The photorealism quest

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The photorealism quest

• An introduction to ray-tracing

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The basic idea
Cast a ray from the viewers eye through each
pixel on the screen to see where it hits some
object
object
view-plane
Apply illumination principles from physics to
determine the intensity of the light that is
reflected from that spot toward the eye
eye of viewer
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Example a spherical object

• Mathematics problem How can we find the spot
  where a ray hits a sphere?
• We apply our knowledge of vector-algebra
• Describe spheres surface by an equation
• Describe rays trajectory by an equation
• Then solve this system of two equations
• There could be 0, 1, or 2 distinct solutions

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Describing the sphere

• A sphere consists of points in space which lie at
  a fixed distance from a given center
• Let r denote this fixed distance (radius)
• Let c ( cx, cy, cz ) be the spheres center
• If w ( wx, wy, wz ) be any point in space,
  then distance( w, c ) is written w c
• Formula w c sqrt( (w - c)(w - c) )
• Sphere is a set w e R3 (w-c)(w-c) r

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Describing a ray

• A ray is a geometrical half-line in space
• It has a beginning point b ( bx, by, bz )
• It has a direction-vector d ( dx, dy, dz )
• It can be described with a parameter t 0
• If w ( wx, wy, wz ) is any point in space,
  then w will be on the half-line just in case
  w b td for some choice of the
  parameter t 0 .

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Computing the hit time

• To find WHERE a ray hits a sphere, we think of w
  as a point traveling in space, and we ask WHEN
  will w hit the shere?
• We can substitute the formula for a point on
  the half-line into our formula for points that
  belong to the spheres surface, getting an
  equation that has t as its only variable
• Its easy to solve such an equation for t to
  find when w hits the spheres surface

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The algebra details

• Sphere w c r
• Half-line w b td, t 0
• Substitution b td c r
• Replacement Let q c b
• Simplification td q r
• Square (td q)(td q) r2
• Expand t2 dd -2tdq qq r2
• Transpose t2 dd -2tdq (qq - r2) 0

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Apply Quadratic Formula

• To solve t2 dd -2tdq (qq - r2) 0
• Format At2 Bt C 0
• Formula t -B/2A sqrt( B2 4AC )/2A
• Notice there could be 0, 1, or 2 hit times
• OK to use a simplifying assumption dd 1
• Equation becomes t2 2tdq (qq r2) 0
• Application We want the earliest hit-time
  t (dq) - sqrt( (dq)2 (qq r2) )

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Interpretations
half-line
w2
w1
b
c
half-line
A half-line that begins inside the sphere will
only hit the spgeres surface once
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Second example A planar surface

• Mathematical problem How can we find the spot
  where a ray hits a plane?
• Again we can employ vector-algebra
• Any plane is determined by a two entities
• a point (chosen arbitrarily) that lies in it, and
• a vector that is perpendicular (normal) to it
• Let p be the point and n be the vector
• Plane is the set w e R3 (w-p)n 0

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When does ray hit plane?

• Plane (w p)n 0
• Half-line w b td, t 0
• Substitution (b td p)n 0
• Replacement Let q p b
• Simplification (td q)n 0
• Expand tdn - qn 0
• Solution t (qn)/(dn)

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Interpretations
half-line
n
p
d
w
If a half-line begins from a point outside a
given plane, then it can only hit that plane once
(and it might possibly not even hit that plane
at all
half-line
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Achromatic light

• Achromatic light brightness, but no color
• Light can come from point-sources, and light can
  come from ambient sources
• Incident light shining on the surface of an
  object can react in three discernable ways
• By being absorbed
• By being reflected
• By being transmitted (into the interior)

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Illumination

• Besides knowing where a ray of light will hit a
  surface, we will also need to know whether that
  ray is reflected or absorbed
• If the ray of light is reflected by a surface,
  does it travel mainly in one direction? Or does
  it scatter off in several directions?
• Various surface materials react differently

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Some terminology

• Scattering of light (diffuse reradiation)
• color may be affected by the surface
• Reflection of light (specular reflection)
• mirror-like shininess, color isnt affected

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Geometric ingredients

• Normal vector to the surface at a point
• Direction vector from point to viewers eye
• Direction vector from point to light-source