©Larry F. Hodges (modified by Amos Johnson) 1 Basic Projections 2D to 3D.

©Larry F. Hodges(modified by Amos Johnson)

1

Basic Basic ProjectionsProjections

2D to 3D2D 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.

(1 0 0 0) (x) (x)

(0 1 0 0) (y) = (y)

(0 0 0 0) (z) (0)

(0 0 1/d 1) (1) (z/d + 1)

(0, 0, -d)

xprojected = xd/(d+z) = x/(1+(z/d))

yprojected = yd/(d+z) = y/(1+(z/d)

-Z

+Z

(x, y, z)

(xproj, yproj, 0)

x

y


8

One- Point Projections

-Z

+Z

(0, 0, 0)

• 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)

(1 0 0 0) (x) (x)

(0 1 0 0) (y) =(y)

(0 0 1 0) (z) (z)

(0 0 1/d 0) (1)(z/d)

Mper

Points plotted are

x/w, y/w where w = z/d

(x, y, z)

(xproj, yproj, d)

x

y

©Larry F. Hodges (modified by Amos Johnson) 1 Basic Projections 2D to 3D.

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