Viewing Transformations

Viewing parameters

Viewing projections

- Parallel

- Parallel orthographic projection

- Parallel oblique projection

Viewing transformations

• Viewing transformations are concerned with projecting 3D objects onto 2D surfaces

View plane

Virtual camera

V

U

N

Xw

Yw Zw

Direction of gaze

View Reference Point(eye position)

Handedness axis

Viewingdistance

View up-direction

Viewing projections

• Parallel– centre of projection in the infinity– the projection lines are parallel

X

Y

Z

Projection plane

Projection lines

Parallel orthographic projection

• Projection lines perpendicular to the projection plane• Transformation matrix

xp = x

yp = yPpar =

1 0 0 0

0 1 0 00 0 0 00 0 0 1

Parallel oblique projection

• Projection lines parallel, but not perpendicular to the projection plane

X

Y

Z

Projection plane

Projection lines

Parallel oblique projection

Pobl =

1 0 0 0

0 1 0 0L cos L sin 0 0

0 0 0 1

Matrix

X

Y

Z

P( 0,0 1)

P'

L

L cos

L sin

Parallel projections

• preserve relative dimensions of the objects

• do not give realistic appearance

Perspective projections

• The projection lines meet in one or more projection centre(s)

X

Y

ZCOP(Centre of Projection)

COP at the centre of the viewing coordinate system

• projection plane is at distance D from the COP

• on the positive part of the z axis (ZCOP = 0)

z

y

D

COP

PP'

P = (x, y, z)

P' = (xp, yp, D )

yp

D = yz xp

D = xz

yp = D yz =

yz/D xp =

D xz =

xz/D

z

y

D

COP

PP'

Transformation matrix

Example point transformation

Pper =

1 0 0 0

0 1 0 00 0 1 00 0 1/D 0

yp = D yz =

yz/D xp =

D xz =

xz/D

Projection plane at the centre of the viewing coordinate system

• projection plane is at distance D from the COP• on the negative part of the z axis (ZCOP = -D)

z

y

D

COPP

P'

P = (x, y, z)

P' = (xp, yp, 0 )

z

y

D

COPP

P'

ypD =

yz+D

xpD =

xz+D

yp = D y

z + D = y

(z/D) + 1 xp = D x

z + D = x

(z/D) + 1

Transformation matrix

Pper =

1 0 0 0

0 1 0 00 0 0 00 0 1/D 1

Example point transformation

yp = D y

z + D = y

(z/D) + 1 xp = D x

z + D = x

(z/D) + 1

Perspective projections

• do not preserve relative dimensions• give realistic appearance

Clipping

• Used to restrict the depth of the projected scene

Back clipping plane

Front clipping plane

View plane