Projection Matrices and Simple Shadows

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Projection Matrices and Simple Shadows
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Normalization

• Rather than derive a different projection matrix
  for each type of projection, we can convert all
  projections to orthogonal projections with the
  default view volume
• This strategy allows us to use standard
  transformations in the pipeline and makes for
  efficient clipping

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Pipeline View
modelview transformation
projection transformation
perspective division
4D ? 3D
nonsingular
clipping
projection
3D ? 2D
against default cube
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Notes

• We stay in four-dimensional homogeneous
  coordinates through both the modelview and
  projection transformations
• Both these transformations are nonsingular
• Default to identity matrices (orthogonal view)
• Normalization lets us clip against simple cube
  regardless of type of projection
• Delay final projection until end
• Important for hidden-surface removal to retain
  depth information as long as possible

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Orthogonal Normalization

• glOrtho(left,right,bottom,top,near,far)

normalization ? find transformation to
convert specified clipping volume to default
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Orthogonal Matrix

• Two steps
• Move center to origin
• T(-(leftright)/2, -(bottomtop)/2,(nearfar)/2))
• Scale to have sides of length 2
• S(2/(left-right),2/(top-bottom),2/(near-far))

P ST
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Final Projection

• Set z 0
• Equivalent to the homogeneous coordinate
  transformation
• Hence, general orthogonal projection in 4D is

Morth
P MorthST
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Oblique Projections

• The OpenGL projection functions cannot produce
  general parallel projections such as
• However if we look at the example of the cube it
  appears that the cube has been sheared
• Oblique Projection Shear Orthogonal Projection

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General Shear
top view

• side view

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Shear Matrix

• xy shear (z values unchanged)
• Projection matrix
• General case

H(q,f)
P Morth H(q,f)
P Morth STH(q,f)
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Equivalency
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Effect on Clipping

• The projection matrix P STH transforms the
  original clipping volume to the default clipping
  volume

object
top view
z 1
DOP
DOP
x -1
x 1
far plane
z -1
clipping volume
near plane
distorted object (projects correctly)
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Simple Perspective

• Consider a simple perspective with the COP at the
  origin, the near clipping plane at z -1, and a
  90 degree field of view determined by the planes
• x ?z, y ?z

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Perspective Matrices

• Simple projection matrix in homogeneous
  coordinates
• Note that this matrix is independent of the far
  clipping plane

M
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Generalization

• N

after perspective division, the point (x, y, z,
1) goes to
x x/z y y/z Z -(ab/z)
which projects orthogonally to the desired point
regardless of a and b
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Picking a and b
If we pick
a
b
the near plane is mapped to z -1 the far plane
is mapped to z 1 and the sides are mapped to x
? 1, y ? 1
Hence the new clipping volume is the default
clipping volume
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Normalization Transformation
distorted object projects correctly

• original clipping
• volume

original object
new clipping volume
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Normalization and Hidden-Surface Removal

• Although our selection of the form of the
  perspective matrices may appear somewhat
  arbitrary, it was chosen so that if z1 gt z2 in
  the original clipping volume then the for the
  transformed points z1 gt z2
• Thus hidden surface removal works if we first
  apply the normalization transformation
• However, the formula z -(ab/z) implies that
  the distances are distorted by the normalization
  which can cause numerical problems especially if
  the near distance is small

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OpenGL Perspective

• glFrustum allows for an unsymmetrical viewing
  frustum (although gluPerspective does not)

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OpenGL Perspective Matrix

• The normalization in glFrustum requires an
  initial shear to form a right viewing pyramid,
  followed by a scaling to get the normalized
  perspective volume. Finally, the perspective
  matrix results in needing only a final orthogonal
  transformation

P NSH
our previously defined perspective matrix
shear and scale
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Why do we do it this way?

• Normalization allows for a single pipeline for
  both perspective and orthogonal viewing
• We stay in four dimensional homogeneous
  coordinates as long as possible to retain
  three-dimensional information needed for
  hidden-surface removal and shading
• We simplify clipping

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Simple Shadows

• A shadow is created when a point is not
  illuminated by any light source
• Default view assumes the flashlight in the eye
  model
• Complete physical reality is difficult so
  alternative methods have be created

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Creating simple shadows

• Lets assume that the shadow is projected onto a
  surface where y 0
• The shadow polygon is the projection of the
  original polygon onto the surface where the COP
  is the light source

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Creating simple shadows

• A projection from a frame where the light source
  is at the origin will give us the vertices of the
  shadow polygon

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Creating simple shadows

• Algorithm
• Draw polygon
• Create a projection matrix for the shadow
• Translate back (from origin)
• Project
• Move light to origin
• Draw polygon again
• Restore previous state

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Creating simple shadows

• GLfloat m16
• for(i0ilt15i) mi0.0
• m0m5m101.0
• m7 -1.0/y1
• glColor3fv(polygon_color)
• glBegin(GL_POLYGON)
• drawStuffHere()
• glEnd()
• glMatrixMode(GL_MODELVIEW)
• glPushMatrix()
• glTranslatef(x1,y1,z1)
• glMultMatrixf(m)
• glTranslate(-x1,-y1,-z1)
• glColor3f(shadow_color)
• glBegin(GL_POLYGON)
• drawStuffHere()
• glEnd()
• glPopMatrix()

