3D-viewing-transform.PDF

7/27/2019 3D-viewing-transform.PDF

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3D Viewing

Projections

AProjectors

Projection PlaneCenter of

Projection

B

B

A

Perspective


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3D Viewing

Projections

Parallel

Projectors

Projection PlaneAt Infinity

A

B

A

B


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3D Viewing

Parallel Projections

Orthographic

Side ViewFront View

Top View

Z X

Y


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3D Viewing


Multiviews

(x=0 or y=0 or z=0 planes),

one View is not adequate

True size and shape for lines

On z=0 plane

Orthographic

=

1000

0000

0010

0001

P


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3D Viewing


Orthographic


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3D Viewing


Axonometric

Additional rotation,translation and then

projection on z=0 plane

[ ][ ] [ ]

=

=

10

10

10

1100

1010

1001

**

**

**

zz

yy

xx

yx

yx

yx

TTU

2*2*2*2*2*2* ;; zzzyyyxxx yxfyxfyxf +=+=+=


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3D Viewing


Three types Trimetric: No foreshortening is the same.

Dimetric: Two foreshortenings are the same.

Isometric: All foreshortenings are the same.

Axonometric


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3D Viewing


Trimetric

zyx fff Dimetric

zy ff =Isometric

zyx fff ==

Axonometric


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3D Viewing


Isometric

Let there be 2 rotations a) about y-axis b) about x-axis

=

1000

0000

0010

0001

1000

0cossin0

0sincos0

0001

1000

0cos0sin

0010

0sin0cos

T


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3D Viewing


Isometric

Let there be 2 rotations a) about y-axis b) about x-axis

=

100000sincossin

00cos0

00sinsincos

T


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3D Viewing


[ ][ ]

=

=

10sincossin

10cos0

10sinsincos

1000

00sincossin

00cos0

00sinsincos

1100

10101001

TU

Isometric


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3D Viewing


sincossin

cos

sinsincos

2222*2*2

22*2*2

2222*2*2

yxf

yxf

yxf

zzz

yyy

xxx

+=+=

=+=

+=+=

Isometric


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3D Viewing


ff

ffffff

zy

yx

zyx

2222

2222

cossincossin

cossinsincos

=+=

=+====

Solving equations find , and f

Isometric


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3D Viewing


Oblique

Non-perpendicular projectorsto the plane of projection

True shape and size for the

faces parallel to the projection

plane is preserved


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3D Viewing


Oblique


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3D Viewing


Oblique

P(x,y,z)

P(x,y)

P(xp,yp)

z

x

y

xp = x + L cos yp = y + L sin

tan =z/L or L = z cotL


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3D Viewing


Oblique

P(x,y,z)

P(x,y)

P(xp,yp)

z

x

y

L

When =45o => CavalierLines perpendicular to the

projection plane are notforeshortened

When cot = => CabinetLines perpendicular to the

projection plane are

foreshortened by half

is typically 300 or 450


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3D Viewing

Perspective Projections

Parallel lines converge

Non-uniform

foreshortening

Helps in depth

perception, important

for 3D viewing

Shape is not preserved

AProjectors

Projection PlaneCenter ofProjection

B

B

A


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3D Viewing



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3D Viewing


Matrix Form

[ ] [ ]

[ ] +++=

+=

11111

1

1000

100

00100001

1

***rz

z

rz

y

rz

xzyx

rzzyxr

zyx


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3D Viewing


Matrix Form

Projection on z=0 plane

[ ][ ]

[ ]

++=

===

1011

11000

0000

0010

0001

1000

100

0010

0001

***

rz

y

rz

xzyx

rPPPT zrrz


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3D Viewing


Geometrically

Z

P(x,y,z) X

Y

P*(x*,y*)

zc

y

l2

l1 y*

c

c

c

z

z

yy

ll

l

zz

z

ll

y

l

y

=

=

=

1

,

*

12

2

122

*


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3D Viewing


Geometrically

Z

P(x,y,z) X

Y

P*(x*,y*)

zc

l2

l1xx*

zc

c

cc

zz

xx

zz

x

z

x

=

=

1

**

When r = - 1/ zc this becomes same

as obtained in matrix form


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3D Viewing


Vanishing Point

Set of parallel lines not parallel to the projection plane

converge to Vanishing Point

VPzY

Z

X


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3D Viewing


Vanishing Point

Point at infinity on +Z axis :

(homogenous)[ ]0100

[ ] [ ]

[ ][ ] [ ]11001

100

1000

100

0010

0001

0100

***

'''

rzyx

r

rwzyx

==

=

Recall r = -1/zc, the vanishing point is at zc


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3D Viewing


Single Point Perspective

[ ] [ ]

[ ]

+++=

+=

11111

1

1000

0100

0010001

1

***

px

z

px

y

px

x

zyx

pxzyx

p

zyx

COP on X-axis

COP (-1/p 0 0 1) VPx (1/p 0 0 1)


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3D Viewing


Single Point Perspective

[ ] [ ]

[ ]

+++=

+=

11111

1

1000

0100

0100001

1

***

qy

z

qy

y

qy

x

zyx

qyzyxq

zyx

COP on Y-axis

COP (0 -1/q 0 1) VPx (0 1/q 0 1)


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3D Viewing


Two Point Perspective

[ ][ ]

==

1000

0100

010

001

q

p

PPP qppq


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3D Viewing


Three Point Perspective

[ ]

=

=

1000

100

010

001PP qp

r

q

pPP rpqr


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3D Viewing



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3D Viewing

Generation of Perspective Views

Additional transformation and then single point

perspective transformation

Simple Translation:Translation (l,m,n),COP=zc ,Projection plane z=0

+

=

=

rnmlrr

nml

T

10000

0010

0001

1000000

0010

0001

10100

0010

0001


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3D Viewing


X

Y

Translation along y=x line:


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3D Viewing


Translation in Z => Scaling

COP

Projection plane


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3D Viewing


Rotation

[ ][ ]

==

1000

000

0010

0001

1000

0cos0sin

0010

0sin0cos

r

PRT rzy

Rotation about Y-axis by


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3D Viewing


Rotation

Rotation about Y-axis by

[ ][ ]

==

1000

cos00sin

0010

sin00cos

r

r

PRT rzy

=> Two Point Perspective Transformation


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3D Viewing


Rotation

Two Rotations a) about Y-axis by

b) about X-axis by[ ][ ]

=

=

1000

000

0010

0001

1000

0cossin0

0sincos0

0001

1000

0cos0sin

0010

0sin0cos

r

PRRT rzxy


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3D Viewing


=

1000

coscos0sincossin

sin0cos0

cossin0sinsincos

r

r

r

T

Two Rotations a) about Y-axis by

b) about X-axis by

Rotation

=> Three Point Perspective Transformation

3D-viewing-transform.PDF

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