposium	O
on	O
Interactive	O
3D	O
Graphics	O
.	O
</s>
<s>
They	O
rendered	O
a	O
35	O
million	O
sphere	O
model	O
at	O
512	O
by	O
512	O
pixel	B-Algorithm
resolution	O
,	O
running	O
at	O
approximately	O
15	O
frames	O
per	O
second	O
on	O
60	O
CPUs	O
.	O
</s>
<s>
The	O
Open	O
RT	O
project	O
included	O
a	O
highly	O
optimized	O
software	O
core	O
for	O
ray	B-Algorithm
tracing	I-Algorithm
along	O
with	O
an	O
OpenGL-like	O
API	O
in	O
order	O
to	O
offer	O
an	O
alternative	O
to	O
the	O
current	O
rasterisation	O
based	O
approach	O
for	O
interactive	O
3D	O
graphics	O
.	O
</s>
<s>
Ray	B-Algorithm
tracing	I-Algorithm
hardware	I-Algorithm
,	O
such	O
as	O
the	O
experimental	O
Ray	B-Algorithm
Processing	I-Algorithm
Unit	I-Algorithm
developed	O
by	O
Sven	O
Woop	O
at	O
the	O
Saarland	O
University	O
,	O
has	O
been	O
designed	O
to	O
accelerate	O
some	O
of	O
the	O
computationally	O
intensive	O
operations	O
of	O
ray	B-Algorithm
tracing	I-Algorithm
.	O
</s>
<s>
The	O
idea	O
that	O
video	O
games	O
could	O
ray-trace	B-Algorithm
their	O
graphics	O
in	O
real	O
time	O
received	O
media	O
attention	O
in	O
the	O
late	O
2000s	O
.	O
</s>
<s>
During	O
that	O
time	O
,	O
a	O
researcher	O
named	O
Daniel	O
Pohl	O
,	O
under	O
the	O
guidance	O
of	O
graphics	O
professor	O
Philipp	O
Slusallek	O
and	O
in	O
cooperation	O
with	O
the	O
Erlangen	O
University	O
and	O
Saarland	O
University	O
in	O
Germany	O
,	O
equipped	O
Quake	B-Application
III	I-Application
and	O
Quake	B-Application
IV	I-Application
with	O
an	O
engine	O
he	O
programmed	O
himself	O
,	O
which	O
Saarland	O
University	O
then	O
demonstrated	O
at	O
CeBIT	O
2007	O
.	O
</s>
<s>
On	O
June	O
12	O
,	O
2008	O
,	O
Intel	O
demonstrated	O
a	O
special	O
version	O
of	O
Enemy	B-Application
Territory	I-Application
:	I-Application
Quake	I-Application
Wars	I-Application
,	O
titled	O
Quake	B-Application
Wars	I-Application
:	O
Ray	O
Traced	O
,	O
using	O
ray	B-Algorithm
tracing	I-Algorithm
for	O
rendering	O
,	O
running	O
in	O
basic	O
HD	O
(	O
720p	O
)	O
resolution	O
.	O
</s>
<s>
ETQW	B-Application
operated	O
at	O
14	O
–	O
29	O
frames	O
per	O
second	O
on	O
a	O
16-core	O
(	O
4	O
socket	O
,	O
4	O
core	O
)	O
Xeon	O
Tigerton	O
system	O
running	O
at	O
2.93GHz	O
.	O
</s>
<s>
At	O
SIGGRAPH	O
2009	O
,	O
Nvidia	O
announced	O
OptiX	B-Algorithm
,	O
a	O
free	O
API	O
for	O
real-time	B-General_Concept
ray	B-Algorithm
tracing	I-Algorithm
on	O
Nvidia	O
GPUs	O
.	O
</s>
<s>
The	O
API	O
exposes	O
seven	O
programmable	O
entry	O
points	O
within	O
the	O
ray	B-Algorithm
tracing	I-Algorithm
pipeline	O
,	O
allowing	O
for	O
custom	O
cameras	O
,	O
ray-primitive	O
intersections	B-Algorithm
