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Viewing Transformations

Viewing parametersViewing projections- Parallel- Parallel orthographic projection

- Parallel oblique projection

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Viewing transformations Viewing transformations are concerned with

projecting 3D objects onto 2D surfaces

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View plane

Virtual camera

V

U

N

Xw

Yw Zw

Direction of gaze

View Reference Point(eye position)

Handedness axis

Viewingdistance

View up-direction

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Viewing projections

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

X

Y

Z

Projection plane

Projection lines

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Parallel orthographic projection

Projection lines perpendicular to the projection plane Transformation matrix

xp = xyp = y

Pp a r =

1 0 0 00 1 0 00 0 0 00 0 0 1

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Parallel oblique projection

Projection lines parallel, but not perpendicular to theprojection plane

X

Y

Z

Projection planeProjection lines

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Parallel oblique projection

Pob l =

1 0 0 00 1 0 0

L c os L sin 0 00 0 0 1

Matrix

X

Y

Z

P( 0,0 1)

P'

L

L cos

L sin

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Parallel projections

preserve relative dimensions of the objects

do not give realistic appearance

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Perspective projections

The projection lines meet in one or more projectioncentre(s)

X

Y

Z COP(Centre of Projection)

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COP at the centre of the viewing coordinatesystem

projection plane is at distance D from the COP on the positive part of the z axis (Z COP = 0)

z

y

D

COP

PP'

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P = (x, y, z)P' = (xp, yp, D )

ypD =

yz

xpD =

xz

yp =D y

z =y

z/ D xp =D x

z =x

z/ D

z

y

D

COP

PP'

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Projection plane at the centre of the viewingcoordinate system

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

z

y

D

COP P

P'

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P = (x, y, z)P' = (xp, yp, 0 )

z

y

D

COP P 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

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Transformation matrix

P per =

1 0 0 00 1 0 0

0 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

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Perspective projections

do not preserve relative dimensions give realistic appearance

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Clipping

Used to restrict the depth of the projected scene

Back clipping

plane

Front clipping

plane

View plane