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Why we need shading

• Suppose we build a model of a sphere using many
  polygons and color it with glColor. We get
  something like
• But we want

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Shading

• Why does the image of a real sphere look like
• Light-material interactions cause each point to
  have a different color or shade
• Need to consider
• Light sources
• Material properties
• Location of viewer
• Surface orientation

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Scattering

• Light strikes A
• Some scattered
• Some absorbed
• Some of scattered light strikes B
• Some scattered
• Some absorbed
• Some of this scattered
• light strikes A
• and so on

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Rendering Equation

• The infinite scattering and absorption of light
  can be described by the rendering equation
• Cannot be solved in general
• Ray tracing is a special case for perfectly
  reflecting surfaces
• Rendering equation is global and includes
• Shadows
• Multiple scattering from object to object

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Global Effects
shadow
multiple reflection
translucent surface
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Local vs Global Rendering

• Correct shading requires a global calculation
  involving all objects and light sources
• Incompatible with pipeline model which shades
  each polygon independently (local rendering)
• However, in computer graphics, especially real
  time graphics, we are happy if things look
  right
• Exist many techniques for approximating global
  effects

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Light-Material Interaction

• Light that strikes an object is partially
  absorbed and partially scattered (reflected)
• The amount reflected determines the color and
  brightness of the object
• A surface appears red under white light because
  the red component of the light is reflected and
  the rest is absorbed
• The reflected light is scattered in a manner that
  depends on the smoothness and orientation of the
  surface

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Light Sources

• General light sources are difficult to work with
  because we must integrate light coming from all
  points on the source

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Simple Light Sources

• Point source
• Model with position and color
• Distant source infinite distance away
  (parallel)
• Spotlight
• Restrict light from ideal point source
• Ambient light
• Same amount of light everywhere in scene
• Can model contribution of many sources and
  reflecting surfaces

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Surface Types

• The smoother a surface, the more reflected light
  is concentrated in the direction a perfect mirror
  would reflected the light
• A very rough surface scatters light in all
  directions

rough surface
smooth surface
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Phong Model

• A simple model that can be computed rapidly
• Has three components
• Diffuse
• Specular
• Ambient
• Uses four vectors
• To source
• To viewer
• Normal
• Perfect reflector

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Ideal Reflector

• Normal is determined by local orientation
• Angle of incidence angle of reflection
• The three vectors must be coplanar

r 2 (l n ) n - l
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Lambertian Surface

• Perfectly diffuse reflector
• Light scattered equally in all directions
• Amount of light reflected is proportional to the
  vertical component of incoming light
• reflected light cos qi
• cos qi l n if vectors normalized
• There are also three coefficients, kr, kb, kg
  that show how much of each color component is
  reflected

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Specular Surfaces

• Most surfaces are neither ideal diffusers nor
  perfectly specular (ideal reflectors)
• Smooth surfaces show specular highlights due to
  incoming light being reflected in directions
  concentrated close to the direction of a perfect
  reflection

specular highlight
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Modeling Specular Relections

• Phong proposed using a term that dropped off as
  the angle between the viewer and the ideal
  reflection increased

Ir ks I cosaf
f
shininess coef
reflected intensity
incoming intensity
absorption coef
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The Shininess Coefficient

• Values of a between 100 and 200 correspond to
  metals
• Values between 5 and 10 give surface that look
  like plastic

cosa f
90
f
-90
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Ambient Light

• Ambient light is the result of multiple
  interactions between (large) light sources and
  the objects in the environment
• Amount and color depend on both the color of the
  light(s) and the material properties of the
  object
• Add ka Ia to diffuse and specular terms

reflection coef
intensity of ambient light
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Distance Terms

• The light from a point source that reaches a
  surface is inversely proportional to the square
  of the distance between them
• We can add a factor of the
• form 1/(ad bd cd2) to
• the diffuse and specular
• terms
• The constant and linear terms soften the effect
  of the point source

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Light Sources

• In the Phong Model, we add the results from each
  light source
• Each light source has separate diffuse, specular,
  and ambient terms to allow for maximum
  flexibility even though this form does not have a
  physical justification
• Separate red, green and blue components
• Hence, 9 coefficients for each point source
• Idr, Idg, Idb, Isr, Isg, Isb, Iar, Iag, Iab

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Material Properties

• Material properties match light source properties
• Nine absorbtion coefficients
• kdr, kdg, kdb, ksr, ksg, ksb, kar, kag, kab
• Shininess coefficient a

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Adding up the Components

• For each light source and each color component,
  the Phong model can be written (without the
  distance terms) as
• I kd Id l n ks Is (v r )a ka Ia
• For each color component
• we add contributions from
• all sources

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Example
Only differences in these teapots are the
parameters in the Phong model