,	O
shaders	O
,	O
shadowing	O
,	O
etc	O
.	O
</s>
<s>
This	O
flexibility	O
enables	O
bidirectional	B-Algorithm
path	I-Algorithm
tracing	I-Algorithm
,	O
Metropolis	B-Algorithm
light	I-Algorithm
transport	I-Algorithm
,	O
and	O
many	O
other	O
rendering	O
algorithms	O
that	O
cannot	O
be	O
implemented	O
with	O
tail	O
recursion	O
.	O
</s>
<s>
OptiX-based	O
renderers	O
are	O
used	O
in	O
Autodesk	O
Arnold	O
,	O
Adobe	B-Operating_System
AfterEffects	I-Operating_System
,	O
Bunkspeed	O
Shot	O
,	O
Autodesk	B-Language
Maya	I-Language
,	O
3ds	B-Language
max	I-Language
,	O
and	O
many	O
other	O
renderers	O
.	O
</s>
<s>
In	O
2014	O
,	O
a	O
demo	O
of	O
the	B-Operating_System
PlayStation	I-Operating_System
4	I-Operating_System
video	O
game	O
The	B-Application
Tomorrow	I-Application
Children	I-Application
,	O
developed	O
by	O
Q-Games	O
and	O
Japan	O
Studio	O
,	O
demonstrated	O
new	O
lighting	O
techniques	O
developed	O
by	O
Q-Games	O
,	O
notably	O
cascaded	O
voxel	B-Algorithm
cone	B-Algorithm
ray	B-Algorithm
tracing	I-Algorithm
,	O
which	O
simulates	O
lighting	O
in	O
real-time	B-General_Concept
and	O
uses	O
more	O
realistic	O
reflections	O
rather	O
than	O
screen	O
space	O
reflections	O
.	O
</s>
<s>
Nvidia	O
introduced	O
their	O
GeForce	B-Device
RTX	I-Device
and	O
Quadro	O
RTX	O
GPUs	O
September	O
2018	O
,	O
based	O
on	O
the	O
Turing	O
architecture	O
that	O
allows	O
for	O
hardware-accelerated	O
ray	B-Algorithm
tracing	I-Algorithm
.	O
</s>
<s>
The	O
unit	O
features	O
BVH	O
traversal	O
,	O
compressed	O
BVH	O
node	O
decompression	O
,	O
ray-AABB	O
intersection	B-Algorithm
testing	O
,	O
and	O
ray-triangle	O
intersection	B-Algorithm
testing	O
.	O
</s>
<s>
The	O
GeForce	B-Device
RTX	I-Device
,	O
in	O
the	O
form	O
of	O
models	O
2080	O
and	O
2080	O
Ti	O
,	O
became	O
the	O
first	O
consumer-oriented	O
brand	O
of	O
graphics	O
card	O
that	O
can	O
process	O
ray	B-Algorithm
tracing	I-Algorithm
in	O
real	O
time	O
,	O
and	O
,	O
in	O
November	O
2018	O
,	O
Electronic	O
Arts	O
 '	O
Battlefield	B-Application
V	I-Application
became	O
the	O
first	O
game	O
to	O
take	O
advantage	O
of	O
its	O
ray	B-Algorithm
tracing	I-Algorithm
capabilities	O
,	O
which	O
it	O
achieves	O
via	O
Microsoft	O
's	O
new	O
API	O
,	O
DirectX	B-Algorithm
Raytracing	I-Algorithm
.	O
</s>
<s>
AMD	O
,	O
which	O
already	O
offered	O
interactive	O
ray	B-Algorithm
tracing	I-Algorithm
on	O
top	O
of	O
OpenCL	B-Application
through	O
its	O
Radeon	O
ProRender	O
,	O
unveiled	O
in	O
October	O
2020	O
the	O
Radeon	B-Device
RX	I-Device
6000	I-Device
series	I-Device
,	O
its	O
second	O
generation	O
Navi	O
GPUs	O
with	O
support	O
for	O
