1. Basic Projections 2D to 3D

2. 3D viewing process Specify a 3D view volume Clip against view volume
Project onto a 2D viewing plane
Define a window on the viewing plane
Apply 2D viewing transformations to map window contents into 2D-image viewport

3. Planar Projections •Perspective: Distance to CoP is finite
•Parallel: Distance to CoP is infinite

4. Parallel Projections
•Orthographic: Direction of projection is orthogonal to the projection plane
•Elevations: Projection plane is perpendicular to a principle axis
•Front
•Top (Plan)
•Side
•Axonometric: Projection plane is not orthogonal to a principle axis
•Isometric: Direction of projection makes equal angles with each principle axis.
•Oblique: Direction of projection is not orthogonal to the projection plane; projection plane is normal to a principle axis
•Cavalier: Direction of projection makes a 45° angle with the projection plane
•Cabinet: Direction of projection makes a 63.4° angle with the projection plane

5. Perspective Projections
•One-point: One principle axis cut by projection plane
One axis vanishing point
•Two-point: Two principle axes cut by projection plane
Two axis vanishing points
•Three-point: Three principle axes cut by projection plane
Three axis vanishing points

6. Perspective Projections

7. One- Point Projections
Center of Projection on the negative z-axis
View-plane parallel to the x-y plane and through the origin. -Z +Z
(x, y, z)
(xproj, yproj, 0) x y
xprojected = xd/(d+z) = x/(1+(z/d))
yprojected = yd/(d+z) = y/(1+(z/d)
( ) (x) (x)
( ) (y) = (y)
( ) (z) (0)
( /d 1) (1) (z/d + 1)
(0, 0, -d)

8. One- Point Projections
(x, y, z)
Center of Projection at the origin
viewplane parallel to the x-y plane a distance d from the origin.
xprojected = dx/z = x/(z/d)
yprojected = dy/z = y/(z/d)
(xproj, yproj, d) y x
( ) (x) (x)
( ) (y) = (y)
( ) (z) (z)
( /d 0) (1) (z/d)
Mper
Points plotted are
x/w, y/w where w = z/d