Ray Tracing

1
Ray Tracing
2
An Ideal Ray Tracer

• Cast light rays from all light sources.
• Follow them to the eye point.
• Reflect off of solid objects.
• Pass through transparent objects (refraction).
• Obtain color by light absorption.
• Diffuse reflection/refraction multiplies rays.
• For rays that hit the eye point, intersect with
  window on view plane, which corresponds to a
  screen pixel.
• Add the light rays color intensities to the
  pixel.
• Problem It could take forever to draw an image!

3
Ray Tracing That Works

• Observation We are only interested in rays that
  cross the eye point.
• Idea Start at eye point and trace backwards.

Light Source
Shadow Ray
Eye
Object
Eye Ray
Shadow Ray
Object
Light Source
4
Visible Surface Determination

• Ray tracing can be used as a visible surface
  determination algorithm
• Often called ray casting.
• (A ray is a uni-directional line segment).
• The algorithm
• For each pixel
• Determine the world coordinates of the pixels
  center.
• Cast a ray from the eye through the pixels
  center.
• Check for intersection of this ray with any
  object.
• For closest intersection, determine color of
  object at that point.

5
Color Determination

• To determine the color at the intersection of an
  eye ray with an object, we can
• Simply assign the color of the object (no
  shading).
• Determine the shading of the object.
• Calculate the contribution of a texture at that
  point.
• Some combination of the above
• Shading
• Cast ray from intersection point to each light
  source.
• The shadow ray.
• Light contributes to shading only if the shadow
  ray does not intersect an object in the scene.

6
Intersections

• Ray tracing algorithms are dominated by
  intersection testing and determination
• Need efficient methods to test for intersection
• Does the eye ray or shadow ray intersect an
  object?
• Need efficient method to calculate intersection
  points and normal vectors (for lighting
  calculations).
• Ray/Sphere
• Ray/Plane
• Ray/Polygon

7
Ray/Sphere Intersection (Algebraic
Solution)

• Ray is defined by R(t) Ro Rdt where t gt 0.
• Ro Origin of ray at (xo, yo, zo)
• Rd Direction of ray xd, yd, zd (unit vector)
• Sphere's surface is defined by the set of points
  (xs, ys, zs) satisfying the equation
• (xs - xc)2 (ys - yc)2 (zs - zc)2 - rs2
  0
• Center of sphere (xc, yc, zc)
• Radius of sphere rs

8
Possible Cases of Ray/Sphere Intersection
1. Ray intersects sphere twice with tgt0 2. Ray
tangent to sphere 3. Ray intersects sphere with
tlt0 4. Ray originates inside sphere 5. Ray does
not intersect sphere
1
2
3
4
5
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Solving For t

• Substitute the basic ray equation
• x xo xdt
• y yo ydt
• z zo zdt
• into the equation of the sphere
• (x0 xdt - xc)2 (y0 ydt - yc)2 (z0 zdt -
  zc)2 - rs2 0
• This is a 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
• Note A1

10
Relation of t to Intersection

• We want the smallest positive t - call it ti

t0
Discriminant 0
t1
t0
t0
t1
t1
Discriminant lt 0
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Actual Intersection

• Intersection point, (xi, yi, zi) (xoxdti,
  yoydti, zozdti)
• Unit vector normal to the surface at this point
  is
• N (xi - xc) / rs, (yi - yc) / rs, (zi - zc) /
  rs
• If the ray originates inside the sphere, N should
  be negated so that it points back toward the
  center.

N
N
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Summary (Algebraic Solution)

• Calculate A, B and C of the quadratic
• Calculate discriminant (If lt 0, then no
  intersection)
• Calculate t0
• If t0 lt 0, then calculate t1 (If t1 lt 0, no
  intersection point on ray)
• Calculate intersection point
• Calculate normal vector at point
• Helpful pointers
• Precompute rs2
• Precompute 1/rs
• If computed t is very small then, due to rounding
  error, you may not have a valid intersection

13
Ray/Sphere Intersection (Geometric Solution)

• In the case of the sphere there are a number of
  tests that we can make to determine that there is
  not an intersection. This approach allows us to
  avoid calculations and result in a faster
  algorithm.
• 1. Find if the ray's origin is outside the sphere.

