Raytracing Implementation

1
Raytracing Implementation

• Raytracing breaks down into two tasks
• Constructing the rays to cast
• Intersecting rays with geometry
• The former problem is simple vector arithmetic
• The intersection problem arises in many areas of
  computer graphics
• Collision detection
• Other rendering algorithms
• Intersection is essentially root finding (as we
  will see)
• Any root finding technique can be applied

2
Constructing Rays

• Define rays by an initial point and a direction
  x(t)x0td
• Eye rays Rays from the eye through a pixel
• Construct using the eye location and the pixels
  location on the image plane. x0eye
• Shadow rays Rays from a point on a surface to
  the light.
• x0point on surface
• Reflection rays Rays from a point on a surface
  in the reflection direction
• Construct using laws of reflection. x0surface
  point
• Transmitted rays Rays from a point on a
  transparent surface through the surface
• Construct using laws of refraction. x0surface
  point

3
Ray-Object Intersections

• Aim Find the parameter value, ti, at which the
  ray first meets object i
• Transform the ray into the objects local
  coordinate system
• Makes ray-object intersections generic
  ray-sphere, ray-plane,
• Write the surface of the object implicitly
  f(x)0
• Unit sphere at the origin is x2-10
• Plane with normal n passing through origin is
  nx0
• Put the ray equation in for x
• Result is an equation of the form f(t)0 where we
  want t
• Now its just root finding

4
Ray-Sphere Intersection

• Quadratic in t
• 2 solutions Ray passes through sphere - take
  minimum value that is gt 0
• 1 solution Ray is tangent - use it if gt0
• 0 solutions Ray does not hit sphere

5
Ray-Plane Intersections

• To do polygons, intersect with plane then do
  point-in-polygon test

6
Point-in-Polygon Testing

• Project point and polygon onto a 2D plane
• Find biggest component of normal vector, and just
  use other two coordinates
• For example, if n(0.2, 0.4, 0.9), just use x,y
  coordinates, which is like projecting down onto
  the x-y plane
• Cast a ray from the point to infinity and count
  the number of edges it crosses
• Odd number means point is inside
• Edge crossing tests are very fast - think clipping

7
More complex tests

• Ray-Polygon test reveals a common strategy
• Intersect with something easy - a superset of the
  actual shape
• Do a bounds check to make sure you have actually
  hit the shape
• Also works for cylinders, disks, cones
• CSG is well suited to raytracing
• Find intersection along ray of all the CSG
  primitives
• Break the ray into intervals according to which
  primitives it is in
• Do set operations to find the first interval that
  is actually inside the CSG object

8
CSG Intersection Testing
Subtract
Subtract intervals
Intersection intervals
9
Ray-Patch Intersection

• Equation in 3 parameters, two for surface and one
  for ray
• Solve using Newtons method for root finding
• Have derivatives from basis functions
• Starting point from control polygon, or random
  guess, or try a whole set of different starting
  values

10
Details

• Must find first intersection of ray from the eye
• Find all candidate intersections, sort them and
  take soonest
• Techniques for avoiding testing all objects
• Bounding boxes that are cheap to test
• Octrees for organizing objects in space
• Take care to eliminate intersections behind the
  eye
• Same rules apply for reflection and transmission
  rays
• Shadow ray just have to find any intersection
  shadowing the light source
• Speedup Keep a cache of shadowing objects - test
  those first

11
Transforming Normal Vectors

• Normal vectors are not transformed the same way
  points are
• Ray directions behave like normal vectors
• Plane equation should still be true with
  transformed points!
• Transform normal vectors with the inverse
  transpose of the transformation matrix
• For rotations, matrix is its own inverse transpose

12
Numerical Issues

• Shadow, reflection and transmission rays have to
  be sure they dont intersect the surface they are
  leaving
• Cant just ignore the objects - some objects
  self-shadow
• Solution Use a tolerance - offset the starting
  point off the surface a little in the normal
  direction
• Finding all the intersections with a spline
  surface patch is difficult
• CSG can have problems when doing set operations
• Make sure pieces being subtracted protrude above
  surfaces

Bad
Good
13
Mapping Techniques

• Raytracing provides a wealth of information about
  the visible surface point
• Position, normal, texture coordinates,
  illuminants, color
• Raytracing also has great flexibility
• Every point is computed independently, so effects
  can easily be applied on a per-pixel basis
• Reflection and transmission and shadow rays can
  be manipulated for various effects
• Even the intersection point can be modified

14
Texture Mapping

• Define texture parameters over the surface
• Determine texture parameters at each hit point
• Use them to look up a texture map, just as with
  OpenGL
• But
• Can also use them to look up other things

15
Bump Mapping

• Vary the surface normal vector according to some
  map
• Influences shading and reflection and refraction
• Variation may be random - for effects like stucco
  - or structured - for effects like tiles

16
Bump Mapping Examples
17
Displacement Mapping

• Bump mapping changes only the normal, not the
  intersection point
• Silhouettes will not show bumps, even though
  shading does
• Displacement mapping actually shifts the
  intersection point according to a map
• Gives bump map effects and also correct
  silhouettes and self shadowing, if implemented
  fully

18
From RmanNotes http//www.cgrg.ohio-state.edu/sma
y/RManNotes/index.html
19
Procedural Texturing

• Define a shading language that can be used to
  compute the color of the intersection point
• A shading language program can be defined for
  each surface
• Every time the surface is hit, the program is run
  to compute the color
• Program has access to the incoming ray, the
  normal, the texture coordinates, the texture
  image, plus lots of other stuff
• Implemented in, for example, RenderMan from Pixar
• Starting to show up in hardware renderers -
  OpenGL extensions