Lecture4 Transformation CoordSys

7/27/2019 Lecture4 Transformation CoordSys

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Computer Graphics

CS630

Lecture 3 Transformation And

Coordinate Systems


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What is a

Transformation? Maps points (x, y) in one coordinatesystem to points (x', y') in another

coordinate system

x' = ax + by + c

y' = dx + ey + f


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Transformations Simple transformation Translation

Rotation

Scaling


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Transformations Deformable transformations Shearing Tapering

Twisting Etc..

Issues Can be combined Are these operations invertible?


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Transformations Why use transformations? Position objects in a scene (modeling)

Change the shape of objects Create multiple copies of objects

Projection for virtual cameras

Animations


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Classes of

Transformation Rigid-Body/ Euclidean transformation Similarity Transforms

Linear Transforms Affine Transforms

Projective Transforms


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Rigid-Body / Euclidean

Transforms Preserves distances

Preserves angles

TranslationRotation

Rigid / Eucl idean

Identity


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How are Transforms

Represented?x' = ax + by + cy' = dx + ey + f

x'

y'

a b

d e

c

f=

x

y+

p' = Mp + t


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Translation

z

y

x

z

y

x

t

t

t

= +

z

y

x

tttT zyx ),,(

z

y

x

z

y

x

t

t

t

= +

'

'

'

z

y

x


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Properties of Translationv=v)0,0,0(T

=v),,(),,( zyxzyx tttTsssT

=

=v),,(1

zyx tttT

v),,(),,( zyxzyx tttTsssT v),,(),,( zyxzyx sssTtttT

v),,( zzyyxx tststsT

v),,( zyx tttT


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Scaling

zs

ys

xs

z

y

x

z

y

x

'

'

'

z

y

x

zyx

s

s

s

sssS

00

00

00

),,(

Uniform scaling i f f zyx sss


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Rotations (2D)

sin

cos

ry

rx

cos)sin(sin)cos('

sin)sin(cos)cos('

rry

rrx

)sin('

)cos('

ry

rx

cossinsincos)sin(

sinsincoscos)cos(

cossin'sincos'

yxyyxx

yx,

',' yx

x

y


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Rotations 2D So in matrix notation

y

x

y

x

cossin

sincos'

'


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Rotations (3D)

100

0cossin

0sincos

)(

cos0sin

010

sin0cos

)(

cossin0

sincos0

001

)(

z

y

x

R

R

R


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Properties of RotationsIRa )0(

)()()()( aaaa RRRR

)()()( aaa RRR

)()()(1 Taaa RRR

)()()()( abba RRRR order matters!


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Combining Translation &

Rotation

)1,1(T

)45( R

)45( R

)1,1(T


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Combining Translation &

RotationT vv'

RTR

TR

R

vv

vv

vv

''

)(''

'''

vv R'

TR

T

vv

vv

''

'''


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Homogeneous Coordinates

Add an extra dimension in 2D, we use 3 x 3 matrices

In 3D, we use 4 x 4 matrices

Each point has an extra value, w

x'

y'

z'w'

=

x

y

zw

a

e

im

b

f

jn

c

g

ko

d

h

lp

p' = Mp


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Homogeneous Coordinates

Most of the time w = 1, and we canignore it

x'

y'

z'

1

=

x

y

z

1

a

e

i

0

b

f

j

0

c

g

k

0

d

h

l

1


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Homogeneous

Coordinates

z

y

x

w

Z

Y

X

can be represented as

wherewZz

wYy

wXx ,,


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Translation Revisited

11000

100010

001

),,(zy

x

tt

t

z

y

x

tttTz

y

x

zyx


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Rotation & Scaling

Revisited

11000

000000

000

),,(zy

x

ss

s

z

y

x

sssSz

y

x

zyx

110000cossin0

0sincos0

0001

)(

z

y

x

z

y

x

Rx


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Combining

Transformations

vvvvvv

vvv

vv

MTRSTRT

RSR

S

'''''''''

'''

'

where TRSM


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Transforming Tangents

t

qp

qp

qpt

qpt

M

M

MM

)(

'''


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Transforming Normals

nnn

nnnn

tntn

tn

tn

tn

TT

T

TT

TT

T

T

T

MM

MM

M

M

11'

''

'

0'

0''

0


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Surface Normal Surface Normal: unit vector that is locallyperpendicular to the surface


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Why is the Normal

important? It's used for shading makes things look 3D!

object color only Diffuse Shading


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Visualization of Surface

Normal

x= Red y= Green z= Blue


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How do we transform

normals?

Object SpaceWorld Space

nOS

nWS


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Transform the Normallike the Ray?

translation?

rotation?

isotropic scale? scale?

reflection?

shear?

perspective?


