Dr' Jinxiang Chai

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CPSC 441 Computer Graphics3-D Viewing

• Dr. Jinxiang Chai

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Review

• 2D Coordinate transform
• Composite transformation
• 3D transformation
• Required readings HB 5-9 to 5-17

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Review 2D Coordinate Trans.

• Transform object description from to

p
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Review 2D Coordinate Trans.

• Transform object description from to

p
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Review Composite 2DTrans.

• Whats the world coordinate of A ?

A
World coordinate frame
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Review 3D Transformation

• Very similar to 2D transformation

3D translation
3D Scaling
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Review 3D Rotation

• Transformation matrix for rotating about an
  arbitrary axis is more complex

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3D Transformation

• Rotation about z
• Rotate about y
• Rotate about x

y
x
z
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Review Composite 3D Trans.
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Outline

• 3D Viewing
• Required readings HB 7-1 to 7-10
• Compile and run the codes in page 388

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Taking Pictures Using A Real Camera

• Steps
• - Identify interesting objects
• - Rotate and translate the camera to a desire
  camera viewpoint
• - Adjust camera settings such as focal length
• - Choose desired resolution and aspect ratio,
  etc.
• - Take a snapshot

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Taking Pictures Using A Real Camera

• Steps
• - Identify interesting objects
• - Rotate and translate the camera to a desire
  camera viewpoint
• - Adjust camera settings such as focal length
• - Choose desired resolution and aspect ratio,
  etc.
• - Take a snapshot
• Graphics does the same thing for rendering an
  image for 3D geometric objects

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3D Geometry Pipeline
Rotate and translate the camera
View space
World space
Object space
Focal length
Aspect ratio resolution
Normalized project space
Image space
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3D Geometry Pipeline

• Before being turned into pixels by graphics
  hardware, a piece of geometry goes through a
  number of transformations...

Model space (Object space)
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3D Geometry Pipeline

• Before being turned into pixels by graphics
  hardware, a piece of geometry goes through a
  number of transformations...

World space (Object space)
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3D Geometry Pipeline

• Before being turned into pixels by graphics
  hardware, a piece of geometry goes through a
  number of transformations...

Eye space (View space)
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3D Geometry Pipeline

• Before being turned into pixels by graphics
  hardware, a piece of geometry goes through a
  number of transformations...

Normalized projection space
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3D Geometry Pipeline

• Before being turned into pixels by graphics
  hardware, a piece of geometry goes through a
  number of transformations...

Image space, window space, raster space, screen
space, device space
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3D Geometry Pipeline
View space
World space
Object space
Normalized project space
Image space
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3D Geometry Pipeline
Translate, scale rotate
World space
Object space
glTranslate(tx,ty,tz)
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3D Geometry Pipeline
Translate, scale rotate
Object space
World space
glScale(sx,sy,sz)
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3D Geometry Pipeline
Translate, scale rotate
Object space
World space
Rotate about r by the angle
glRotate
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3D Geometry Pipeline
View space
World space
Object space
Normalized project space
Image space
Screen space
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3D Geometry Pipeline

• Now look at how we would compute the world-gteye
  transformation

View space
World space
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3D Geometry Pipeline

• Now look at how we would compute the world-gteye
  transformation

Rotatetranslate
View space
World space
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Camera Coordinate

• Canonical coordinate system
• - usually right handed (looking down z
  axis)
• - convenient for project and clipping

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Camera Coordinate

• Mapping from world to eye coordinates
• - eye position maps to origin
• - right vector maps to x axis
• - up vector maps to y axis
• - back vector maps to z axis

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Viewing Transformation

• We have the camera in world coordinates
• We want to transformation T which takes object
  from world to camera

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Viewing Transformation

• We have the camera in world coordinates
• We want to transformation T which takes object
  from world to camera
• Trick find T-1 taking object from camera to
  world

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Viewing Transformation

• We have the camera in world coordinates
• We want to transformation T which takes object
  from world to camera
• Trick find T-1 taking object from camera to
  world

?
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Review 3D Coordinate Trans.

• Transform object description from to

p
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Review 3D Coordinate Trans.

• Transform object description from to

p
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Review 3D Coordinate Trans.

