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Summary Orthographic Projection: Projectors Orthog 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 Viewing, projections Hofstra University 16 Viewing, projections Hofstra University 17 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. Viewing, projections Hofstra University 18 Viewing, projections Hofstra University 19 Orthographic Projection: projectors orthogonal to projection plane Orthographic Projections same for all points DOP (direction of projectors) DOP is perpendicular to the view plane Viewing, projections Hofstra University 20 Viewing, projections Hofstra University 21 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 Viewing, projections Hofstra University 22 Viewing, projections Hofstra University 23 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 Viewing, projections Hofstra University 24 Viewing, projections Hofstra University 25 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.) Viewing, projections Hofstra University 26 Viewing, projections Hofstra University 27 2 Vanishing Points Vanishing Points Viewing, projections Hofstra University 28 Viewing, projections Hofstra University 29 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 Viewing, projections Hofstra University 30 Viewing, projections Hofstra University 31 3.
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