l6_3dtrans.pdf

7/27/2019 L6_3dtrans.pdf

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3D TRANSFORMATIONS

1. Linear 3D Transformations:Translation, Rotation, Scaling

Shearing, Reflection

2. Perspective Transformations

AML710 CAD LECTURE 6

Transformations in 3 dimensions

Geometric transformations are mappings fromone coordinate system onto itself.

The geometric model undergoes changerelative to its MCS (Model CoordinateSystem)

The Transformations are applied to an objectrepresented by point sets.

Rigid Body Motion: The relative distances

between object particles remain constantAffine and Non-Affine maps

Transformed point set X* = f(P, transformationparameters)


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Homogeneous coordinates in 3 dimensions

A point in homogeneous coordinates (x, y, z,

h), h 0, corresponds to the 3-D vertex (x/h,y/h, z/h) in Cartesian coordinates.

Homogeneous coordinates in 3D give rise to 4

dimensional position vector.

y

h

x

(x, y, z, h)

Generalized 4 x 4 transformation matrix inhomogeneous coordinates

=

snml

rjfc

qieb

pgda

T][

Perspective transformations

Linear transformations local scaling, shear,rotation / reflection

Translations l, m, n along x, y, and z axis

Overall scaling

LocalScaling


3/12

3D Scaling

=

1000

000000

000

][j

e

a

Ts

=

1111111100001111

33003300

02200220

][][ sTX

Ex: Required scaling to scale theRPP to a unit cube is , 1/3, 1

Overall Scaling

=

s

Ts

000

0100

0010

0001

][

=

11111111

00002222

22002200

02200220

][X

Ex: Uniformly scale the unit cube by a factor of 2requires s=

[ ] [ ]

[ ]T

T

s

szsysx

szyxXT

1/'/'/'

'''][

=

=

If S1 Reduction

in volume


4/12

3D SHEARING

=

11111111

7.095.025.007.195.125.11

115.085.00215.115.01

75.025.01025.075.05.15.0

][X

Ex: Uniformly scale the unit cube by a factor of 2requires d= -0.75, g=0.5,i=1, b=-0.85,c=0.25,f=0.7

[ ] [ ]Ts zfycxizybxgzydxXT 1][ ++++++=

=

11000

01

01

01

][z

y

x

fc

ib

gd

TSH

3D ROTATIONS (i) Rotation about z-axis

We are already familiar with rotation

about z-axis ( in 2D rotations)

=

1000

0100

00cossin

00sincos

][

RzT

Det[T]=+1

The position vector is assumed to be a row vector in right handed system


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3D ROTATIONS (ii) Rotation about x-axis

Similarly we can obtain rotation matrix

about x-axis

=

1000

0cossin0

0sincos0

0001

][

RxT

3D ROTATIONS (iii) Rotation about y-axis

We can obtain rotation

about y-axis as

=

1000

0cos0sin

0010

0sin0cos

][

RyT


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Example for the RPP given by the matrix belowobtain -90 rotation about x -axis

=

11111111

00001111

22002200

03300330

][X

=

1000

0010

0100

0001

][RxT

=

11111111

2200220000001111

03300330

][X

Example for the RPP given by the matrix belowobtain 90 rotation about y -axis

=

11111111

00001111

22002200

03300330

][X

=

1000

0001

0010

0100

][Ry

T

=

11111111

03300330

22002200

00001111

][X


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3D REFLECTIONS As in 2D, we can perform 3Dtransformations about a plane now.

Rotation of 180about an axis passing throughorigin out into 4-D space and projection back onto3D space.