• Transform object description from camera to
  world

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Viewing Transformation

• Trick find T-1 taking object from camera to
  world
• - eye position maps to origin
• - back vector maps to z axis
• - up vector maps to y axis
• - right vector maps to x axis

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Viewing Transformation

• Trick find T-1 taking object from camera to
  world

HB equation (7-4)
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Viewing Trans gluLookAt

• Mapping from world to eye coordinates

gluLookAt (x0,y0,z0,xref,yref,zref,Vx, Vy,Vz)
HB equation (7-1)
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3D Geometry Pipeline
3D-3D viewing transformation
World space
View space
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Projection

• General definition
• transform points in n-space to m-space (mltn)
• In computer graphics
• map 3D coordinates to 2D screen coordinates

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Taxonomy of Projections
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Taxonomy of Projections
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Parallel Projection

• Center of projection is at infinity
• Direction of projection (DOP) same for all
  points

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Orthographic Projection

• Direction of projection (DOP) perpendicular to
  view plane

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Orthographic Projection

• Direction of projection (DOP) perpendicular to
  view plane

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Oblique Projection

• DOP not perpendicular to view plane

A
A
O
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Oblique Projection

• DOP not perpendicular to view plane

A
A
O
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Properties of Parallel Projection

• Not realistic looking
• Good for exact measurement
• Are actually affine transformation
• - parallel lines remain parallel
• - ratios are preserved
• - angles are often not preserved
• Most often used in CAD, architectural drawings,
  etc. where taking exact measurement is important

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Perspective Projection

• Maps points onto view plane along projectors
  emanating from center of projection (COP)

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Perspective Projection

• Maps points onto view plane along projectors
  emanating from center of projection (COP)

Whats relationship between 3D points and
projected 2D points?
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Camera-gtScreen

• Remember Object-gtcamera-gtscreen

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Camera-gtScreen

• Remember Object-gtcamera-gtscreen
• Screen is z-d plane for some constant d

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Camera-gtScreen

• Remember Object-gtcamera-gtscreen
• Screen is z-d plane for some constant d
• Coordinates of origin of screen is (0,0,-d)

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Camera-gtScreen

• Remember Object-gtcamera-gtscreen
• Screen is z-d plane for some constant d
• Coordinates of origin of screen is (0,0,-d)
• Its x and y axes is parallel to the x and y axes
  of eye coordinate system

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Camera-gtScreen

• Remember Object-gtcamera-gtscreen
• Screen is z-d plane for some constant d
• Coordinates of origin of screen is (0,0,-d)
• Its x and y axes is parallel to the x and y axes
  of eye coordinate system
• All these coordinates are in camera space now

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Camera-gtScreen

• Consider the projection of a point on the camera
  plane

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Camera-gtScreen

• Consider the projection of a point on the camera
  plane

By similar triangles, we can compute how much the
x and y coordinates are scaled
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Camera-gtScreen

• Consider the projection of a point on the camera
  plane

By similar triangles, we can compute how much the
x and y coordinates are scaled
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Homogeneous Point Revisited

• Remember how we said 2D/3D geometric
  transformations work with the last coordinate
  always set to one
• What happens if the coordinate is not one
• We divide all coordinates by w

If w1, nothing happens Sometimes, we call this
division step the perspective divide
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The Perspective Matrix

• Now we can rewrite perspective projection
  equation as a matrix equation

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The Perspective Matrix

• Now we can rewrite perspective projection
  equation as a matrix equation

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The Perspective Matrix

• Now we can rewrite perspective projection
  equation as a matrix equation

After the division by w, we have
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Viewing Angle

• An alternative to specifying the distance between
  COP and PP is to specify viewing angle

Given the height of the image h and ? , what is
d?
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Viewing Angle

• An alternative to specifying the distance between
  COP and PP is to specify viewing angle

Given the height of the image h and ? , what is
d?
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Viewing Angle

• An alternative to specifying the distance between
  COP and PP is to specify viewing angle

Given the height of the image h and ? , what is
d?
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Viewing Angle

• An alternative to specifying the distance between
  COP and PP is to specify viewing angle

Given the height of the image h and ? , what is
d?
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Viewing Angle

• An alternative to specifying the distance between
  COP and PP is to specify viewing angle

Given the height of the image h and ? , what is
d?
What happens to d as ? increases (keep d
constant)?
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Vanishing Points

• What happens to parallel lines they are not
  parallel to the projection plane?

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Vanishing Points

• What happens to parallel lines they are not
  parallel to the projection plane?
• The equation of the line

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Vanishing Points

• What happens to parallel lines they are not
  parallel to the projection plane?
• The equation of the line
• After perspective transformation, we have

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Vanishing Points (cont.)

