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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

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Projections and perspectives l.jpg

Projections and Perspectives

Dr Nicolas Holzschuch

University of Cape Town

e-mail: [email protected]

Modified by Longin Jan Latecki

[email protected]


Projections and perspectives2 l.jpg

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 l.jpg

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 l.jpg

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 l.jpg

Parallel Projection Matrix

  • Parallel projection onto z=0 plane:

    x’=x, y’=y, w’=w

  • Matrix for this projection:


Perspective projection matrix l.jpg

Perspective Projection Matrix

  • Projection onto plane z=0, with center of projection at z=-d:


Distance to projection plane l.jpg

Distance to projection plane

d

Center of

Projection

Projection

Plane


Field of view l.jpg

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 l.jpg

Field of view

d

Center of

Projection

1

a

-1

Projection

Plane


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Homogeneous coordinates

  • Essential for perspective projection

  • Note the shortening of distances uses w

  • Impossible todo without homogeneous


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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 l.jpg

Translate

Projection

Rotate

3D Model

Coordinates

2D Pixel

Coordinates

Canonical

3D coord.

Canonical Coordinates

  • After translation, before projection: canonical 3D coordinates.


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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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