hardware-accelerated	O
ray	B-Algorithm
tracing	I-Algorithm
at	O
an	O
online	O
event	O
.	O
</s>
<s>
Current	O
home	O
gaming	O
consoles	O
implement	O
dedicated	O
ray	B-Algorithm
tracing	I-Algorithm
hardware	I-Algorithm
components	I-Algorithm
in	O
their	O
GPUs	O
for	O
real-time	B-General_Concept
ray	B-Algorithm
tracing	I-Algorithm
effects	O
,	O
which	O
began	O
with	O
the	O
ninth-generation	B-Device
consoles	O
PlayStation	B-Device
5	I-Device
,	O
Xbox	B-Device
Series	I-Device
X	I-Device
and	I-Device
Series	I-Device
S	I-Device
.	O
</s>
<s>
Various	O
complexity	O
results	O
have	O
been	O
proven	O
for	O
certain	O
formulations	O
of	O
the	O
ray	B-Algorithm
tracing	I-Algorithm
problem	O
.	O
</s>
<s>
In	O
particular	O
,	O
if	O
the	O
decision	O
version	O
of	O
the	O
ray	B-Algorithm
tracing	I-Algorithm
problem	O
is	O
defined	O
as	O
follows	O
–	O
given	O
a	O
light	O
ray	O
's	O
initial	O
position	O
and	O
direction	O
and	O
some	O
fixed	O
point	O
,	O
does	O
the	O
ray	O
eventually	O
reach	O
that	O
point	O
,	O
then	O
the	O
referenced	O
paper	O
proves	O
the	O
following	O
results	O
:	O
</s>
<s>
Ray	B-Algorithm
tracing	I-Algorithm
in	O
3D	O
optical	B-Algorithm
systems	O
with	O
a	O
finite	O
set	O
of	O
reflective	O
or	O
refractive	O
objects	O
represented	O
by	O
a	O
system	O
of	O
rational	O
quadratic	O
inequalities	O
is	O
undecidable	O
.	O
</s>
<s>
Ray	B-Algorithm
tracing	I-Algorithm
in	O
3D	O
optical	B-Algorithm
systems	O
with	O
a	O
finite	O
set	O
of	O
refractive	O
objects	O
represented	O
by	O
a	O
system	O
of	O
rational	O
linear	O
inequalities	O
is	O
undecidable	O
.	O
</s>
<s>
Ray	B-Algorithm
tracing	I-Algorithm
in	O
3D	O
optical	B-Algorithm
systems	O
with	O
a	O
finite	O
set	O
of	O
rectangular	O
reflective	O
or	O
refractive	O
objects	O
is	O
undecidable	O
.	O
</s>
<s>
Ray	B-Algorithm
tracing	I-Algorithm
in	O
3D	O
optical	B-Algorithm
systems	O
with	O
a	O
finite	O
set	O
of	O
reflective	O
or	O
partially	O
reflective	O
objects	O
represented	O
by	O
a	O
system	O
of	O
linear	O
inequalities	O
,	O
some	O
of	O
which	O
can	O
be	O
irrational	O
is	O
undecidable	O
.	O
</s>
<s>
Ray	B-Algorithm
tracing	I-Algorithm
in	O
3D	O
optical	B-Algorithm
systems	O
with	O
a	O
finite	O
set	O
of	O
reflective	O
or	O
partially	O
reflective	O
objects	O
represented	O
by	O
a	O
system	O
of	O
rational	O
linear	O
inequalities	O
is	O
PSPACE-hard	O
.	O
</s>
<s>
For	O
any	O
dimension	O
equal	O
to	O
or	O
greater	O
than	O
2	O
,	O
ray	B-Algorithm
tracing	I-Algorithm
with	O
a	O
finite	O
set	O
of	O
parallel	O
and	O
perpendicular	O
reflective	O
surfaces	O
represented	O
by	O
rational	O
linear	O
inequalities	O
is	O
in	O
PSPACE	O
.	O
</s>