• After perspective transformation, we have
• Divided by w
• Letting t go to infinity

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Vanishing Points

• What happens to parallel lines they are not
  parallel to the projection plane?
• The equation of the line

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Vanishing Points

• What happens to parallel lines they are not
  parallel to the projection plane?
• The equation of the line
• How about the line

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Vanishing Points

• What happens to parallel lines they are not
  parallel to the projection plane?
• The equation of the line
• How about the line

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Vanishing Points

• What happens to parallel lines they are not
  parallel to the projection plane?
• The equation of the line
• How about the line

Same vanishing point!
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Vanishing Points

• What happens to parallel lines they are not
  parallel to the projection plane?
• Each set of parallel lines intersect at a
  vanishing point on the PP

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Vanishing Points

• What happens to parallel lines they are not
  parallel to the projection plane?
• Each set of parallel lines intersect at a
  vanishing point on the PP
• How many vanishing points are there?

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Vanishing Points

• What happens to parallel lines they are not
  parallel to the projection plane?
• Each set of parallel lines intersect at a
  vanishing point on the PP
• How many vanishing points are there?

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Properties of Perspective Proj.

• Perspective projection is an example projective
  transformation
• - lines maps to lines
• - parallel lines do not necessary remain
  parallel
• - ratios are not preserved

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Properties of Perspective Proj.

• Perspective projection is an example projective
  transformation
• - lines maps to lines
• - parallel lines do not necessary remain
  parallel
• - ratios are not preserved
• One of advantages of perspective projection is
  that size varies inversely proportional to the
  distance-looks realistic

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Properties of Perspective Proj.

• Perspective projection is an example projective
  transformation
• - lines maps to lines
• - parallel lines do not necessary remain
  parallel
• - ratios are not preserved
• One of advantages of perspective projection is
  that size varies inversely proportional to the
  distance-looks realistic
• We can not judge distances exactly as we can
  with parallel projection

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3D Geometry Pipeline
View space
World space
Object space
Normalized project space
Image space
Screen space
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Perspective Projection Volume

• The center of projection and the portion of
  projection plane that map to the final image form
  an infinite pyramid. The sides of pyramid called
  clipping planes
• Additional clipping planes are inserted to
  restrict the range of depths

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Normalized Perspective-Projection
Normalized project space
View space
glFrustum(xwmin, xwmax,ywmin, ywmax ,dnear,dfar)
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Normalized Perspective-Projection
B(xwmax,ywmax,zfar)
A(xwmin,ywmin,znear)
View space
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Normalized Perspective-Projection
B(1,1,1)
B(xwmax,ywmax,zfar)
A(-1,-1,-1)
A(xwmin,ywmin,znear)
Normalized project space (left-handed)
View space (right-handed)
A maps to A, B maps to B Keep the directions of
x and y axes!
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Normalized Perspective-Projection
B(1,1,1)
B(xwmax,ywmax,zfar)
A(-1,-1,-1)
A(xwmin,ywmin,znear)
HB equation (7-40)
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Normalized Perspective-Projection
B(1,1,1)
B(xwmax,ywmax,zfar)
A(-1,-1,-1)
A(xwmin,ywmin,znear)
Normalized project space
View space
glFrustum(xwmin, xwmax,ywmin, ywmax ,dnear,dfar)
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3D Geometry Pipeline
View space
World space
Object space
Normalized project space
Image space
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Viewport Transformation
B(1,1,1)
B(xvmax,yvmax,1)
Normalized project space
A(xvmin,yvmin,0)
A(-1,-1,-1)
Image space
Normalized project space
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Viewport Transformation
B(1,1,1)
B(xvmax,yvmax,1)
Normalized project space
A(xvmin,yvmin,0)
A(-1,-1,-1)
Image space
Normalized project space
HB equation (7-42)
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Viewport Transformation
(xmin,ymin)
Image space-.gtImage space
gluViewport(xmin, ymin, width, height)
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Summary 3D Geometry Pipeline
View space
World space
Object space
Normalized project space
Image space
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Next Lecture

• Hierarchical Models
• Animation with motion capture data
• - skeletal model (.asf)
• - motion data (.amc)
• Required readings
• - HB chapter 14,
• - OpenGL Programming Guide, chapter 3