Projections Perspective projection n Projecting: mapping from 3D viewing coordinates to 2D coordinates in projection plane. In homogeneous coordinates it is a map from 4D viewing coordinates to 3D. n Projections
n Parallel: orthogonal and oblique
n Perspective n Canonical views: orthographic and perspective projections Projectors intersect at COP
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Parallel Projections Parallel projections: summary
n Center of projection is at infinity.
n Projectors are parallel.
n Parallel lines stay parallel
n There is no forshorthening
n Distances and angles are transformed consistently
n Used most often in engineering design, CAD systems. Used for top and side drawings from Projectors parallel. which measurements could be made. COP at infinity.
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Orthographic Projection: projectors orthogonal to projection plane Orthographic Projections
same for all points DOP (direction of projectors) DOP is perpendicular to the view plane
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1 Multiview Parallel Projection Isometric Projection
Projector makes equal angles with all three principal axes Faces are parallel to the projection plane All three axes are equally foreshortened
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Oblique Parallel Projections Mechanical Drawing n Most general parallel views n Projectors make an arbitrary angle with the projection plane n Angles in planes parallel to the projection plane are preserved
isometric
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Oblique Projections: projectors are not Perspective Projection
orthogonal to image plane n Most natural for people Cavalier n In human vision, perspective projection of the world is Angle between projectors and projection plane is 45°. Lines orthogonal to the projection plane created on the retina (back of the eye) Retain their exact length. Perpendicular faces are projected at full scale n Used in CG for creating realistic images n Perspective projection images carry depth cues n Foreshorthening causes distant objects to appear smaller n Relative lengths and angles are not preserved
Cabinet n A perspective image cannot be used for metric Angle between projectors and projection plane is arctan(2)=63.4°. Lines orthogonal to the measurements of the 3D world projection plane are projected at half length. Perpendicular faces are projected at 50% scale. Looks like forshorthening. n Parallel lines not parallel to the image plane converge at a vanishing point n An axis (principal) vanishing point is a point of convergence for lines parallel to a principal axis of the object. We distinguish one-, two-, three-point projections.)
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2 Vanishing Points Vanishing Points
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Math of Projections:Overview Early Perspective n Math of perspective projection, standard configuration Giotto n OpenGL perspective projections n Math of orthographic projection n OpenGL orthographic projections n Viewport transformations and setting them in OpenGL n Summary
n Viewing transformations
n Orthographic projection canonical viewing volume
n Perspective projection canonical viewing volume n Hidden surface removal Not systematic—parallel lines do not converge to a single "vanishing" point
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