Through x-y plane

Similarly through y-z and x-z planes are

and

=

1000

0100

0010

0001

][xyT

=

10000100

0010

0001

][yz

T

=

10000100

0010

0001

][xz

T

Det[T]=-1

Example for the RPP given by the matrix belowobtain 3D reflection through xy - plane

=

11111111

22221111

11001100

12211221

][X

=

1000

0100

0010

0001

][xy

T

=

11111111

22221111

1100110012211221

][X


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COMBINATION OF TRANSFORMATIONS As in2D, we can perform a sequence of 3D linear

transformations.This is achieved by concatenation of

transformation matrices to obtain a combinedtransformation matrix

A combined matrix

Where [Ti] are any combination of

Translation

Scaling

Shearing linear trans. but not perspective

Rotation transformation

Reflection (Results in loss of info)

nTTTTTXXT 4321][]][[ =

ExampleTransform the given position vector [ 3 2 1 1]by the following sequence of operations(i) Translate by 1, -1, -1 in x, y, and z respectively(ii) Rotate by +30about x-axis and +45about y axisThe concatenated transformation matrix is:

==

1000

0cos0sin

0010

0sin0cos

1000

0cossin0

0sincos0

0001

1000

1100

1010

1001

])45(

][)30(

][[][

ryTrxTtrTT

=

=

1

061.1

866.0

768.1

1

1

2

3

1000

259.0612.0354.0707.0

366.05.0866.00

673.1612.0354.0707.0

]][[ XT


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Rotation about an axis parallel to a coordinate axis

Translate the axis(line) to coincide with the axis to

which it is parallel Rotate the object by required angle

Translate the object back to its original position

Example Consider the following cube. Rotate itby 30 about an axis x passing through its centroid

l=3/2,m3/2 and n=3/2

=

11111111

11122222

22112211

12211221

1000

100

010

001

1000

0cossin0

0sincos0

0001

1000

100

010

001

][n

m

l

n

m

l

T

=

=

11111111

317.1317.1817.0817.0183.2183.2683.1683.1

183.2183.2317.1317.1683.1683.1817.0817.0

12211221

11111111

11122222

22112211

12211221

1000

549.0866.05.00

951.05.0866.00

0001

][T


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ROTATION ABOUT AN ARBITRARY AXIS IN SPACE

Make the arbitrary axis coincide with one of the

coordinate axes.Consider an arbitrary axis passing through a point

(x0, y0 ,z0)

Procedure

Translate (x0, y0 ,z0) so that the point is at origin

Make appropriate rotations to make the linecoincide with one of the axes, say z-axis

Rotate the object about z-axis by required angle

Apply the inverse of step 2

Apply the inverse of step 1 Coinciding the arbitrary axis with any axis

the rotations are needed about other two axes


To calculate the angles of rotations about the x andy axes consider direction cosines (cx, cy ,cz)

d

c

d

cccd

yzzy ==+= sincos22

xcd == sincos

Y

X

Z

cx

1

d

Y

X

Z

cy

cz

cx

d


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The complete sequence of operations can be

summarised as follows:

111][][]][][][][[][= TrRRRRRTrT

=

1

0100

0010

0001

][

000 zyx

Tr

=

1000

0)cos(0)sin(

0010

0)sin(0)cos(

][

R

=

1000

0cossin0

0sincos0

0001

][

R

=

1

01000010

0001

][

000

1

zyx

Tr

=

1000

0)cos()sin(00)sin()cos(0

0001

][1

R

=

1000

0cos0sin0010

0sin0cos

][ 1

R


Substituting for the respective angles we obtain thefollowing matrices:

111][][]][][][][[][= TrRRRRRTrT

=

=

1000

00

0010

00

1000

0)cos(0)sin(

0010

0)sin(0)cos(

][dc

cd

Rx

x

=

=

1000

0//0

0//0

0001

1000

0cossin0

0sincos0

0001

][dcdc

dcdcR

zy

yz

=

1000

0100

00cossin

00sincos

][

R


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Direction cosines of a an arbitrary axis when two

points on the line are known:

)()()(][ 010101 zzyyxxV =

),,( 000 zyx

),,( 111 zyx

V

++

= 2

012

012

01

010101

)()()(

)()()(

][ zzyyxxzzyyxx

zyx ccc


Find the transformation matrix for reflection withrespect to the plane passing through the origin andhaving normal vector n = i+j+k

[ ] 3111|| 2/1222 ==n

n

=

3

111][ zyx ccc

l6_3dtrans.pdf

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