Ray Tracing

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Ray Tracing

• Matthew W. Campbell
• Senior, Computer Science Dept.

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Ray tracing overview

• Technique for rendering 3D graphics.
• Point sampling technique.
• rays pixels (assuming normal sampling)
• Models light interaction among surfaces and the
  environment.
• Uses illumination models to determine color value
  at the intersection of a ray and an object.
• Supports transparency, reflections, refractions,
  etc.

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Ray tracing concept and samples
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Ray tracing algorithm

• Begin tracing.
• Construct a ray through each pixel in the scene.
• Pixel color result of tracing the ray
• Tracing the ray
• Find intersection of ray with closest object in
  the scene.
• Compute intersection point and normal at the
  intersection point.
• Shade the point.
• Shade the point
• For each light
• Determine if point is in shadow relative to light
  source.
• Using illumination model, determine color at the
  intersection point.

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Constructing primary rays

• Ray is defined by R(t) Ro Rdt where tgt0.
• Ro Origin of ray at (xo, yo, zo)
• Rd Direction of ray xd, yd, zd (unit vector).
• Determine world location of pixel (i,j).
• Origin (x,y,z) camera position.
• Direction (x,y,z)
• j (WIDTH/2)
• i (HEIGHT/2)
• 255.0 Fairly abitrary, just interpret as FOV.
• Normalize direction vector.

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Ray Plane intersection

• Plane equation
• Ax By Cz D 0 (point on plane)
• (A,B,C) is normal of the plane (unit vector)

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Ray/Plane cont.

• Substitute ray equation into plane equation.
• A(xo xdt) B(yo ydt) C(zo zdt) D 0
• Solve for t.
• t -(Axo Byo Czo D) / (Axd Byd Czd)
• t -(N Ro D) / (N Rd)

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Ray/Plane Summary

• Compute N Rd
• If equals zero, ray and plane are parallel,
  therefore, no intersection.
• Compute t
• t -(N Ro D) / (N Rd)
• If t lt 0, no intersection
• Compute intersection point.
• (x,y,z) R0 (Rd t)

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Ray Sphere Intersection

• Algebraic solution
• (x0 xdt - xc)2 (y0 ydt - yc )2 (z0 zdt
  - zc)2 - rs2 0
• Quadratic equation in t At2 Bt C 0 where
• A xd2 yd2 zd2
• B 2xd(x0 - xc) yd(y0 - yc) zd(z0 - zc)
• C (x0 - xc)2 (y0 - yc)2 (z0 - zc)2 - rs2
• Intersection point
• (x,y,z) (xo xdt, yo ydt, zo zdt)
• N (x - xc) / rs, (y - yc) / rs, (z - zc) / rs

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Ray-Sphere Cont.

• Geometric Solution
• Lots of ways to determine that no intersection
  exists, why does this help?

• 6 Steps
• 1 Find if the rays origin is inside the
  sphere.
• Origin to center vector, OC C Ro
• Length squared of OC OC2
• If OC2 lt rs2 then ray origin is inside of the
  sphere. (go to step 4). Else if OC2 gt rs2
  then ray origin is outside of the sphere.

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Ray-Sphere Cont.

• Step 2
• Calculate ray distance which is closest to the
  center of the sphere.
• L OC cos ß OC Rd cos ß OCRd
• If the ray is outside of the sphere and points
  away from the sphere, L must be lt 0, and the ray
  misses the sphere.

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Ray-Sphere Cont.

• Step 3
• Calculate the squared distance from the closest
  approach along the ray to the spheres center
  to the spheres surface. (HC2 half coord
  distance squared).
• D2 OC2 L2
• HC2 rs2 D2
• If HC2 lt 0, ray misses sphere.

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Ray-Sphere Cont.

• Step 4
• Find t.
• t L HC, if ray originates outside the sphere.
• t L HC, if ray originates inside the sphere.
• Step 5
• Calculate the intersection point.
• (x,y,z) Ro (tRd) I
• Step 6
• Calculate the normal vector.
• N (I-Sc) / Sr, negate if ray originates inside
  of the sphere.

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Ray-Polygon Intersection

• We define a polygon as a list of n points
• (x0, y0, z0), (x1, y1, z1),(xn-1, yn-1, zn-1).
• The polygon is in a plane, i.e. we use planar
  polygons.
• Algorithm overview
• Intersect the ray with the plane of the polygon.
• Project the polygon into 2D space.
• Translate the polygon so the intersection point
  of the ray and the plane sits at the origin.
• Determine whether the origin is within the 2D
  polygon.

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Ray-Polygon Cont.

• Project the polygon into 2D space.
• Planar projection.
• Take the polygons normal and find the dominate
  component.
• Throw away the corresponding coordinate of each
  point in the polygon.
• Ex Polygon normal (0,0,1) Lies on the XY
  plane.
• Throw away the z component of each point in the
  polygon resulting in a 2D polygon.

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Ray-Polygon Cont.

• Translate the polygon and check to see if origin
  is inside.
• Translate the polygon by the intersection point
  of the ray and the plane ( I).
• For all points in the polygon, Pnew Pold I
• To determine whether the origin is within a 2D
  polygon, we only need to count the number of
  times the polygons edges cross the positive
  u-axis as we walk around the polygon. Odd
  number crossings origin is within.

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Ray-Polygon Cont.
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Ray tracing optimizations

• Adaptive grid sampling
• Instead of using one ray per pixel, use one ray
  per block of pixels if the detail in that part of
  the image is small enough.
• Spatial Partitioning!!
• Uniform grids.
• Octrees, KD-trees, BSP-trees.
• Any BVH or bounding volume hierarchy.
• Reduce recursion depth when calculating
  reflections and refractions.

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