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Three - Dimensional

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Three-Dimensional Gra hics

Use of a right-handed coordinate system(consistent with math)

-

.To transform from right to left, negate the

z values.

Ri ht Handed S ace Left Handed S ace

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Homogeneous representation of aoint in 3D s ace:

wzyx T P

point)3Dafor1,w(

represented by 4x4 matrices:P’ = A.P

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Transformation Matrix in 3D:

pcba

r i g A

snml where,

cbaproduces linear transformations:

i

e sca ng, s ear ng, re ec onand rotation.

K = [p q r] T, produces translation

= [l m n] T, yields perspective transformation

while, = s, is responsible for uniform scaling

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x 010

x t

000 z S 100 z t 10001000

ScaleTranslation

001 x Sh Shear

010 z

y

Sh 1000

by scale and shear

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3D Reflection:

The following matrices:0001 0001 0001

0100 XY T

0100YZ T

0100 ZX T

1000

1000

1000

produce reflection about:

plane plane plane

res ectivel .

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Rotation Matrices alon an axis:

0sin0cos 0010

0)sin()cos(0

1000 1000

X-axis Y-axis

00γsinγcos

0100γ

cosγ

s n Z-axis

Why is the sign reversed in one case ?

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Y Around-

X Around-

Z Around-

X ZY

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Z 0001 X

X-axis0)sin()cos(0

Y

Y1000

Y 00γsinγcos X

Z-axis 0100

00γcosγsin Z

X1000

X Around 0)sin(0)cos( X -ax s

0)cos(0)sin( Z

Z

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Rotation About an Arbitrary Axis in Space

Assume, we want to perform a rotation byegrees, a ou an ax s n space pass ng

through the point (x 0 , y 0 , z 0 ) with direction

cos nes c x , c y , c z . . ,

|T| = - (x0, y

0, z

0) T

. ,principle axes, let's pick, Z (|R x |, |R y |).

. z .

4. Then we undo the rotations to align the.

5. We undo the translation: translate by- 0 , - 0 , - 0

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in step (2), as given before.

This is going to take 2 rotations:

i About x-axis(to place the axis in the xz plane)

and

ii) About y-axisto p ace t e resu t co nc ent w t t e

z-axis).

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z First ste of Rotation:

c yd

c z

0

xc x

o a on a ou x y :

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Pro ect the unit vector alon OP intothe yz plane.

The y and z components, c y and c z , arethe direction cosines of the unit vector

a ong e ar rary ax s.It can be seen from the dia ram that :

22 C

z y c

d cos s n 22 cc d sin

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z

zAfter first step

c

d

cP c z

0

x, ,x

ydx

x

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Final Transformation matrix for 3D rotation,a out an ar trary ax s:

M = |T| |R x | |R | |R z| |R | -1 |R x | -1 |T| -1where: 0001

010 00

y ;00 d d R

y z

x

100 0 z 00 d d z y

0010 x

00cossin

00 d C x y 0100 z

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= x y z y - x - -

- x y z x y

= R -1

A special case of 3D rotation:

o a on a ou an ax s para e o acoordinate axis (say, parallel to X-axis):

M = T R T -1

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Rotation About an Arbitrary Axis in Space

Assume, we want to perform a rotation byegrees, a ou an ax s n space pass ng

through the point (x 0 , y 0 , z 0 ) with direction

cos nes c x , c y , c z . . ,

|T| = - (x 0 , y 0 , z 0 ) T . ,

principle axes, let's pick, Z (|R x |, |R y |). . z .

4. Then we undo the rotations to align the.

5. We undo the translation: translate by- 0 , - 0 , - 0

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z

zAfter first step

c

d

cP c z

0

x, ,x

ydx

x

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Final Transformation matrix for 3D rotation,a out an ar trary ax s:

M = |T| |R x | |R | |R z| |R | -1 |R x | -1 |T| -1where: 0001

010 00

y ;00 d d R

y z

x

100 0 z 00 d d z y

0010 x

00cossin

00 d C x y 0100 z

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you are g ven po n s ns ea on e

axis of rotation), you can calculate the directioncos nes o e ax s as o ows:

z z y y x x V T

101

c x 01

y

V.vector theof lenght theisV where

z

||

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Reflection throu h an arbitrar lane

about an arbitrary axis.

M = |T| |R x | |R y | |R fl| |R y | -1 |R x | -1 |T| -1

T does the job of translating the origin to

the lane.R x and R y will rotate the vector normal to

t e re ect on p ane at t e or g n , unt t scoincident with the +Z axis.

R fl is the reflection matrix about X-Y

.

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definition of objects. Also called

o e ng space.

W o r l d S p a c e :

where the scene and viewinspecification is made

E y e s p a c e (Normalized Viewing Space):

looking down the Z axis.

