Projection-Computer Graphics

7/27/2019 Projection-Computer Graphics

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Projection


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Projection


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Projection

Conceptual model of 3D viewing process


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In general, project ionstransform points in a coordinatesystem 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.

We will deal with planar geometr ic p ro ject ionswhere:

The projection is onto a plane rather than a curved surface

The projectors are straight lines rather than curves

Projection


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key terms Projection from 3D to 2D is defined by straightprojection rays

(pro jectors) emanating from the 'center of project ion', passing

through each point of the object, and intersecting the 'project ion

plane' to form a projection.

Projection


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Planer Geometric Projection

2 types of projections perspective andparallel.

Key factor is the center of p roject ion.

if distance to center of projection is finite : perspective

if infinite : parallel

A

B

A'

B'

Center of

projection

Projectors

Projection

plane

Perspective projection

A

BA'

B'

Center of

projection

at infinity

Projectors

Projection

plane

Direction

of

projection

Parallel projection


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Perspective v Parallel

Perspective: visual effect is similar to human

visual system...

has 'perspective foreshortening'

size of object varies inversely withdistance from the center of

projection.

Parallel lines do not in general

project to parallel lines

angles only remain intact for

faces parallel to projection plane.


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Parallel: less realistic view because of no

foreshortening

however, parallel lines remain

parallel. angles only remain intact for faces

parallel to projection plane.

Perspective v Parallel


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Perspective projection- anomalies

Perspective foreshorteningThe farther an object isfrom COP the smaller it appears

A

BA'

B'

Center of

projection

Projectors

Projection

plane

C

D

C'

D'

Perspective foreshortening


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Vanishing Points:Any set of parallel lines not parallel to the viewplane appear to meet at some point.

There are an infinite number of these, 1 for each of the infinite

amount of directions line can be oriented

x

y

z

z-axis vanishing point

Vanishing point



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View Confusion :Objects behind the center of projection areprojected upside down and backward onto the view-plane

Topologica l distor t ion: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'1P'2

View Plane

Plane containing

Center of Projection (C)


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Vanishing Point

Vanishing Point

COP

View Plane


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Vanishing Point

If a set of lines are parallel to one of the three axes, the vanishing point is called an axis vanishing

point (Principal Vanishing Point).

There are at most 3 such points, corresponding to the number of axes cut by the projection

plane

One-point:

One principle axis cut by projection plane

One axis vanishing point

Two-point:

Two principle axes cut by projection plane

Two axis vanishing points

Three-point:

Three principle axes cut by projection plane

Three axis vanishing points


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One point perspective projection of a cube X and Y parallel lines do not converge

Vanishing Point


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Vanishing Point


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Two-point perspective

projection:

This is often used inarchitectural, engineering and

industrial design drawings.

Three-point is used less

frequently as it adds little extrarealism to that offered by two-

point perspective projection.

Vanishing Point


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Vanishing Point

VPL VPRH

VP1 VP2

VP3


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Projective Transformation

y

x

z

view direction

center of

projection

plane of

projection

d

Settings for perspective projection


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y

P(y,z)y

z

P'(yp,z

p)

d-z

plane ofprojection

1,,,1,,, ddz

y

dz

xzyx

dz

dz

yy

d

y

z

yp

p



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1

/

/

1????

????

????

????

ddz

ydz

x

z

y

x

d

zz

y

x

z

y

x

1????

????

????

????

d

zz

yx

z

y

x

d10

100

0100

0010

0001

1

d

dz

ydz

x

d

zz

yx

divisioneperspectiv



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Parallel projection

2 principle types: orthographicand

oblique.

Orthographic : direction of projection =

normal to the projectionplane.

Oblique : direction of projection !=

normal to the projectionplane.

n

n


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Orthographic (or orthogonal) projections:

front elevation, top-elevation and side-elevation.

all have projection plane perpendicular to a principleaxes.

Useful because angle and distance measurements canbe made...

However, As only one face of an object is shown, it can

be hard to create a mental image of the object, evenwhen several view are available

Orthographic projection


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Orthographic projection


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Orthogonal Projection Matrix

1

0

11000

0000

0010

0001

y

x

z

y

x

y

x

zview direction

plane of

projection

direction of

projection


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Axonometric projection

Axonometr ic Project ionsuse projection planes that arenot normal to a principal axis.On the basis ofprojection

planenormalN = (dx , dy, dz) subclasses are:

o Isometr ic: | dx | =| dy | =| dz |i.e. Nmakes equal angles withall principal axes.

o Dimetr ic: | dx | =| dy |

o Trimetr ic:| dx | != | dy | != | dz |


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Axonometric vs Perspective

Axonometric projection shows several faces of an objectat once like perspective projection.

