CS 551/651: Advanced Computer Graphics Accelerating Ray Tracing David Luebke 1 4/27/2017

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 2 4/27/2017

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 3 4/27/2017

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 4 4/27/2017

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 5 4/27/2017

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 6 4/27/2017

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 7 4/27/2017

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 8 4/27/2017

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 9 4/27/2017

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 10 4/27/2017

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 11 4/27/2017

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 12 4/27/2017

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 13 4/27/2017

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 14 4/27/2017

The End David Luebke 15 4/27/2017