3D Projection Transformations

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3D Projection Transformations

• Soon Tee Teoh
• CS 147

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3D Projections
Rays converge on eye position
Rays parallel to view plane
Perspective
Parallel
Orthographic
Oblique
Cabinet
Cavalier
Elevations
Axonometric
Isometric
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Perspective and Parallel Projections
View plane
Perspective
Parallel
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3D Projections
Rays converge on eye position
Rays parallel
Perspective
Parallel
Rays at angle to view plane
Rays perpendicular to view plane
Orthographic
Oblique
Cabinet
Cavalier
Elevations
Axonometric
Isometric
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Parallel ProjectionsOrthographic and Oblique
View plane
Oblique
Orthographic
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Principal Axes

• Man-made objects often have cube-like shape.
  These objects have 3 principal axes.

From www.loc.gov/ jefftour/cutaway.html
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One point, two point, three point perspective

• Depends on how many principal axes intersect with
  view plane.
• Parallel lines not parallel to view plane have
  the same vanishing point.

One point perspective One principal axis
intersects view plane
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One point, two point, three point perspective
Two point perspective two principal axes
intersect view plane
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One point, two point, three point perspective
Three point perspective Three principal axes
intersect view plane
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One point, two point, three point perspective
View Plane
Three point
Two point
One point
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3D Projections
Rays converge on eye position
Rays parallel
Perspective
Parallel
Rays at angle to view plane
Rays perpendicular to view plane
Orthographic
Oblique
View plane aligned with principal axes
View plane not aligned with principal axes
Cabinet
Cavalier
Elevations
Axonometric
Isometric
Trimetric
Dimetric
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Front Elevation

• Parallel Orthogonal Elevation

Front elevation of tallest buildings in the world
From members.iinet.net.au/ paulkoh
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Elevations
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Isometric View

• In isometric view, the three principal axes of
  the object intersect the view plane at equal
  distance. Therefore, when projected, they are
  120o apart.

http//sucod.shef.ac.uk/sucod/gallery/arc320/2003/
p2/Rachel/images/colour.JPG
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3D Projections
Rays converge on eye position
Rays parallel
Perspective
Parallel
Rays at angle to view plane
Rays perpendicular to view plane
Orthographic
Oblique
Cabinet
Cavalier
Elevations
Axonometric
Isometric
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Oblique projections

• Projection lines are at an angle to the view
  plane.
• Let the angle be a be the angle the projection
  line makes with the view plane.
• tan a 1 (or, a 45o) called cavalier
  projection
• tan a 2 (or, a 63.4o) called cabinet
  projection

1
2
a
a
1
1
cavalier
cabinet
1
1/2
1
1
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Orthographic Parallel Projection Matrix

• Transform each vertex from Viewing Coordinates
  into Normalized Coordinates using orthographic
  projection
• Suppose that a point is (x,y,z) in Viewing
  Coordinates, whats the transformation necessary
  to transform it to (x,y,z) in Normalized
  Coordinates?
• Given the dimensions of the view window xwmin,
  xwmax, ywmin, ywmax
• Orthogonal Projection Matrix on p. 362.
• Basically Translate center of view to origin and
  then Scale to (-1,1) cube
• Translate by -(minmax)/2, then scale by
  2/(max-min).

xwmax xwmin xwmax - xwmin
2 xwmax - xwmin
0 0 -
2 ywmax - ywmin
ywmax ywmin ywmax - ywmin
0 -
0
M
-2 znear - zfar
znear zfar znear - zfar
0 0
0 0 0
1
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Perspective Projection Matrix

• Suppose camera position is at origin (0,0,0)
• Suppose view plane is at distance d from origin
• Consider top view

Original point in viewing coordinates
View plane
x
Point projected to view plane
x
x
(0,0,0)
z
d
z
The projected coordinate, x dx/z Similarly, y
dy/z
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Perspective Projection Matrix
1 0 0 0 0 1 0 0 0 0 1
0 0 0 1/d 0
Perspective Projection Matrix M
In homogeneous coordinates,
1 0 0 0 x x 0
1 0 0 y y 0 0
1 0 z z 0 0
1/d 0 1 z/d

Point in viewing coordinates
Point projected to view plane
In normal coordinates, (x,y,z) (dx/z,dy/z,d)