Computer%20Graphics%20(Spring%202003)

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Computer Graphics (Spring 2003)

• COMS 4160, Lecture 9 Transformations 2
• Ravi Ramamoorthi

http//www.cs.columbia.edu/cs4160
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Motivation

• Modeling
• Objects in convenient coordinate system
• Transform into world space, eye space
• Viewing project world space into 2D rectangle
• Virtual cameras
• Types of projections (perspective, orthographic)
• Window system

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Rotations

• Very brief overview (not in detail)
• For little more detail, some useful websites
• www.cs.berkeley.edu/ug/slide/pipeline/assignments
  /as5/rotation.html
• www.cs.cmu.edu/afs/cs/academic/class/16741-s02/www
  /lecture6.pdf

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Rotations

• 2D (simple)
• Matrix representation, commutative
• 3D (complicated)
• Non-commutative
• Counterclockwise
• Matrix reps for axis transforms

5
Euler Angles

• Rotate about Z, then new X, then
  new Z
• Goldstein pp 107
• Product of axis transforms
• Efficient, 3 parameters
• Bad for numerics, splining

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Axis-Angle

• Single axis, angle about it
• 3 params (axis is normalized)
• Somewhat intuitive
• Many of same numerical problems as Euler-Angles
• Formula (axis-angle to rotation matrix)

Dual Matrix of v
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Viewing

• 3D
• Pipeline (pp 230 FvDFH)
• Projection 3D -gt 2D
• Clip near and far planes (viewing volume)

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The Whole Viewing Pipeline
Eye coordinates
Model coordinates
Perspective transformation
Model transformation
Screen coordinates
Viewport transformation
World coordinates
Camera transformation
Window coordinates
Raster transformation
Device coordinates
Slide courtesy Greg Humphreys
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Projections

• To lower dimensional space
• Preserve straight lines
• Trivial example Drop one coordinate
• We are concerned with 3D -gt 2D
• Taxonomy Fig 6.13, pp 237
• Perspective realistic images, no proportions
  preserved
• Parallel preserves proportions
• Orthographic (common in graphics)

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Perspective

• Center of projection, rays, projection plane
• Distance from center of projection to projection
  plane is finite. If infinite, projection rays
  parallel
• Foreshortening Key feature
• Size inverse proportional to distance
• Cameras, visual system are similar

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Perspective or Parallel?

• If the distance between the center of
  projection and the projection plane is finite,
  the projection is called a perspective
  projection. Otherwise, its a parallel
  projection.

A
A
A
A
B
B
B
B
C (at infinity)
C
Parallel
Perspective
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Perspective

• Distortions
• Lines remain lines, everything else is distorted
• Vanishing points for parallel lines
• Shape distortions
• Parallel Projections (parallel lines are
  parallel)
• Not realistic but preserve size (useful in
  mechanical drawings)

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Parallel (Orthographic)

• Projection vector (direction of rays)
• Perpendicular to projection plane Orthographic
• Otherwise, oblique.
• Far off objects in perspective are approx.
  orthographic
• Uniform foreshortening
• Parallelism preserved, angles are not
• More info in book
• We now concentrate on perspective

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Perspective mathematically 1

• Intuition divide by z
• Eye as a point (or pinhole camera)
• Easy in eye space (homogeneous coordinates)
• Simple form equation 6.4 page 255
• Full mathematics next week

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Overhead View of Our Screen
Looks like weve got some nice similar triangles
here!
Next few slides courtesy Greg Humphreys
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The Perspective Matrix

• This division by z can be accomplished by a 4x4
  matrix
• Were in homogeneous coordinates, so if we
  multiply some point by P, we
  get .
• Recall that to get the actual 3D point, you
  divide by w, giving ,
  which was the point on our screen. Now just drop
  the third coordinate!

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Viewing in OpenGL

• OpenGL has multiple matrix stacks -
  transformation functions right-multiply the top
  of the stack
• Two most important stacks GL_MODELVIEW and
  GL_PROJECTION
• Points get multiplied by the modelview matrix
  first, and then the projection matrix
• This is only really necessary for convenience --
  you could ignore all of this and put your entire
  viewing matrix into one or the other. However,
  if one is changing and the other one isnt,
  youll save time

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OpenGL Example
void SetUpViewing() // The viewport isnt a
matrix, its just state... glViewport( 0, 0,
window_width, window_height ) // Set up camera
transformation first glMatrixMode( GL_PROJECTION
) glLoadIdentity() gluPerspective( 60, 1, 1,
1000 ) // fov, aspect, near, far gluLookAt( 3,
3, 2, // eye point 0, 0, 0,
// look at point 0, 0, 1 )
// up vector // Set up the model
transformation glMatrixMode( GL_MODELVIEW
) glRotatef( theta, 0, 0, 1 ) // rotate the
model glScalef( zoom, zoom, zoom ) // scale the
model