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Ray Tracing
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Basics
Intersection Computations
General Issues:
Ray: express in parametric form
where
E
is the eyepoint and
P
is the pixel point
Scene Object: direct implicit form
express as
f
(
Q
)=0 when
Q
is a surface point, where
f
is a given formula
intersection computation is an equation to solve:
find
t
such that
f
(
E
+
t
(
P

E
))=0
Scene Object: procedural implicit form
f
is not a given formula
f
is only defined procedurally
yields
t
, where A is a root finding method
(secant, Newton, bisection, ...)
Quadric Surfaces:
Surface given by
Ray given by
Substitute ray
x
,
y
,
z
into surface formula
quadratic equation results for
t
organize expression terms for numerical accuracy; ie. to avoid
cancellation
combinations of numbers with widely different magnitudes
Polygons:
The plane of the polygon should be known
(
A
,
B
,
C
,0) is the normal vector
pick three succesive vertices
should subtend a ``reasonable'' angle
(bounded away from 0 or 180 degrees)
normal vector is the cross product
D
= (
Ax
+
By
+
Cz
) for any vertex (
x
,
y
,
z
,1) of the polygon
Substitute ray
x
,
y
,
z
into surface formula
linear equation results for
t
Solution provides planar point
is this inside or outside the polygon?
Planar Coordinates:
Take origin point and two independent vectors on the plane
Express any point in the plane as
intersection point is
clipping algorithm against polygon edges in these
coordinates
Alternatives:
Must this all be done in world coordinates?
Alternative: Intersections in model space
form normals and planar coordinates in model space and store with model
backtransform the ray into model space using inverse modeling transformations
perform the intersection and illumination calculations
Alternative: Intersections in world space
form normals and planar coordinates in model space and store
forward transform using modeling transformations
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Shading
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Ray Tracing
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Basics
CS488/688: Introduction to Interactive Computer Graphics
University of Waterloo
Computer Graphics Lab
cs488@cgl.uwaterloo.ca