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CS475m - Computer Graphics

Lecture 6 : Viewing

CS475m: Lecture 6 Parag Chaudhuri

Perspective Projection

Image Planeor Projection Plane

Object

Projectors

Centre of Projection


Parallel Projection


Object

ProjectorsCentre of Projection?


Parallel ProjectionOrthographic Projection

Top View

Side View

Front View

● Multiviews (x=0, y=0 or z=0 or principal planes).

● True size or shape for lines.

● For projection on the z=0 plane we get the projection matrix as

P z=0=[1 0 0 00 1 0 00 0 0 00 0 0 1 ]


Parallel ProjectionAxonometric Projection

● Transform and then project using an orthographic projection such that at multiple adjacent faces are visible – better representation of a 3D object using 1 view. Face parallel to projection plane shows true shape and size.

● If be the matrix formed by stacking up the unit vectors along the three axes, and be the axonometric projection, then

T.U=T .[1 0 00 1 00 0 11 1 1

]=[xx ' x y ' x z 'yx ' y y ' y z '0 0 01 1 1

]T

U




T.U=T .[1 0 00 1 00 0 11 1 1

]=[x x ' x y ' x z 'y x ' y y ' y z '0 0 01 1 1

]T

U

● The foreshortening ratios for each projected principal axes are then given by:

f x= x x '2yx '2 f y= x y ' 2

y y ' 2 f z= xz ' 2yz ' 2




T.U =T .[1 0 00 1 00 0 11 1 1

]=[x x ' xy ' x z 'y x ' yy ' y z '0 0 01 1 1

]T

U

● The foreshortening ratios for each projected principal axes are then given by:

f x= x x '2y x '2 f y= x y '2

yy ' 2 f z= x z ' 2yz ' 2



● Depending on the kind of foreshortening they cause we can have three types of axonometric projections

— Trimetric (all foreshortenings are different)

f x≠ f y≠ f z





— Dimetric (two foreshortenings are the same)

f x≠ f y≠ f z f x= f z





— Dimetric (two foreshortenings are the same)

— Isometric (all foreshortenings are the same)

f x≠ f y≠ f z f x= f z f x= f y= f z



● Assuming we rotate by and before we do the projection on the z=0 plane.

T=P z=0 . R x . R y

=[1 0 0 00 1 0 00 0 0 00 0 0 1 ] . [

1 0 0 00 cos −sin 00 sin cos 00 0 0 1 ] . [

cos 0 sin 00 1 0 0−sin 0 cos 00 0 0 1 ]

=[cos 0 sin 0sin sin cos −sin cos 00 0 0 00 0 0 1 ]

Ry R x



● Now we apply this axonometric projection to

T.U=[cos 0 sin 0sin sin cos −sin cos 00 0 0 00 0 0 1

] .[1 0 00 1 00 0 11 1 1

]=[

cos 0 sin

sin sin cos −sin cos0 0 01 1 1

]

UT



● So the foreshortening ratios become

f x2=cos2

sin2sin2

f y2=cos2

f z2=sin2

sin2cos2

● For Isometric projections, if we solve for then we get and

f x= f y= f z=35.26 =±45


Parallel ProjectionOblique Projection

● The projectors are parallel to each other but they are not perpendicular to the plane of projection.

● Only planes parallel to plane of projection show true shape and size.



P x , y , z

P' x , y

P ' ' x p , y p

l

xp=xl cos

yp=yl sin

tan=zl

or l=z cot


● When we get a Cavalier projection. Lines perpendicular to the projection plane are not foreshortened.

● When we get Cabinet projections. Lines perpendicular to the projection plane are foreshortened by half.

● is typically 30° or 45°.


P x , y , z

P ' x , y

P' ' x p , y p

l

cot =1 /2

=45




Object

Projectors

Centre of Projection

● Projectors converge at a finite centre of projection.

● Parallel lines converge.

● We get nonuniform foreshortening.

● Shape is not preserved.

● We see in perspective – so perspective viewing seems natural and helps in depth perception.


A digression into art

13th century , Arezzo by Giotto



Early 15th century , The Little Garden of Paradise



15th century , The Baptistry in Florence, Filippo Brunelleschi



15th century , Fresco of Holy Trinity, Masaccio



15th century , School of Athens, Raphael



P x , y , z

P' x ' , y '

zc

y y 'l1

l2

y 'l2

=y

l 2−l1

z c

l2

=zc−z

l2−l1

⇒y '=y

1−zz c



P x , y , z

P ' x ' , y '

zc

x x '

l1l2

x 'l 2

=x

l2−l1

zc

l2

=z c−z

l2−l1

⇒ x '=x

1−zz c

z c

z



● First we apply a perspective transform to a point that takes it to

[x 'y 'z 'w ' ]=[

1 0 0 00 1 0 00 0 1 00 0 r 1

] . [xyz1 ]=[

xyz

rz1]x '=

xrz1

, y '=y

rz1, z '=

zrz1

X '=Pr . X

X X '


● If then we get


● First we apply a perspective transform to a point that takes it to

● Now we add projection on the plane.