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More Normal

Visualizations

Incorrect Normal Transformation Correct Normal Transformation


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Transforming Normals

nnn

nnnn

tntn

tn

tn

tn

TT

T

TT

TT

T

T

T

MM

MM

M

M

11'

''

'

0'

0''

0


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Rotations about an

arbitrary axisRotate by around a unit axis r

r


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An Alternative View We can view the rotation around anarbitrary axis as a set of simpler

steps We know how to rotate and translatearound the world coordinate system

Can we use this knowledge toperform the rotation?


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Rotation about an

arbitrary axis Translate the space so that the origin ofthe unit vector is on the world origin

Rotate such that the extremity of the

vector now lies in the xz plane (x-axisrotation)

Rotate such that the point lies in the z-axis (y-axis rotation)

Perform the rotation around the z-axis Undo the previous transformations


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Rotation about an

arbitrary axis Step 1Rotate x-axis

x

y

z

(a,b,c)

x

(a,b,c)


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Closer Look at Y-Z Plane Need to rotate degrees around thex-axis

y

z


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Equations for

||)1,0,0(||||),,0(||

)1,0,0(),,0()cos(

||),,0(||||)1,0,0(||

||),,0()1,0,0(||)sin(

cb

cb

cb

cb


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Rotation about the Y-

axis Using the same analysis as before, weneed to rotate degrees around the

Y-axis y

z

x

(a,b,c)=Rx () (a,b,c)T


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Equations for

||)1,0,0(||||)',','(||

)1,0,0()',','(

)cos(

||)',','(||||)1,0,0(||

||)',','()1,0,0(||)sin(

cba

cba

cba

cba


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Rotation about the Z-

axis Now, it is aligned with the Z-axis,thus we can simply rotate degrees

around the Z-axis. Then undo all the transformations wejust did


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Equation summary

c

ba

TRRRRRT

c

ba

rot xyzyxaxis )()()()()(

'

''

)( 111


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DeformationsTransformations that do not preserve

shape

Non-uniform scaling Shearing

Tapering

Twisting Bending


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Shearing

11000

01

01

01

1

'

'

'

z

y

x

ss

ss

ss

z

y

x

zyzx

yzyx

xzxy

0

0

0

1

zyzx

yzyx

xz

xy

ss

ss

s

s

x

y

x

y


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Tapering

11000

0)(00

00)(0

0001

1

'

'

'

z

y

x

xf

xf

z

y

x


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Twisting

11000

0))(cos(0))(sin(

0010

0))(sin(0))(cos(

1

'

'

'

z

y

x

yy

yy

z

y

x


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Bending

11000

0)()(0

0)()(0

0001

1

'

'

'

z

y

x

ykyh

ygyf

z

y

x


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Quick Recap Computer Graphics is using acomputer to generate an image from

a representation.Model Image

computer


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Modeling What we have been studying so far isthe mathematics behind the creation

and manipulation of the 3Drepresentation of the object.

Model Image

computer


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What have we seen so

far? Basic representations (point, vector) Basic operations on points and

vectors (dot product, cross products,etc.)

Transformation manipulative

operators on the basicrepresentation (translate, rotate,deformations) 4x4 matrices toencode all these.


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Why do we need this? In order to generate a picture from amodel, we need to be able to not only

specify a model (representation) butalso manipulate the model in order tocreate more interesting images.


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Overview The next set of slides will deal withthe other half of the process.

From a model, how do we generate animage

Model Imagecomputer


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Scene Description

Scene

LightsCamera ObjectsMaterialsBackground


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Graphics Pipeline


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Graphics Pipeline Modeling transforms orientthe models within a

common coordinate frame

(world space)


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Graphics Pipeline


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Graphics Pipeline Maps world space to eye space Viewing position is transformed

to origin & direction is oriented

along some axis (usually z)


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Graphics Pipeline Transform to Normalized DeviceCoordinates (NDC)

Portions of the object outside the

view volume (view frustum) areremoved

G hi


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Graphics

Pipeline The objects areprojected to the2D image place

(screen space)


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Graphics Pipeline

G hi


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Graphics

Pipeline Z-buffer - Eachpixel remembersthe closest object

(depth buffer)


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Graphics Pipeline Almost every step in the graphicspipeline involves a change of

coordinate system. Transformationsare central to understanding 3Dcomputer graphics.


