three dimensional transformationspeople.scs.carleton.ca/~nussbaum/courses/comp3501/notes... ·...

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Three dimensional transformations

Doron Nussbaum COMP 3501 - 3D transformations 1

Object Manipulation

• Graphics primitives allows us to draw a variety f i t i d hof pictures, images and graphs

• In many application there is a also need to manipulate the displays – Rearranging the display

– Changing the size of the different objects


Changing the size of the different objects

• This is accomplished by using geometric transformations

2


Rotate right by 90o

Translate to position


Scale down to 2/3 of sizeRotate right by 60o

Translate to position

3

Basic Geometric Transformation

• The basic geometric transformation are:– Translation

– Scaling

– Rotation

• Other


• Other – Reflection

– Shear

Rotation

• Rotation of an object is repositioning it along a circular path in the xy planecircular path in the xy plane.

• A rotation is specified by a rotation angle and a pivot point (rotation point) p

20


10

15

5

201510

p

4

2D Rotation

• The angle is defined in a counter clockwise d d th i torder around the point p (counter clockwise is positive angle

and clockwise is negative angle)

• The rotation can also be viewed as rotation around an axis line perpendicular to the xyplane through the point p


• Assuming rotation around the origin (0,0)

(x,y)

(x’,y’)

r

r


cossinsincos)sin('

sinsincoscos)cos('

rrry

rrrx

5

cossinsincos)sin('

sinsincoscos)cos('

rrry

rrrx

cossin'

sincos'

obtain we

sin and cos

Since

cossinsincos)sin(

yxy

yxx

ryrx

rrry


cossin yxy

cossin

sincos),()','( yxyx

Rotation around point p

coscos x-αrrx

cos)(sin)('

sin)(cos)(

sinsincoscos

)cos('

sin sin

cos cos

y-x-αy

y-x-α

rr

rx

y-rry

x αrrx

(x,y)

(x’,y’)

r

r

( )


()

),(cossin

sincos),()','(

yxyx

6

Matrix Transforms

• For "simplicity", we want to represent our t f t i titransforms as matrix operations

• Why is this simple?– single type of operation for all transforms

– IMPORTANT: collection of transforms can be expressed as a single matrix


expressed as a single matrix• efficiency

Homogenous CoordinatesMotivation

• In many cases, e.g., animation, we need to combine a number of basic transformations such as a rotationnumber of basic transformations such as a rotation followed by a translation

P’=RP + T

• R is a 3x3 matrix while T is a 3x1 matrix


• The rotation operation is multiplication operation while the translation operation is an addition

7

Homogenous Computations

• How to speed up the process and to avoid long i ( ll th t th b th dprocessing (recall that there can be thousands

of vertices in an image)?

• Idea:use a single matrix operation for all types of


– use a single matrix operation for all types of transformations (rotation, scaling, translation)

• Instead of using triplet of coordinates (x, y, z) f h t ( ) hfor each vertex use a (xw,yw,zw,w) where

zw

yw

xw

z

y

x

zyx wz

wy

wx www toleads which , ,


• There are many choices for w (normally w=1)

w

8

Translation

• Purpose: to move an object along a line from one location to another locationlocation to another location


Translation Matrix

0010

0001

• This is a rigid transformation - objects are not d f d

1TTT

0100

0010)1,,,()1,',','(

zyx

zyxzyx

}|{' ptp


deformed.

• Enough to translate the vertices (end points)

9

Rotation around point p2D Homogenous Coordinates

coscos x-αrrx

cos)(sin)('

sin)(cos)(

sinsincoscos

)cos('

sin sin

cos cos

y-x-αy

y-x-α

rr

rx

y-rry

x αrrx


1

010

001

100

0cossin

0sincos

1

010

001

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

yxyx

3D Rotation Matrices

0100

00cossin

00sincos

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

yxyxRotation around Z axis

1000

0cos0sin

0010

0sin0cos

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

yxyx

1000

0100

Rotation around Y axis


1000

0cossin0

0sincos0

0001

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

yxyxRotation around X axis

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Scaling

• A scaling transformation alters the size of an object.

• The operation is accomplished by multiplying each coordinate by scaling factors sx, sy sz :

• When sx (sy, sz) > 1 we enlarge the object

• When sx (sy, sz) < 1 we reduce the object

000xS


1000

000

000)1,,()1,','(

z

y

S

Syxyx

3D Matrices

• Pay attention to:Concatenating matrices (e g rotation translation) to– Concatenating matrices (e.g., rotation, translation) to which side you should concatenate.

– Note Direct3D multiplies the vector on the left side (VT*M) vs. open GL which multiplies the vector on the right hand side similar to what is done in mathematics (M*V), where V is a 4x1 vector and M is a 4x4 matrixa 4x4 matrix.

– The difference in the sign before the sin(θ) when the rotation is around the y-axis vs. rotation around the x-axis and z-axis

Doron Nussbaum 3D transformations 20

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Compound Objects Concatenation

• Done on the board in class


Transformation between coordinate systems

• Motivation:– objects are often defined within their own local coordinate– objects are often defined within their own local coordinate

systems which must be converted into a scene coordinate system.

