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CS 551/651: Advanced Computer Graphics

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

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  1. CS 551/651: Advanced Computer Graphics Accelerating Ray Tracing David Luebke 111/17/2014

  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 211/17/2014

  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 + Oz2 - 2(l Ox + m Oy + n Oz + r2) David Luebke 311/17/2014

  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 411/17/2014

  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 511/17/2014

  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 611/17/2014

  7. 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

  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 811/17/2014

  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 911/17/2014

  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! ShadowRay Light Buffer Occluding Polys Current Intersection Point David Luebke 1011/17/2014

  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 1111/17/2014

  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 1211/17/2014

  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 1311/17/2014

  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 1411/17/2014

  15. The End David Luebke 1511/17/2014

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