Projection Our 3-D scenes are all specified in 3-D world coordinates To display these we need to generate a 2-D image --- project objects onto a picture plane Projection is just one part of the process of converting from 3-D world coordinates to a 2-D image 3-D world coordinate output primitives Clip against view volume Project onto projection plane Transform to 2-D device coordinates 2-D device coordinates

Projection defined as a mapping of point p(x,y,z) to it’s image point p’(x’y’z’) in projection plane. P(x,y,z) P(x’,y’,z’) Projection plane/ view plane The mapping is defined by: Projector Projection plane

There are two broad classes of projection: Parallel: Typically used for architectural and engineering drawings Perspective: Realistic looking and used in computer graphics Parallel Projection Perspective Projection

Projection Perspective Parallel One - Point Oblique Orthographic Projector Parallel One - Point Projector Oblique Orthographic Two - Point Cavalier Projector Three - Point Cabinet Multi view Axonometric Projection Plane Projection Plane Isometric Elevation Planner Diametric Front Top Trimetric Rear Projection Plane Side Projection Plane

Perspective Projection Object position’s are transformed to the view plane along the lines that converge at a pt (C.O.P) So how do we figure out these projections? Picture Plane Objects in World Space

Depending on orientation of the projection plane it can be -: A. One vanishing point – : B. Two vanishing point – : C. Three vanishing point – : Vanishing pt / C.O.P Z-axis Vanishing Pt X-axis Vanishing Pt

Parallel Projection Object position’s are transformed to the view plane along the lines that are parallel to each other Projector is perpendicular To view plane A. Orthographic projection P(x’,y’,z’) : - - P(x,y,z) Projector makes any angle other than 90’ with view plane Oblique projection : - - P(x,y,z) P(x’,y’,z’)

A. Orthographic Projection View plane is normal to any of the principal axis OR When direction of projection (projector) is parallel to any of the principal axis Multi View :-- Rear view

A. Orthographic Projection View plane is not normal to any of the principal axis OR Direction of projection (projector) is not parallel to any of the principal axis Axonometric Projection :-- When view plane normal (projector) makes equal angle with each principal axis a. Isometric :-- b. Diametric :- When view plane normal (projector) makes equal angle with two principal axis c. Trimetric :-- When view plane normal (projector) makes different angle with each principal axis Q*Bert

Comparison of isometric, dimetric & trimetric

When direction of projection(projector) is not perpendicular B. Oblique Projection When direction of projection(projector) is not perpendicular to the view plane When direction of projection(projector) makes 45’ angle with the view plane 1. Cavalier :-- When direction of projection(projector) makes ? angle with the view plane where tan(?) = 2 1. Cabinet :--

Examples of Oblique Projection

Advantages and Disadvantages (perspective) Objects further from viewer are projected smaller than the same sized objects closer to the viewer (diminution) Looks realistic Equal distances along a line are not projected into equal distances (nonuniform foreshortening) Angles preserved only in planes parallel to the projection plane More difficult to construct by hand than parallel projections (but not more difficult by computer)

Advantages and Disadvantage (parallel) Preserves both distances and angles Shapes preserved Can be used for measurements Building plans Manuals Cannot see what object really looks like because many surfaces hidden from view Often we add the isometric