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CS 551/651: Advanced Computer Graphics

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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

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CS 551/651: Advanced Computer Graphics

Accelerating Ray Tracing

David Luebke 111/17/2014

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 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 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 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
• 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

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 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
• 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
• 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
• 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
• 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 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 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

David Luebke 1511/17/2014