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CS 551/651: Advanced Computer Graphics. Accelerating Ray Tracing. Ray-Sphere Intersection. Ray R = O + tD x = O x + t * D x y = O y + t * D y z = O z + t * D z Sphere at ( l, m, n ) of radius r is: ( x - l ) 2 + ( y - m ) 2 + ( z - n ) 2 = r 2

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cs 551 651 advanced computer graphics

CS 551/651: Advanced Computer Graphics

Accelerating Ray Tracing

David Luebke 111/17/2014

ray sphere intersection
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 211/17/2014

ray sphere intersection1
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 + Oz2 - 2(l Ox + m Oy + n Oz + r2)

David Luebke 311/17/2014

ray sphere intersection2
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 411/17/2014

ray sphere intersection3
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 511/17/2014

ray polygon intersection
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 611/17/2014

ray polygon intersection1

y

N

P2

P1

d

x

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 ?)

David Luebke 711/17/2014

ray polygon intersection2
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 811/17/2014

ray box intersection
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 911/17/2014

shadow ray problems too much computation
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!

ShadowRay

Light Buffer

Occluding Polys

Current Intersection Point

David Luebke 1011/17/2014

shadow ray problems sharp shadows
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 1111/17/2014

shadow ray problems area light sources
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 1211/17/2014

shadow ray problems area light sources1
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 1311/17/2014

shadow ray problems area light sources2
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 1411/17/2014

the end
The End

David Luebke 1511/17/2014