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IntuitivelyObjectSpace

WorldSpace

Camera

Space+Projection

+NDC

Rasterization


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Coordinate Systems Object coordinates World coordinates

Camera coordinates Normalized device coordinates

Window coordinates


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Object CoordinatesConvenient place to model the object

O


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World CoordinatesCommon coordinates

for the scene

O

O

W

TSRMwo

Positioning Synthetic Camera


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What are our degrees of freedom in camerapositioning?To achieve effective visual simulation, we want:1) the eye point to be in proximity of modeled scene2) the view to be directed toward region of interest,and3) the image plane to have a reasonable twist

Eye Coordinates


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Eye Coordinates

Eyepoint at originu axis toward right of image planevaxis toward top of image planeview direction along negativen axis

Transformation to Eye


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Transformation to EyeCoordinates

Our task: construct the transformation Mthat re-expresses world coordinates in theviewer frame

Machinery: Changing


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Machinery: ChangingOrthobases

Suppose you are given an

orthobasis u, v, nWhat is the action of thematrix M withrows u, v, and n as below?

A l i M t


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Applying M to u, v, n

Two equally valid interpretations, depending onreference frame:1: Think of uvn basis as a rigid object in a fixedworld spaceThen M rotates uvn basis into xyz basis

2: Think of a fixedaxis triad, with labels fromxyz spaceThen M reexpresses an xyz point p in uvncoords!It is this second interpretation that we use todayto relabel world-space geometry with eye space

coordinates

P iti i S th ti C


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Given eyepoint e, basis u, v, nDeduce M that expresses world in eye coordinates:Overlay origins, then change bases:

P i i i S h i C


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Check: does M re-express world geometry in eye coordinates?

P iti i S th ti C


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Camera specification must include:World-space eye position eWorld-space lookat direction -n

Are e and -n enough to determine the camera DOFs(degrees of freedom)?

Positioning Synthetic


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Positioning SyntheticCamera

Are e and -n enough to determine the camera DOFs?No. Note that we were notgiven u and v!(Why not simply require the user to specify them?)

We must also determine u and v, i.e., camera twist about n.Typically done by specification of a world-space up vectorprovided by user interface, e.g., using gluLookat(e, c, up)Twist constraint: Align vwith world up vector (How?)

Positioning Synthetic


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Positioning SyntheticCamera

Where are we?


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Where are we?

What is Projection?


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What is Projection?

Any operation that reduces dimension (e.g., 3D to 2D)

Orthographic Projection

Perspective Projection

O th hi P j ti


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Orthographic Projection

focal point at infinity

rays are parallel and orthogonal to the image plane

Image

World

F

F

Image

World

I

W


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Comparison

Si l P ti C


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Simple Perspective Camera

camera looks along z-axis focal point is the origin

image plane is parallel to xy-plane at distance d

dis call focal length

Similar Triangles


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Similar TrianglesY

Z

[0, d][0, 0]

[Y, Z]

[(d/Z)Y, d]

Similar situation with x-coordinate

Similar Triangles:point [x,y,z] projects to [(d/z)x, (d/z)y, d]

Projection Matrix


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Projection MatrixProjection using homogeneous coordinates:

transform [x, y, z] to [(d/z)x, (d/z)y, d]

2-D image point: discard third coordinate

apply viewport transformation to obtainphysical pixel coordinates

d 0 0 0

0 d 0 0

0 0 d 0

0 0 1 0

x

y

z

1

dx dy dz z[ ] dzx

d

zy d

Divide by 4th coordinate

(the w coordinate)



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z= 0 not allowed (what happens to points on plane z= 0?)Operation well-defined for all other points



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Perspective ProjectionMatrix formulation using homogeneous 4-vectors:

Finally, recover projected point using homogenous convention:Divide by 4thelement to convert 4-vector to 3-vector:


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Camera CoordinatesCoordinate system with the camera in a

convenient pose

1000

nr

vr

ur

zyx

zyx

zyx

cwnnn

vvvuuu

Mu

v

n

x

y

z

ew o ume anNormalized Device


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Normalized DeviceCoordinates

Normalized Device


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Normalized DeviceCoordinates

Device independent coordinates

Visible coordinate usually range from:

11

11

11

z

y

x


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Perspective ProjectionTaking the camera coordinates to NDC

z

x

near


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z

x

near'p

p

z

xnearx

z

x

near

x

'

'

Perspective Projection +


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Perspective Projection +NDC

0100

200

02

0

002

nearfar

nearfar

nearfar

nearfar

bottomtop

bottomtop

bottomtop

near

leftright

leftright

leftright

near

Mpc


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Window Coordinates

Adjusting the NDC to fit the window

),( 00 yx is the lower left of the window


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Window Coordinates

1000

01002

02

0

200

2

0

0

_heightyheight

widthx

width

M pw

Summ : Obj ct C din t


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Summary : Object Coordinateto Device Coordinate

Take your representation (points) andtransform it from Object Space to WorldSpace (Mwo)

Take your World Space point andtransform it to Camera Space (Mcw) Perform the remapping and projection onto

the image plane in Normalized Device

Coordinates (Mw_pMpc) Perform this set of transformations on

each point of the polygonal object (M=M M M M )

Lecture4 Transformation CoordSys

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