CSE452 Computer Graphics

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CSE452 Computer Graphics

• Lecture 8 Computer Projection 2

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Review

• In the last lecture
• We set up a Virtual Camera
• Position
• Orientation
• Clipping planes
• Viewing angles
• Orthographic/Perspective
• We are ready to project!

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Preview

• The perspective view frustum (i.e., a truncated
  pyramid) can be hard for both clipping and
  projection
• Lets first consider a parallel, canonical volume

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Canonical View Volume
Near Clipping Plane
Far Clipping Plane
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Canonical View Volume

• Canonical view volume makes things easy
• Easy clipping Clip against the coordinates range
• Easy projecting drop the z coordinate! (because
  viewing plane is the xy plane, and projectors are
  parallel to z axis)

Coordinates in the canonical volume
Projected 2D coordinates
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Viewing Transformation

• For any viewing volume transform it into the
  canonical volume!
• The transformation brings the blue box into the
  red box
• The same transformation brings world coordinates
  xw,yw,zw into canonical coordinates within the
  canonical volume xc,yc,zc
• Clip and project

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Camera Coordinate System

• Three orthogonal axes setting up the cameras
  world
• u (right), v (straight-up), n (negative look)
• Unit vectors forming an orthonormal basis
• Observing the right-hand rule

Y
X
Z
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Camera Coordinate System

• Computing n
• Opposite to look vector L, normalized

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Camera Coordinate System

• Computing v
• Projection of up vector U onto the camera plane,
  normalized

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Camera Coordinate System

• Computing u

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Camera Coordinate System

• Summary
• Three axes, computed from look vector L and up
  vector U
• They are unit vectors
• u,v,n follow the right-hand rule

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Computing Viewing Transformation

• Two steps
• Step 1 aligning camera coordinate system P,u,v,n
  with world coordinate system O,x,y,z
• Step 2 scale and stretch the viewing volume to
  the cuboid

Y
X
Z
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Step 1

• First, translate the eye point P to the origin
• Homogenous coordinates! Easy to compose
  transformation as matrix product

O (P)
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Step 1 (cont)

• Rotate the three axes u,v,n to x,y,z
• First, set up the equation to solve for the
  rotation matrix R we know that vectors u,v,n are
  rotated to three canonical vectors, and the
  original stays put.
• In matrix form

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Step 1 (cont)

• Simplify our notation

• Solve for the rotation matrix
• M is ortho-normal, meaning .
  Therefore

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After Step 1

• Eye point at origin, looking down negative z axis

(v)
O (P)
(n)
(u)
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Step 2

• Scale the viewing volume to the correct size
• Here is a look down the Y axis
• First, force width/height angle to be right
  angles
• Scaling in x,y coordinates

Z
X
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Step 2 (cont)

• Scale the viewing volume to the correct size
• Next, push the far plane to the xy plane at z-1
• Scaling in all three coordinates

Z
X
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Step 2 (cont)

• Now, the far plane corners are at

(v)
O (P)
(n)
(u)
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Step 2 (cont)

• Perspective transformation
• Stretching the truncated pyramid to the cuboid
• Stretch in XY plane Non-uniform stretching based
  on z
• Change Z range (-dn/df,-1 -gt 0,1)

y
y
1
z
z
-1
1
0
-dn/df
-1
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Step 2 (cont)

• Perspective transformation
• Lets see what matrix D does
• When converting from homogeneous coordinates
  (x,y,z,w) to Cartesian coordinates, we need to
  divide x,y,z by w !!

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Step 2 (cont)

• Perspective transformation
• Stretching the truncated pyramid to the cuboid
• Stretch in XY plane Non-uniform stretching based
  on z
• Change Z range (-dn/df,-1 -gt 0,1)

y
y
1
z
z
-1
1
0
-dn/df
-1
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Putting Together

• Translation
• Rotation
• Scaling
• Perspective transformation

World-to-camera transformation

• Final viewing transformation to bring a point
  to the canonical volume

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Clipping

• After transformation into the canonical volume,
  each object will be clipped against 6 cuboid
  faces.
• Point clipping checking coordinates range
• Edge clipping computing line/plane intersections
• We will discuss a simplified, 2D clipping problem
  in next lecture.
• Triangle clipping Need to consider face/face
  intersections

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Projecting

• Dropping z coordinate
• Resulting points have range

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Viewport Transform

• Get viewport (pixel) coordinates
• Viewport coordinate 0,0 is at top-left corner
• If the viewport is a pixels wide and b pixels
  high, the pixel coordinates for a projected point
  x,y is

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Summary

• To compute projection
• Set up camera parameters
• Compute viewing transformation
• Clipping against the canonical volume
• Projecting to viewport

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Practice Questions

• Camera projection using following parameters eye
  point at 0,5,0, look vector 0,-1,0, up vector
  1,1,1, height and width angle 90 degrees, near
  and far plane at distances 0.1 and 10 from the
  eye point.
• Compute camera coordinate system axes u,v,n. Show
  that u,v,n form an orthonormal basis.
• Write down the world-to-camera matrix and the
  perspective matrix.

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Practice Questions

• 2. Place a unit-radius sphere at the origin. Draw
  the spheres location in the camera coordinate
  system before perspective transformation. That
  is, where does the sphere end up after applying
  the World-to-camera matrices? Draw the canonical
  coordinate system from the side, with y up and z
  to the left, x out of the plane. Give the
  distance along the z axis and the height/size of
  the sphere in the z and y direction.
• Distance along z (apply D.WTC to the origin, then
  take z coordinate) -1/2
• Size along z (apply D.WTC to n and -n, then take
  z difference) radius 1/5
• Size along y (apply D.WTC to v and -v, then take
  z difference) radius 1/5

0,1,-1
Y
Z
0,-1,-1