But the foreshortening is uniform rather than being

related to the distance from the COP.

y

z

x

Projection Plane

Isometric proj


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Oblique parallel projection

Oblique parallel projections Objects can be visualized better then with

orthographic projections

Can measure distances, but not angles* Can only measure angles for faces of objectsparallel to the plane

2 common oblique parallel projections:

Cavalierand Cabinet


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n

Projection Plane Normal

Projector

Projection Plane

x

y

z


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Cavalier:

The direction of the projection makes a 45 degree

angle with the projection plane.

There is no foreshortening



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Cabinet:

The direction of the projection makes a 63.4 degree

angle with the projection plane. This results in

foreshortening of the z axis, and provides a more

realistic view


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Cavalier, cabinet and orthogonal projectionscan all be specified in terms of(, )or(, )since

tan() = 1/

P=(0, 0, 1)

P ( cos(), sin(),0)

cos()

sin()


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l=1 b = 45 Cavalier projection a = 0 - 360

l=0.5 b = 63.4 Cabinet projection a = 0 360

l=0 b = 90 Orthogonal projection a = 0 360


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PP = ( cos(), sin(),-1) = DOP

Proj(P) = ( cos(), sin(),0)

Generally

multiply by z and allow for (non-zero) x and yx = x + z lcos a

y = y + z lsin a

1

.

1000

0000

0sin100cos01

1

0 z

yx

yx

alal

x

y

a),( yx

l

),( pp yx

al

al

sin

cos

yy

xx

p

p


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Generalized Projection Matrix

x or y

Center of Projection

(COP)

z

P = (x, y, z)

Pp

= (xp, y

p, z

p)

(dx, d

y, d

z)

(0, 0, zp)

Q

Plane of Projection

zyxp

p

dddQzCOP

tCOPPtCOPP

,,,0,0

10,

zpzp

yy

xx

QdzztQdzz

QdytQdy

QdxtQdx

zyxP

,,


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zpzpp

zpzpp

QdzzQdzzt

QdzztQdzz

11

1

1

1

2

z

p

z

zpp

z

p

z

p

z

p

pp

z

p

z

yp

z

y

p

z

p

z

xp

z

x

p

Qd

zz

Qd

Qdzz

Qd

zz

Qd

zz

Qd

zz

zz

Qd

zz

ddz

ddzy

y

Qd

zz

ddz

ddzx

x


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11

00

00

10

01

2

z

p

z

p

z

p

z

p

z

yp

z

y

z

xp

z

x

gen

Qd

z

Qd

zQd

z

Qd

z

d

dz

d

d

d

dz

d

d

M

1

1

1

2

z

p

p

z

p

z

p

p

z

p

z

yp

z

y

p

z

p

z

xp

z

x

p

Qd

zz

zQd

z

Qd

zz

z

Qd

zz

d

d

zd

d

zyy

Qd

zz

ddz

ddzx

x


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01

00

0100

0010

0001

1,0,0,,

d

Mperddd

dQdz

zyx

p

11

00

00

10

01

2

z

p

z

p

z

p

z

p

z

y

p

z

y

z

xp

z

x

gen

Qd

z

Qd

zQdz

Qdz

d

dz

d

dddz

dd

M


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1000

0000

00100001

1,0,0,,0

par

dddQz

Mzyxp

11

00

00

10

01

2

z

p

z

p

z

p

z

p

z

y

p

z

y

z

xp

z

x

gen

Qd

z

Qd

zQdz

Qdz

d

dz

d

dddz

dd

M


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Taxonomy of Projection


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OpenGLs Perspective Specification

w

h

y

x

z

near

far

aspect = w / h

y field-of-view / fovyaspect ratio

near and far clipping planesviewing frustum

gluPerspective(fovy, aspect, near, far)

glFrustum(left, right, bottom, top, near, far)

P i i h D h


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Perspective without Depth

dzz

yx

z

y

x

d 101

00

0100

00100001

1

d

dz

y

dz

x

d

zz

y

x

divisioneperspectiv

The depth information is

lost as the last two

components are same But dept information of the

projected points is essential

for hidden surface removal

and other purposes like

blending, shading etc.

P ti ith t D th


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Perspective without Depth

1

101

0000

0010

0001

z

zd dz

y

dz

x

d

zz

y

x

z

y

x

d

divisioneperspectiv

bababa

zddz ba

For < 0, z is a monotonically increasing function of depth.


