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Projections

n Projecting: mapping from 3D viewing coordinates to 2D coordinates in projection plane. In homogeneous coordinates it is a mapfrom 4D viewing coordinates to 3D.

n Projectionsn Parallel: orthogonal and obliquen Perspective

n Canonical views: orthographic and perspective projections

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Perspective projection

Projectors intersect at COP

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Parallel Projections

Projectors parallel.COP at infinity.

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Parallel projections: summary

n Center of projection is at infinity.n Projectors are parallel.n Parallel lines stay paralleln There is no forshortheningn Distances and angles are transformed

consistentlyn Used most often in engineering design, CAD

systems. Used for top and side drawings from which measurements could be made.

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Orthographic Projection: projectors orthogonal to projection plane

DOPsame for all points

(direction of projectors)(direction of projectors)

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Orthographic Projections

DOP is perpendicular to the view plane

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Multiview Parallel Projection

Faces are parallel to the projection plane

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Isometric Projection

Projector makes equal angles with all three principal axes

All three axes are equally foreshortened

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isometric

Mechanical Drawing

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Oblique Parallel Projectionsn Most general parallel viewsn Projectors make an arbitrary angle with the

projection planen Angles in planes parallel to the projection plane are

preserved

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Oblique Projections: projectors are not orthogonal to image plane

CavalierAngle between projectors and projection plane is 45°. Lines orthogonal to the projection planeRetain their exact length. Perpendicular faces are projected at full scale

CabinetAngle between projectors and projection plane is arctan(2)=63.4°. Lines orthogonal to the projection plane are projected at half length. Perpendicular faces are projected at 50% scale.Looks like forshorthening.

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Perspective Projection

n Most natural for peoplen In human vision, perspective projection of the world is

created on the retina (back of the eye)n Used in CG for creating realistic imagesn Perspective projection images carry depth cuesn Foreshorthening causes distant objects to appear smallern Relative lengths and angles are not preservedn A perspective image cannot be used for metric

measurements of the 3D worldn Parallel lines not parallel to the image plane converge at a

vanishing pointn 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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Vanishing Points

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Vanishing Points

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Early Perspective

Not systematic—parallel lines do not converge to a single "vanishing" point

Giotto

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Math of Projections:Overviewn Math of perspective projection, standard

configurationn OpenGL perspective projectionsn Math of orthographic projectionn OpenGL orthographic projectionsn Viewport transformations and setting them in

OpenGLn Summary

n Viewing transformationsn Orthographic projection canonical viewing volumen Perspective projection canonical viewing volume

n Hidden surface removal