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A 3D Pro ective s ace.Dimensions: [-1:1] in X & Y, [0:1] in Z.

This is where image space hidden.

S c r e e n S p a c e ( 2 D ) :

Range of Coordinates - ,

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Classification of Geometric Projections

Planar geometricprojections

Parallel Perspective

Orthographic Oblique One-point Two-

pointTop Cabinet

Other Three-

Front Side Axonometric Cavalier pointe eva on e eva on

Isometric Other

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Distance from COP to projection

parallel & we specify a center of.

Center of Projection is also called

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The size of the perspectiveprojection of the object varies inverselywith the distance of the object from thecenter of projection.

The perspective projections of any

to the projection plane converge to avanishing point .

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Projectionplane

Center ofprojection

Plane normal

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Projection plane

x-axis z-axis

en er o ro ec on

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X or Y P(X,Y,Z)

PP

x p or y p

O (COP)

d d l

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Perspective Geometr and Camera Models

F P(X,Y,Z)X or Y

IP X or YP X,Y,Z

PP x or y

(COP) ZO

Perspective Geometry and Camera Models

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Perspective Geometry and Camera Models

X or Y P(X,Y,Z)

x p or y p X or Y P(X,Y,Z)

O (COP) PP

COP

x p or y pZO

f

Equations of Perspective geometry, next ->

i f

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E uations of ;

Zf ;

Zf

pp

Perspective geometry

;Yy ;Xx pp

M per

0f 100 0001

0010 MP´ = M .P

T,

Generalized formulation of

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Generalized formulation of perspective projection:

(COP)X or YPP

P´(x p , y p, Zp)Q

P(X,Y,Z)(d x, d y, d z)

(0, 0, Z p )

Parametric eqn. of the line L betweenan :

COP + t(P-COP); 0 < t < 1.

Let the direction vector from (0 0 Z ) to

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Let the direction vector from (0, 0, Z ) toCOP be (d x , d y , d z),

and Q be the distance from (0, 0, Z p ) to COP.

Then COP = (0, 0, Z p ) + Q(d x , d y , d z).

The coordinates of any point on line L is:´ -x x

Y´ = Qd y + (Y- Qd y)t;

´

Z´ = (Z p + Qd z) + (Z - (Z p + Qd z))t;

s ng e con on = p , a e n ersec onof line L and plane PP:

Qd t z

Now subsitute to

z p , p p .

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d Z

d Z X x x d d x z z

1 p

Z Z Y p

y z z p1

d

Generalized formula of perspective

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dz

d01 xp

x

dd yydd

z

p

z

ZZZ

00 ppgen

QdQd zz

100 pzz

Special cases from the generalized formulation

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Special cases from the generalized formulationof the perspective projection matrix

a r xType Zp Q [d x , d y , d z]

Morth 0 Infinity [0, 0, -1]

per , , -

M’per 0 d [0, 0, -1]

If Q is finite, M gen defines a one-point

.

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ara e ro ec on

Distance from COP to projectionlane is infinite.

Therefore, the projectors are parallel lines& we need to specify a:

direction of projection (DOP)Orthographic:

e rec on o pro ec on an enormal to the projection plane are the same.

projection plane).

Classification of Geometric Pro ections

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Classification of Geometric Pro ections

Planar geometricprojections

Parallel Perspective

Orthographic Oblique One-point Two-

pointTop Cabinet

Other Three-

Front Side Axonometric Cavalier pointe eva on e eva on

Isometric Other

Pro ection

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Pro ection

ProjectorsPlane(top view)

Projectors forforside view

op v ew

Planeside view

ProjectionPlane

ro ec ors orfront view

(front view)

xamp e o r ograp c ro ec on

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Example of Isometric Projection:Projection

planero ec or

ro ec on-plane normal

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use planes of projection that are nottherefore show multiple face of an

object.) sometr c pro ect on: pro ect on p ane

normal makes equal angles with eachprinciple axis. DOP Vector: [1 1 1].

All 3 axis are equally foreshortened

axes to be made with the same scale.

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direction of projection differ.

Principle axis

Projectors are not normal to theprojection plane

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plane y

Projector

zx

Projection-plane normal

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y

P´l

xz P 0 0 1

General obli ue ro ection of a oint/line:

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Projection Plane : x-y plane; P´ is thepro ect on o P , , 1 onto x-y p ane.

` ´vector onto x-y plane and is the angle

e pro ec on ma es w e x-ax s.

` ´ , vary.

Coordinates of P´: ( l c o s , l s i n , 0).As given in the figure: DOP is:

(d x , d y , -1) or ( l c o s , l s i n , -1).

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a s y

P´l

xz P 0 0 1

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v

uVPN

n COP PRP

-for perspective projection

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CW

VRP

VPNPRP

n

Infinite parallelopiped view volumefor parallel projection

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