– Often occurs when one buys 3D models from a modeller

• Example– For example a cubical consists of a desk, a workstation and

a chair is defined in a local coordinate system When an

Doron Nussbaum COMP 3501 - 3D transformations 22Doron Nussbaum 22

a chair is defined in a local coordinate system. When an office is designed the layout consists of multiple cubicles which are positioned in the office. The location of the desks, chairs and workstations must be defined using with respect to the office

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Office Elements


Workstation

Office Layout

Office

Y-axis

Workstation

X-axis

Y-axis

X-axis

Y-a

xis

Doron Nussbaum COMP 3501 - 3D transformations 24X-axis

Workstation

X-axis

Y-a

xis Workstation

X-axis

Y-a

xis

13

• We want to know what will be the coordinates of p=(x y z) in the new coordinate systemp=(x,y,z) in the new coordinate system

• In order to accomplish this we need to– translate the new coordinate system to the origin of the

current coordinate system

Rotate the new coordinate system so that the axis overlap


– Rotate the new coordinate system so that the axis overlap

– Use the two transformation matrices to transform the object to the new coordinate system


Scaling using sx=sy=3

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• Uniform scaling sx=sy

• Differential scaling sxsy

• Transformed objects using the matrix S are not only scaled but also change their position


• Show example

Composite Transformation

• What happens when we combine several t f titransformations (concatenation/composition)

– compute the product of two or more matrices to obtain the final transformation matrix

321 )))(((' MMPMP


321

321

)(

))((

PM

MMP

MMMP

15

Concatenation of Similar Transformations

• Translation

Hence,

100

10

01

100

10

01

100

10

01

21

21

1

1

2

2

yy

xx

y

x

y

x

tt

tt

t

t

t

t


),(),(),( 21211122 yyxxyxyx ttttTttTttT

• Rotation

PRR

PRRP

)()(

)()('

12

12

Show a picture of two rotations


)()()(

Hence,

1212 RRR

rotations

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• Scaling

Hence

100

00

00

100

00

00

100

00

00

12

12

1

1

2

2

yy

xx

y

x

y

x

ss

ss

s

s

s

s


),(),(),(

Hence,

21211122 yyxxyxyx ssssSssSssS

Pivot Point Rotation

• So far all the transformations assumed that they are done around the origin (0 0)done around the origin (0,0)

• This can be generalized to any arbitrary point as follows– Translate the object to the origin such that the pivot point is

at the origin

– Transform the object around the origin (e g rotation)


Transform the object around the origin (e.g., rotation)

– Use inverse translation used in step 1 to move the object back to its original position

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Example Rotate by 45o around p

p =(x,y) p

Translate to the origin - T(-x,-y)


p

Rotate by 45o - R(45)T(-x,-y)

p

Translate back to p - T(x,y) R(45)T(-x,-

Example Rotate by 45o around p

( )p =(x,y)

Translate to the origin - T(-x,-y)

p =(x,y)

Doron Nussbaum COMP 3501 - 3D transformations 34Rotate by 45o - R(45)T(-x,-y) Translate back to p - T(x,y) R(45)T(-x,-y)

p =(x,y) p =(x,y)

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2D example

010sincos01

i)1(i

100

10

01

100

0cossin

0sincos

100

10

01

r

r

r

r

y

x

y

x


100

sin)cos1(cossin

sin)cos1(sincos

rr

rr

xy

yx

Scaling with respect to a fixed point

• Similar procedure is used to scale with respect t fi d i t

100

10

01

100

00

00

100

10

01

r

r

y

x

r

r

y

x

S

S

y

x

to a fixed point

p =(x,y)


100

)1(0

)1(0

yrx

xrx

SyS

SxS

Translate to the origin -T(-x,-y)

Rotate by 45o - R(45)T(-x,-y) Translate back to p -T(x,y) R(45)T(-x,-y)

p =(x,y) p =(x,y)

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General Scaling -scaling with respect to a line

• The scaling matrix allows only scaling along the x-axis and the y-scaling along the x axis and the yaxis.

• How to scale with respect to any line?

y s2


x

s1

General Scaling

• We use a similar technique to use any scaling in any direction.– Rotate the object such that the scaling direction coincide with the x-yRotate the object such that the scaling direction coincide with the x y

axis.

– Scale the object

– rotate back

Doron Nussbaum COMP 3501 - 3D transformations 38Scale at = 45o and sx=1 sy=2

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Transformation are not commutative


Translate and then rotate - R()T(x,y)e.g. R(45)T(5,0)

Rotate and then Translate - T(x,y) R()e.g. T(5,0) R(45)

Other Transformations

• Reflection– A transformation that produces a mirror image of

an object

– it is generated by rotating the object 180o around the axis of reflection


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Reflection about the axis(x-axis or y-axis)

• a reflection about the x-axis

• a reflection about the y-axis

100

010

001


100

010

001

• Show a picture of reflection about the x-axis d i

Reflection about the origin(x-axis and the y-axis)

and y-axis

100

010

001


• This transformation can also be viewed as a rotation where =180o

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Reflection about lines

• Line y=x

• It can be viewed as

100

001

010


– a rotation by -45o (CCW 45o)

– Reflection about the x-axis

– a rotation by 45o (CW 45o)

Reflection about a general line

• Reflection about a general line y=ax+ b is achieved as:– translate the object and the line such that the line passes through thetranslate the object and the line such that the line passes through the

origin - T(0,-b)

– rotate the object and the line such that it coincide with one of the axis

– apply the reflection transformation of that axis

– rotate the object and the line back using the inverse rotation

– translate the object and the line back using inverse transformation


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Shear

• Shear transformation distorts the shape of the bj tobject.

• The effect causes the object to be pushed to one side as if it was constructed of layers that slide on top of each other


• (show picture)

• Shearing transformation along the x-axis

• example

yyyshxx'

sh

x

x

'

100

010

01


100

010

021 1,1

1

3,12,1

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• Shearing transformation along the x-axis using a reference line

• example using a shearing along the x-axis of ½ and a reference line y=-1

yyyyshxx

yshsh

refx

refxx

' )('

100

010

1


100

010

1 21

21

-1

-1

1,12,11,1

1.5,00.5,0

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