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C i l Vi V l


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Canonical View Volume

There is a reversal of the z- coordinates, in the sense that before

the transformation, points further from the viewer have smaller z-

coordinates

y

z

z=-far

z=-near

+1

-1+1 -1

y

zviewer

z=-1 (near)z=1 (far)

P ti M t i


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Perspective Matrix

The matrix to performperspective transformation:

1

)/()(

)/(

)/(

10100

00

000

000

zz

zy

zx

z

z

y

x

z

y

x

ba

ba

ba

P ti M t i


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Perspective Matrix

w

h/2

y

xz

a = w / hz = -near

z = -far

zc

z

c

azzyx

c

azyax

y

x

h

wa

c

zy

y

zc

,,,,

2cot

P ti M t i


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Perspective Matrix

1,1,1,,

1,1,1,,

1,1,1,,

1,1,1,,

nc

n

c

an

nc

n

c

an

nc

n

c

an

n

c

n

c

an

1,1,1,,

1,1,1,,

1,1,1,,

1,1,1,,

fc

f

c

af

fc

f

c

af

fc

f

c

af

f

c

f

c

af

w

h/2

y

xz

a = w / h z = -near

z = -far

P ti M t i


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Perspective Matrix

1

1

1

1

n

nc

nc

an

ba

nn

c

a

c

ba

11

1

1

10100

00

000

000

1,1,1,,

nc

nc

an

ncn

can

ba

11

1

1

1n

nc

c

a

ba

P ti M t i


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Perspective Matrix

11

1

1

10100

00

000

000

1,1,1,,

fc

fc

af

fc

f

c

af

ba

1

1

1

1

1

f

fc

c

a

ba

1

1

1

1

f

fc

fc

af

ba

fn

fn

fn

nf

nn

ff

2b

a

ba

ba

P ti M t i


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Perspective Matrix

The matrix to performperspective transformation:

0100

2

00

000000

fn

fn

fn

nfc

a

c

Ta onom of projection


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Taxonomy of projection

Generalized Projection


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Using the origin as the center of projection, derive the perspective

transformation onto the plane passing through the point R0(x0, y0, z0)

and having the normal vector N = n1I + n2J + n3K.

x

y

z

P(x, y, z)

P'(x', y', z')

N = n1I + n2J + n3K

R0=x0 ,y0, z0

O



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znynxn

d

321

0

a

x

y

z

P(x, y, z)

P'(x', y', z')

N = n1I + n2J + n3K

R0=x0 ,y0, z0

O

P'O = PO

x' = x, y' = y, z ' = z

N. R0P' = 0

n1x ' + n2y ' + n3z '

=n1x0 + n2y0 + n3z0 = d0

0

000

000

000

321

0

0

0

nnn

d

d

d

Per



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Derive the general perspective transformation onto a plane with reference

point R0 and normal vector N and using C(a,b,c) as the center of projection.

x

y

z

P(x, y, z)

P'(x', y', z')

N = n1I + n2J + n3K

R0=x0 ,y0, z0

C



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)()()( 321 cznbynaxnd

a

P'C = PCx' = (x-a) + a

n1x' + n2y' + n3z = d0

d = (n1x0 + n2y0 + n3z0) (n1a + n2b + n3c)

= d0 d

1

x

y

z

P(x, y,

P'(x', y', z')

N = n1I + n2J + n3K

R0=x0 ,y0, z0

C



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Follow the steps Translate so that C lies at the origin

Per

Translate back

1321

0321

0321

0321

dnnn

cdcndcncn

bdbnbndbn

adananand



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Find (a) the vanishing points for a given perspectivetransformation in the direction given by a vector U (b)principal vanishing point.

Family of parallel lines having the direction U(u1,u2,u3)

can be written in parametric form as x = u1t+p, y = u2t+q, z = u3t+r

here (p, q, r) is any point on the line

Let, proj(x,y,z,1) = (x, y, z, h) x'= (d+an1)(u1t+p) + an2(u2t+q) + an3(u3t+r) ad0 y' = bn1(u1t+p) + (d+bn2)(u2t+q) + bn3(u3t+r) bd0 z' = cn1(u1t+p) + cn2(u2t+q) + (d+cn3)(u3t+r) cd0 h = n1(u1t+p) + n2(u2t+q) + n3(u3t+r) d1



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The vanishing point (xv,y

v, z

v) is obtained when t=

xu = (x/h) at t=

= a + (du1/k)

yu = b + (du2/k)

zu = c + (du3/k)

k = N.U = n1u1 + n2u2 + n3u3

If k=0 then ?

Principal vanishing point when

U = I xu = a + d / n1, yu = b, zu = c,

U = J

U = k

Ref


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Ref.

FV: p. 229-237, 253-258

Sch: prob. 7.1 7.15

Perspective Proj.pdf

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Projection-Computer Graphics