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Projections and Perspectives

Projections and Perspectives. Dr Nicolas Holzschuch University of Cape Town e-mail: holzschu@cs.uct.ac.za Modified by Longin Jan Latecki latecki@temple.edu. 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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  1. Projections and Perspectives Dr Nicolas Holzschuch University of Cape Town e-mail: holzschu@cs.uct.ac.za Modified by Longin Jan Latecki latecki@temple.edu

  2. 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.

  3. 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

  4. 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

  5. Parallel Projection Matrix • Parallel projection onto z=0 plane: x’=x, y’=y, w’=w • Matrix for this projection:

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

  7. Distance to projection plane d Center of Projection Projection Plane

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

  9. Field of view d Center of Projection 1 a -1 Projection Plane

  10. Homogeneous coordinates • Essential for perspective projection • Note the shortening of distances uses w • Impossible todo without homogeneous

  11. 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

  12. Translate Projection Rotate 3D Model Coordinates 2D Pixel Coordinates Canonical 3D coord. Canonical Coordinates • After translation, before projection: canonical 3D coordinates.

  13. 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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