[x 'y 'z 'w ' ]=[

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

]. [1 0 0 00 1 0 00 0 1 00 0 r 1

] .[xyz1 ]=[

xy0

rz1]x '=

xrz1

, y '=y

rz1, z '=0

X '=P z=0 . Pr . X

X X '

z=0

r=−1z c

x '=x

1−zzc

, y '=y

1−zzc


Perspective Projection● Vanishing point in the z

direction.

● Set of lines not parallel to the projection plane converge at a vanishing point.

VP z


● If then we get , i.e., the vanishing point lies an equal distance on the opposite side of the projection plane as the center of projection.


● To find the vanishing point along the z direction we apply the perspective transformation to the point at infinity along the z direction.

[x 'y 'z 'w ' ]=[

1 0 0 00 1 0 00 0 1 00 0 r 1

] .[0010]=[

001r] x '=0, y '=0, z '=

1r

X '=Pr . X

r=−1 / zc z '=−zc


● CoP is at , VP is at


● Centre of projection (CoP) on x axis

[x 'y 'z 'w ' ]=[

1 0 0 00 1 0 00 0 1 0p 0 0 1

] .[xyz1 ]=[

xyz

px1]x '=

xpx1

, y '=y

px1, z '=

zpx1

X '=P p . X

−1 / p ,0,0,1 1 / p , 0, 0,1

Single point perspective


● CoP is at , VP is at


● Centre of projection (CoP) on y axis

[x 'y 'z 'w ' ]=[

1 0 0 00 1 0 00 0 1 00 q 0 1

]. [xyz1 ]=[

xyz

qy1]x '=

xqy1

, y '=y

qy1, z '=

zqy1

X '=Pq . X

0,−1/q ,0,1 0,1 /q ,0,1

Single point perspective



P pq=P p . Pq=[1 0 0 00 1 0 00 0 1 0p 0 0 1

] .[1 0 0 00 1 0 00 0 1 00 q 0 1

]=[1 0 0 00 1 0 00 0 1 0p q 0 1

]Two point perspective

x '=x

pxqy1, y '=

ypxqy1

, z '=z

pxqy1

● Two vanishing points

● Two CoPs?

From http://gatewayhsart.wordpress.com CS475m: Lecture 6 Parag Chaudhuri


P pqr=P p . P q . Pr=[1 0 0 00 1 0 00 0 1 0p q r 1

]Three point perspective

x '=x

pxqyrz1, y '=

ypxqyrz1

● Three vanishing points

● Three CoPs?

z '=z

pxqyrz1

M. C. Esher, Ascending and Descending CS475m: Lecture 6 Parag Chaudhuri


T =Pr z=0 . T l , m , n=[1 0 0 00 1 0 00 0 0 00 0 r 1

] . [1 0 0 l0 1 0 m0 0 1 n0 0 0 1

]=[1 0 0 l0 1 0 m0 0 0 00 0 r rn1

]

Generation of perspective views

● Transform and then apply single point persepective.

● Let us try to translate, apply a perspective and project to z=0


Perspective ProjectionGeneration of perspective views

● Translation along x=y line.



● Translation along the z axis causes change in scaling.

z −z

y

−y CS475m: Lecture 6 Parag Chaudhuri


● Rotate about y axis and then apply single point persepective projection.

T=Pr z=0 . Ry =[1 0 0 00 1 0 00 0 0 00 0 r 1

] .[cos 0 sin 0

0 1 0 0−sin 0 cos 0

0 0 0 1]



● Rotate about y axis and then apply single point persepective projection.

T =Pr z=0 . Ry =[1 0 0 00 1 0 00 0 0 00 0 r 1

] .[cos 0 sin 0

0 1 0 0−sin 0 cos 0

0 0 0 1]

=[cos 0 sin 0

0 1 0 00 0 0 0

−r sin 0 r cos 1]

● We get a two point perspective.



● Rotate about y axis, x axis and then apply single point persepective projection.

T=Pr z=0 . R x . R y

=[1 0 0 00 1 0 00 0 0 00 0 r 1

] .[1 0 0 00 cos −sin 00 sin cos 00 0 0 1

] .[cos 0 sin 0

0 1 0 0−sin 0 cos 0

0 0 0 1]



● Rotate about x axis, y axis and then apply single point persepective projection.

T =Pr z=0 . R y , R x

=[cos 0 sin 0

sin sin cos −sin cos 00 0 0 0

−r cos sin r sin r cos cos 1]

● We get a three point perspective.


Taxonomy

Parallel Perspective

Orthographic ObliqueAxonometric

Trimetric

Dimetric

Isometric

Cavalier

Cabinet

Front

Top

Side

Planar Projections

One Point

Two Point

Three Point