Projection

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Projection

• Types of Projection• Simple Projections

• Generalized Projection

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Clip againstview volume

Projectonto projection

plane

Transform intoviewport in 2D

devicecoordinates

3D worldcoordinates

Clipped worldcoordinates

2D devicecoordinates

Conceptual model of the 3D viewing processConceptual model of the 3D viewing process

Viewing In 3D

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ProjectionsIn general, projectionsprojections transform points in a coordinate system of dimension n into points in a coordinate system of dimension less than n.

We shall limit ourselves to the projection from 3D to 2D.

• The projection is onto a plane rather than a curved surface

• The projectors are straight lines rather than curves

We will deal with planar geometric projectionsplanar geometric projections where:

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A

BA'

B'

Center ofprojection

Projectors

Projectionplane

The projectionprojection of a 3D object is defined by straight projection rays (called projectorsprojectors) emanating from a center of projectioncenter of projection, passing through each point of the object, and intersecting a projection planeprojection plane to form the projection.

Projections

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Planer Geometric Projections

Two basic classes (on the basis of the distance of the centercenter of projectionof projection from the projection planeprojection plane) :

A

BA'

B'

Center ofprojection

Projectors

Projectionplane

Perspective projectionPerspective projection

• perspective projectionperspective projection : the distance is finite

A

BA'

B'

Center ofprojectionat infinity

Projectors

Projectionplane

Directionof

projection

Parallel projectionParallel projection

• Parallel projectionParallel projection : the distance is infinite

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A

BA'

B'

Center ofprojection

Projectors

Projectionplane

C

DC'

D'

1. 1. Perspective foreshorteningPerspective foreshortening The farther an object is from COP the smaller it appears

Perspective foreshorteningPerspective foreshortening

Perspective Projection-Anomalies

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2. 2. Vanishing PointsVanishing Points:: Any set of parallel lines not parallel to view plane appear to meet at some point.


x

y

z

z-axis vanishing point

Vanishing pointVanishing point

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3. 3. View ConfusionView Confusion:: Objects behind the center of projection are projected upside down and backward onto the view-plane.


x

y

z

P1

P2P3

P1`P2`

P3`C

O

RAC/RA


4. 4. Topological distortionTopological distortion: : A line segment joining a point which lies in front of the viewer to a point in back of the viewer is projected to a broken line of infinite extent.

P1

P3

P'3

C

Y

X

Z

P2

P'2P'1

View Plane

Plane containingCenter of Projection (C)

RAC/RASubclasses of planar geometric projectionsSubclasses of planar geometric projections

Planner Geometric Projections

V = ±N

Planar geometricprojections

Parallel Perspective

Orthographic Oblique

Top

Front

Side

Axonometric

Isometric Other

Cavaliertan -1 (2)

Other

One-point

Two-point

Three-point

COPCOP = COP = d

N,V

N V

N=axis

Cavaliertan -1 (1)

N V

zx

yProjection Plane

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Two types (on the basis of the relation between the directiondirection of projection Vof projection V and the normal to the projection plane Nnormal to the projection plane N) :

• orthographicorthographic : VV and NN are the same or the reverse of each other.

• obliqueoblique : VV and NN are neither same nor reverse.

Parallel Projections

Y

X

Z

Orthographic

Oblique

N

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Projection plane(top view)

Projection plane(side view)Projection plane

(front view)

Orthographic parallel projectionsOrthographic parallel projections

Orthographic Projections

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Axonometric ProjectionsAxonometric Projections use projection planes that are not normal to a principal axis.On the basis of projection planeprojection plane normal N = (dnormal N = (dxx, d, dyy, d, dzz) subclasses are:

• IsometricIsometric : | ddx x | = | = | ddy y | = | = | ddz z || i.e. NN makes equal angles with all principal axes.

• othersothers : NN makes unequal angles with one or more principal axes.

Axonometric Projections

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Oblique Projections

: is the angle the projection makes with x-axis : angle between view plane and direction of projectionl : original length of a line perpendicular to view planel : projected length of a line perpendicular to view plane

z

y

x

l

l'

l'sin

l'cos

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Cavalier & Cabinet

x

z

y

x

z

y

= 30 = 45

1

1

1

1

1

1

x

z

y

x

z

y

= 30 = 45

11/2

1 1

11/2

Cavalier projection = 45l = l

Cabinet projection = 63.4l = l/2

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Projective Projection

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Perspective Projections

3-Vanishing Point

1-Vanishing Point

2-Vanishing Point

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y

x

z

view direction

center ofprojection

plane ofprojection

d

Settings for perspective projectionSettings for perspective projection

Projective Transformations

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y

P(y,z)y

zP'(y p ,z p)

d-z

plane ofprojection

1,,,1,,, ddz

ydz

xzyx

dzdz

xxdx

zx

dzyy

dy

zy

pp

pp


z

RAC/RA

1

/

/

1????????????????

ddz

ydz

x

zyx

dzzyx

zyx

1????????????????

dzzyx

zyx

d 10100010000100001

Matrices for Projective Trans.

RAC/RA

11000000000000

01

0,,

zyx

d

dd

dz

dydx

zpypxp

zdz

dxxdz

dyy pp


C(0,0,-d)

y

x

z xp

P(x,y,z)

P(xp,yp,0)

x

Alternative approach,

Projection plane at Z=0

And Center at

Z=-d

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10

11000000000100001

yx

zyx

y

x

zview direction

plane ofprojection

direction ofprojection

Orthogonal Projection Matrix

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Projection

• To studyFoley: 6.1, 6.1.1, 6.1.2Schaum: 7.1, 7.2, 7.3

• Problems: 7.3, 7.4, 7.5, 7.10, 7.11, 7.12

RAC/RA

Projection

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