SI31 Advanced Computer Graphics AGR

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SI31Advanced Computer GraphicsAGR

• Lecture 4
• Projection
• Clipping
• Viewport Transformation

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Interlude

• Why does a mirror reflect left-right and not
  up-down?

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• Projection

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Viewing Co-ordinate System

• The viewing transformation has transformed
  objects into the viewing co-ordinate system,
  where the camera position is at the origin,
  looking along the negative z-direction

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View Volume
yV
zV
near plane
far plane
camera
q
xV
dFP
We determine the view volume by - view angle,
q - aspect ratio of viewplane - distances to near
plane dNP and far plane dFP
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Projection
We shall project on to the near plane. Remember
this is at right angles to the zV direction, and
has z-coordinate zNP - dNP
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Perspective Projection Calculation
zNP
looking down x-axis towards the origin
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Perspective Projection Calculation
P
By similar triangles, yP / yQ ( - zNP) / ( -
zQ) and so yP yQ (- zNP) / ( - zQ) or yP yQ
dNP / ( - zQ)
Similarly for the x-coordinate of P xP xQ
dNP / ( - zQ)
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Using Matrices and Homogeneous Co-ordinates

• We can express the perspective transformation in
  matrix form
• Point Q in homogeneous coordinates is (xQ, yQ,
  zQ, 1)
• We shall generate a point H in homogeneous
  co-ordinates (xH, yH, zH, wH), where wH is not 1
• But the point (xH/wH, yH/wH, zH/wH, 1) is the
  same as H in homogeneous space
• This gives us the point P in 3D space, ie xP
  xH/wH, simly for yP

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Transformation Matrix for Perspective

Thus in Homogeneous co-ordinates xH xQ yH
yQ zH zQ wH (-1/dNP)zQ
In Cartesian co-ordinates xP xH / wH
xQdNP/(-zQ) yP similar zP -dNP zNP
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Exercises

• Suppose the camera is not at the origin, but at
  point zC
• Calculate the formulae for perspective projection
  and work out the resulting transformation matrix
  in homogeneous co-ordinates
• In practice the perspective division is delayed -
  why might this be done? -what useful information
  is lost?

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OpenGL

• Perspective projection achieved by
• gluPerspective (angle_of_view, aspect_ratio,
  near, far)
• aspect ratio is width/height
• near and far are positive distances

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Vanishing Points

• When a 3D object is projected onto a view plane
  using perspective, parallel lines in object NOT
  parallel to the view plane converge to a
  vanishing point

vanishing point
one-point perspective projection of cube
view plane
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One- and Two-Point Perspective Drawing
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One-point Perspective
This is Trinity with the Virgin, St John and
Donors, by Mastaccio in 1427
Said to be the first painting in perspective
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Two-point Perspective
Edward Hopper Lighthouse at Two
Lights -see www.postershop.com
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Parallel Projection - Two types

• Orthographic parallel projection has view plane
  perpendicular to direction of projection

• Oblique parallel projection has view plane at an
  oblique angle to direction of projection

P1
P1
P2
P2
view plane
view plane
We shall only consider orthographic projection
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Parallel Projection Calculation
looking down x-axis
Q
P
yV
zQ
zNP
zV
view plane
yP yQ similarly xP xQ
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Parallel Projection Calculation

• So this is much easier than perspective!
• xP xQ
• yP yQ
• zP zNP
• The transformation matrix is simply

1 0 0 0 0 1 0 0 0
0 zNP/zQ 0 0 0 0 1
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• Clipping

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View Frustum and Clipping
yV
zV
near plane
far plane
camera
q
xV
dFP
The view volume is a frustum in viewing
co-ordinates - we need to be able to clip objects
outside of this region
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Clipping to View Frustum

• It is quite easy to clip lines to the front and
  back planes (just clip in z)..
• .. but it is difficult to clip to the sides
  because they are sloping planes
• Instead we carry out the projection first which
  converts the frustum to a rectangular
  parallelepiped (ie a cuboid)

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Clipping for Parallel Projection

• In the parallel projection case, the viewing
  volume is already a rectangular parallelepiped

far plane
view volume
near plane
zV
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Normalized Projection Co-ordinates

• Final step before clipping is to normalize the
  co-ordinates of the rectangular parallelepiped to
  some standard shape
• for example, in some systems, it is the cube with
  limits 1 and -1 in each direction
• This is just a scale transformation
• Clipping is then carried out against this
  standard shape

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Viewing Pipeline So Far

• Our pipeline now looks like

NORMALIZATIONTRANSFORMATION
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• Viewport Transformation

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And finally...

• The last step is to position the picture on the
  display surface
• This is done by a viewport transformation where
  the normalized projection co-ordinates are
  transformed to display co-ordinates, ie pixels on
  the screen

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Viewing Pipeline - The End

• A final viewing pipeline is therefore

device co-ordinates
DEVICETRANSFORMATION