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Projections and Perspectives. Dr Nicolas Holzschuch University of Cape Town e-mail: [email protected] Modified by Longin Jan Latecki [email protected] 3D Model Coordinates. 2D Pixel Coordinates. Projections and Perspectives. We have to display our 3D model Screen is 2D

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projections and perspectives

Projections and Perspectives

Dr Nicolas Holzschuch

University of Cape Town

e-mail: [email protected]

Modified by Longin Jan Latecki

[email protected]

projections and perspectives2

3D Model

Coordinates

2D Pixel

Coordinates

Projections and Perspectives
  • We have to display our 3D model
  • Screen is 2D
  • Transformation from 3D model coordinates to 2D pixels coordinates.
projections
Projections
  • Representation of a 3D scene, for a virtual observer
  • One viewpoint:
    • position of the observer
  • One direction of view:
    • direction where the observer is looking
  • One “up vector”:
    • vertical for the observer
different projections
Different projections
  • Parallel projections:
    • no shortening due to distance
    • several kinds, depending on orientation:
      • isometric, cavalier,…
  • Perspective projections:
    • shortening of objects in the distance
    • several kind, depending on orientation:
      • one, two, three vanishing points
parallel projection matrix
Parallel Projection Matrix
  • Parallel projection onto z=0 plane:

x’=x, y’=y, w’=w

  • Matrix for this projection:
perspective projection matrix
Perspective Projection Matrix
  • Projection onto plane z=0, with center of projection at z=-d:
distance to projection plane
Distance to projection plane

d

Center of

Projection

Projection

Plane

field of view
Field of view
  • Distance to projection plane not intuitive
  • Easier notion: field of view
  • FOV= angle, in degrees
  • Expresses how wide is my vision
field of view9
Field of view

d

Center of

Projection

1

a

-1

Projection

Plane

homogeneous coordinates
Homogeneous coordinates
  • Essential for perspective projection
  • Note the shortening of distances uses w
  • Impossible todo without homogeneous
other viewpoints
Other viewpoints
  • If we are viewing from another point?
  • Translations:
    • until viewpoint is at origin
  • Rotations:
    • until direction of viewing is on z-axis
  • Back to the previous case
canonical coordinates

Translate

Projection

Rotate

3D Model

Coordinates

2D Pixel

Coordinates

Canonical

3D coord.

Canonical Coordinates
  • After translation, before projection: canonical 3D coordinates.
projections discussion
Projections: discussion
  • Projections have several goals:
    • exactitude (e.g. plans for architects)
    • realism (view of a scene, VR)
    • visualization
    • artistic view: non-linear perspectives (e.g. Rubber Soul)
  • An important part of realism in 3D rendering
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