Projections and Perspectives

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Presentation transcript:

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 Sep. 11, 2002

Projections and Perspectives We have to display our 3D model Screen is 2D Transformation from 3D model coordinates to 2D pixels coordinates. 2D Pixel Coordinates 3D Model

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 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 onto z=0 plane: x’=x, y’=y, w’=w Matrix for this projection:

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

Distance to projection plane Center of Projection Projection Plane

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 view d Center of Projection 1 a -1 Projection Plane

Homogeneous coordinates Essential for perspective projection Note the shortening of distances uses w Impossible to do without homogeneous

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

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