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Ray Tracing Intersections
CS
Lecture
January 26, 2009
CS510CS510 -- Computer Graphics, Spring 2009Computer Graphics, Spring 2009 © Bruce A. Draper & J. Ross Beveridge, January 26, 2009
Recursive Ray Tracing:
The General Idea
z How do we see objects?
- Light leaves a light source, bounces of an object, and onto our
retina (or film).
- Or, it leaves a light source, bounces of one object, then off
another, and onto our retina.
- Or, it leaves a light source, bounces of one surface, then another,
…., and onto our retina.
z We want to emulate this process, starting at the camera (or
retina) and working backwards….
CS510 -CS510- Computer Graphics, Spring 2009Computer Graphics, Spring 2009 © Bruce A. Draper & J. Ross Beveridge, January 26, 2009
Computing Intersections
z Ray-to-Surface Intersection Calculations are Key.
- This is most of the computation in Ray Tracing
z Algorithms tied to surface type.
- Sphere
- Polygon
- Bicubic Surfaces (really hard! we won’t do these)
z First pass: General Mathematics
z Second pass: Highly Optimized Codes
CS510CS510 -- Computer Graphics, Spring 2009Computer Graphics, Spring 2009 © Bruce A. Draper & J. Ross Beveridge, January 26, 2009
Representing a Ray
z Defined by a point and a unit
vector.
z Where E is the focal point
(Eye) and L is a pixel point
(Lens Point)
R s ( )= L + sU
U =
L − E
L − E
Sphere Intersection
Parametric Ray - Brute Force
z Consider a sphere centered at P c with radius r.
P − P
c
2
= r
2
L + sU − P
c
2
− r
2
substitute
T = P
c
− L
Yielding a quadratic equation
(^ sU^ −^ T ) ⋅^ ( sU^ −^ T ) −^ r
2
L
Sphere Intersection
Brute Force (II)
z Expand to see what is happening. Let
T =
x c − x 0
y (^) c − y 0
zc − z 0
s −
u (^) x
u (^) y
u z
t (^) x
t (^) y
t z
s −
u (^) x
u (^) y
u z
t (^) x
t (^) y
t z
r
2 0
W ⋅ W − r
2 = (^0) W=
s u x − t x
s u (^) y − t (^) y
s uz − t (^) z
( s ux − t (^) x )+ + − =
2 ( s uy − t (^) y )
2 ( s u (^) z − t (^) z )
2 r
2 0
CS510 -CS510- Computer Graphics, Spring 2009Computer Graphics, Spring 2009 © Bruce A. Draper & J. Ross Beveridge, January 26, 2009
Sphere Intersection (III)
z After multiplying out, expand and collect s terms.
( u x + + ) +
2 u (^) z
2 u (^) y
2 s
2
( (^) − 2 t (^) y u (^) y − 2 t (^) x ux − 2 t (^) z uz ) s +
t x − + + =
2 r
2 t (^) z
2 t (^) y
2 0
z U is a unit vector, so (u x
2
2
2 ) = 1
z So the equation may be written as:
s
2
− 2 ( U ⋅ T ) s + T ⋅ T − r
2
CS510CS510 -- Computer Graphics, Spring 2009Computer Graphics, Spring 2009 © Bruce A. Draper & J. Ross Beveridge, January 26, 2009
Sphere Intersection
Reduces to Quadratic
Consider that we now have a simple quadratic equation.
2 2
2
where
c T r
b UT
a
as bs c
2 2 2
2 2 2
2
Therefore:
s UT UT T r
UT UT T r s
a
b b ac s
CS510 -CS510- Computer Graphics, Spring 2009Computer Graphics, Spring 2009 © Bruce A. Draper & J. Ross Beveridge, January 26, 2009
Actual Sphere
Intersection Points
z Compute the two s values for the
two intersections:
(^222) 2
(^222) 1
s UT UT T r
s UT UT T r
8 multiplies/squares
(caching results; r 2
previously stored)
8 additions
1 square root
z Compute the actual positions along the ray for the smallest
positive s:
s
= min( s 1 , s 2
P
= P 0 + s
U
3 multiplies
3 additions
1 min
CS510CS510 -- Computer Graphics, Spring 2009Computer Graphics, Spring 2009 © Bruce A. Draper & J. Ross Beveridge, January 26, 2009
But Wait
z The proceeding follows naturally from parametric
approach to intersecting objects.
z But is it smart?
z Might there be a more efficient way?
z Yes.
- Glassner, A. (ed) An Introduction to Ray Tracing. Academic Press, 1989
- http://www.cs.unc.edu/~rademach/xroads-RT/RTarticle.html
z Let’s see it …
Sphere Ray Intersection
Faster Method
v d
b r
C
v
2
2 = c
2
d
2
2 = r
2
d
2 = r
2 − c
2 − v
2
d = r
2 − c
2 − v
2
E
O
R
• If d
2
less than zero, no intersection.
• Otherwise, Q = E + (v-d)R
Q
Faster Method
How Fast?
z Assume R is given - unit length ray direction.
z Need to compute v: v = (O-E)•R
- 3 multiplies, 2 additions.
z Both r 2 and c 2 are already computed.
- c 2 computed for case of ray coming from focal point
z Test if r
2 is greater than (c
2
2 )
- 1 mulitply, one conditional.
z Only if intersection, Q = E+(v-d)R.
- 1 subtract, 3 multiplies, 6 additions.
z Another reference on this approach:
- http://www.groovyvis.com/other/raytracing/basic.html
Odd/Even Intersections (II)
z Key calculation: ray to ray intersection
- Note that both rays have a free variable T
- Three dimensions ⇒ 3 equations
(overconstrained)
- For an intersection to be valid:
z 0 ≤ T (^) IPRay
z 0 ≤ T edge < 1
Is this efficient?
z No!
- But its easy ☺
- It’s general : works for non-convex and self-
intersecting polygons.
z More efficient methods for convex polygons
and for triangles