Origin to center vector, OC C - Ro Length
squared of OC OC2 If OC2 lt rs2 then
the ray origin is inside the sphere (go to step
4) Else if OC2 gt rs2 then the ray origin is
outside the sphere
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R/S Geometric Solution (cont.)

• 2. Find the length of the ray to the point along
  the ray closest to the sphere's center.

L OC cos ß OC Rd cos ß OCRd
If the ray is outside and points away from the
sphere (L lt 0), the ray must miss the sphere.
15
R/S Geometric Solution (cont.)

• 3. Else find the squared distance from the
  closest approach along the ray to the sphere's
  center to the sphere's surface (HC2 half chord
  distance squared).

D2 OC2 - L2 HC2 rs2 - D2 rs2 -
OC2 L2
If HC2 lt 0, the ray misses the sphere.
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R/S Geometric Solution (cont.)

• 4. Else find t for the intersection
• ti L - HC for rays originating outside the
  sphere
• ti L HC for rays originating inside or on
  the sphere
• 5. Calculate intersection point (xi, yi, zi) by
  plugging ti into Ro tRd
• 6. Calculate normal vector
• (xi - xc)/rs (yi - yc)/rs, (zi - zc)/rs
• (negate if ray originates inside sphere)

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Summary (Geometric Solution)

• Find distance squared between ray origin and
  center of sphere, test for ray outside sphere.
• Calculate ray distance which is closest to the
  center of the sphere.
• Test if the ray points away from the sphere.
• Find square of half chord intersection distance.
• Test if square is negative.
• Calculate intersection distance.
• Find intersection point.
• Calculate normal vector at point.
• These techniques are no slower than the
  algebraic solution and often allow you to
  determine that there is no intersection using
  fewer calculations.

18
Ray/Plane Intersection

• Ray is defined by R(t) Ro Rdt where t gt 0
• Ro Origin of ray at (xo, yo, zo)
• Rd Direction of ray xd, yd, zd (unit
  vector)
• Plane is defined by A, B, C, D
• Ax By Cz D 0 for a point in the plane
• Normal Vector, N A, B, C (unit vector)
• A2 B2 C2 1

19
Ray/Plane (cont.)

• Substitute the ray equation into the 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)
• Note The normal vector of the plane should
  usually (except for backface removal) be for the
  surface facing the ray. If it isn't then it
  should be reversed before illumination is
  calculated.
• If N Rd gt 0 then use N -A, B, C

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What Can Happen?
21
Ray/Plane Summary

• Intersection point
• (xi, yi, zi) (xo xdti, yo ydti, zo zdti)
• Calculate N Rd and compare it to zero.
• Calculate ti and compare it to zero.
• Compute intersection point.
• Compare N Rd to zero and reverse normal if
  appropriate

22
Ray/Polygon Intersection

• A polygon is defined by a set of p points Gn
  (xn, yn, zn) n 0, 1, ..., (p-1)
• The polygon is in a plane, Ax By Cz D 0,
  Normal N A, B, C
• Find the intersection of the ray with the plane
  of the polygon.
• Throw away the coordinate of each point for which
  the corresponding normal component is of the
  greatest magnitude (dominant coordinate).
• You now have a 2-D polygon defined by the set of
  points
• (un, vn) n 0, 1, .., p-1
• Translate the polygon so that the intersection
  point of the ray with the plane of the polygon
  goes to the origin of the u,v system. Call these
  translated points
• (un, vn) n 0, 1, .., p-1
• Determine whether the origin is within the 2D
  polygon.

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Ray/Polygon (cont.)
24
Ray/Polygon Algorithm

• 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.
• num_crossings 0
• sign_holder sign(v0)
• for(a0 a lt p a)
• b (a1) p
• next_sign_holder sign(vb)
• if (sign_holder ! next_sign_holder)
• if (ua gt 0 and ub gt 0)
• num_crossings num_crossings 1
• else if (ua gt 0 or ub gt 0)
• int ua - va(ub - ua) / (vb - va)
• if(int gt 0)
• num_crossings num_crossings 1
• sign_holder next_sign_holder
• if(num_crossings 2 1) ray intersects polygon