1. CS 551/651: Advanced Computer Graphics
Accelerating Ray Tracing
David Luebke /27/2017

2. Ray-Sphere Intersection
Ray R = O + tD
x = Ox + t * Dx
y = Oy + t * Dy
z = Oz + t * Dz
Sphere at (l, m, n) of radius r is:
(x - l)2 + (y - m)2 + (z - n)2 = r 2
Substitute for x,y,z and solve for t…
David Luebke /27/2017

3. Ray-Sphere Intersection
Works out as a quadratic equation:
at2 + bt + c = 0
where
a = Dx2 + Dy2 + Dz2
b = 2Dx (Ox - l) + 2Dy (Oy - m) + 2Dz (Oz - n)
c = l2 + m2 + n2 + Ox2 + Oy2 + Oz (l Ox + m Oy + n Oz + r2)
David Luebke /27/2017

4. Ray-Sphere Intersection
If solving for t gives no real roots: ray does not intersect sphere
If solving gives 1 real root r, ray grazes sphere where t = r
If solving gives 2 real roots (r1, r2), ray intersects sphere at t = r1 & t = r2
Ignore negative values
Smallest value is first intersection
David Luebke /27/2017

5. Ray-Sphere Intersection
Find intersection point Pi = (xi, yi, zi) by plugging t back into ray equation
Find normal at intersection point by subtracting sphere center from Pi and normalizing:
(When might we need the normal? When not?)
David Luebke /27/2017

6. Ray-Polygon Intersection
Polygons are the most common model representation (Why?)
Basic approach:
Find plane equation of polygon
Find intersection of ray and plane
Does polygon contain intersection point?
David Luebke /27/2017

7. Ray-Polygon Intersection
Find plane equation of polygon: ax + by + cz + d = 0
How?
N = [a, b, c]
d = N  P1
(How to find N ?) x y N P1 P2 d
David Luebke /27/2017

8. Ray-Polygon Intersection
Find intersection of ray and plane:
t = -(aOx + bOy + cOz + d) / (aDx + bDy + cDz)
Does poly contain intersection point Pi ?
Book’s algorithm:
Draw line from Pi to each polygon vertex
Measure angles between lines (how?)
If sum of angles between lines is 360°, polygon contains Pi
David Luebke /27/2017

9. Ray-Box Intersection
Often want to find whether a ray hits an axis-aligned box (Why?)
One way:
Intersect the ray with the pairs of parallel planes that form the box
If the intervals of intersection overlap, the ray intersects the box.
David Luebke /27/2017

10. Shadow Ray Problems: Too Much Computation
Light buffer (Haines/Greenberg, 86)
Precompute lists of polygons surrounding light source in all directions
Sort each list by distance to light source
Now shadow ray need only be intersected with appropriate list!
Shadow Ray
Light Buffer
Occluding Polys
Current Intersection Point
David Luebke /27/2017

11. Shadow Ray Problems: Sharp Shadows
Why are the shadows sharp?
A: Infinitely small point light sources
What can we do about it?
A: Implement area light sources
How?
David Luebke /27/2017

12. Shadow Ray Problems: Area Light Sources
Could trace a conical beam from point of intersection to light source:
Track portion of beam blocked by occluding polygons:
30% blockage
David Luebke /27/2017

13. Shadow Ray Problems: Area Light Sources
Too hard! Approximate instead:
Sample the light source over its area and take weighted average:
50% blockage
David Luebke /27/2017

14. Shadow Ray Problems: Area Light Sources
Disadvantages:
Less accurate (50% vs. 30% blockage)
Oops! Just quadrupled (at least) number of shadow rays
Moral of the story:
Soft shadows are very expensive in ray tracing
David Luebke /27/2017

15. The End
David Luebke /27/2017