Three-Dimensional Geometric and Modeling Transformations

By

Dr. Rajeev Srivastava

CSE, IIT(BHU)

Computer Graphics

Three-Dimensional Geometric and Modeling Transformations

Some Basics

3D Translations.

3D Scaling.

3D Rotation.

3D Reflections.

Transformations.

Some Basics

Basic geometric types.

Scalars s

Vectors v

Points p

Transformations

Types of transformation:

rotation, translation, scale, Reflections, shears.

Matrix representation

Order

P=T(P)

3D Point

We will consider points as column vectors. Thus, a typical point with coordinates (x, y, z) is represented as:

z

y

x

3D Translations.

P is translated to P' by T:

1000

100

010

001

z

y

x

t

t

t Called the translation

matrix T =

6

3D Translations.

11000

100

010

001

1

'

'

'

z

y

x

t

t

t

z

y

x

z

y

x

PTP '

7

3D Translations.

An object is translated in 3D dimensional by transforming each of the defining points of the objects.

3D Translations.

zyx tttT ,,

zyx ,,

',',' zyx

3D Scaling

P is scaled to P' by S:

Called the

Scaling matrix

S =

1000

000

000

000

z

y

x

s

s

s

10

3D Scaling

Scaling with respect to the coordinate origin

11000

000

000

000

1

'

'

'

z

y

x

s

s

s

z

y

x

z

y

x

PSP '

3D Scaling

Scaling with respect to a selected fixed position (xf, yf, zf)

1. Translate the fixed point to origin

2. Scale the object relative to the coordinate origin

3. Translate the fixed point back to its original position

1000

)1(00

)1(00

)1(00

),,(),,(),,(fzz

fyy

fxx

fffzyxfffzss

yss

xss

zyxTsssSzyxT

3D Scaling

3D Reflections

About an axis: equivalent to 180rotation about that axis

14

3D Reflections

1000

0100

0010

0001

zRF

15

3D Shearing

Modify object shapes

Useful for perspective projections:

E.g. draw a cube (3D) on a screen (2D)

Alter the values for x and y by an amount

proportional to the distance from zref

16

3D Shearing

1000

0100

10

01

refzyzy

refzxzx

zshear

zshsh

zshsh

M

17

Shears

1000

0100

010

001

b

a

SH z

mshe1990@hotmail.com && 18

Rotation Positive rotation angles produce counterclockwise

rotations about a coordinate axis

mshe1990@hotmail.com &&

Rotation

mshe1990@hotmail.com && 20

Coordinate-Axes Rotations

11000

0100

00cossin

00sincos

1

'

'

'

z

y

x

z

y

x

sincos' yxx

cossin' yxy

zz '

PRP z )('

mshe1990@hotmail.com &&

Coordinate-Axes Rotations

xzyx

mshe1990@hotmail.com && 22

Coordinate-Axes Rotations

11000

0cossin0

0sincos0

0001

1

'

'

'

z

y

x

z

y

x

sincos' zyy

xx '

PRP x )('

cossin' zzz

mshe1990@hotmail.com && 23

Coordinate-Axes Rotations

11000

0cos0sin

0010

0sin0cos

1

'

'

'

z

y

x

z

y

x

sincos' xzz

cossin' xzx

yy '

PRP y )('

mshe1990@hotmail.com && 24

General Three-Dimensional Rotations

An object is to be rotated about an axis that is parallel to one of the coordinate axes

1. Translate the object so that the rotation axis coincides with the parallel coordinate axis

2. Perform the specified rotation about that axis

3. Translate the object so that the rotation axis is moved back to its original position

PTRTP x )(' 1

TRTR x )()( 1

mshe1990@hotmail.com && 25

General Three-Dimensional Rotations

An object is to be rotated about an axis that is not parallel to one of the coordinate axes

1. Translate the object so that the rotation axis passes through the coordinate origin.

2. Rotate the object so that the axis of rotation coincide with one of the coordinate axes.

3. Perform the specified rotation about that coordinate axis.

4. Apply inverse rotations to bring the rotation axis back to its original orientation.

5. Apply the inverse Translation to bring the rotation axis back to its original position.

26

1 2

2 1 2 1 2 1 2 1

The vector from to is:

, ,x x y y z z

P P

V P P

2 1

2 1

2 1

2 2 2

Unit rotation vector: | | , ,

| |

| |

| |

1

a b c

a x x

b y y

c z z

a b c

u V V

V

V

V

1

1

1

1 0 0

0 1 0

0 0 1

0 0 0 1

x

y

z

T

, ,a b cu

x

y

z

27

Rotating to coincide with axiszu

2 2

2 2

First rotate around axis to lay in plane.

Equivqlent to rotation 's projection on plane around axis.

cos , sin .

We obtained a unit vector ,0, in plane.

x x z

y z x

c b c c d b d

a b c d x z

u

u

w

1 0 0 0

0 0

0 0

0 0 0 1

x

c d b d

b d c d

R

, ,a b cu

x

y

z

u

28

, ,a b cu

x

y

z

,0,a dw

2 2

Rotate counterclockwise around axis.

is a unit vector whose component is , component is 0,

hence component is .

y

x a y

z b c d

w

w

cos , sind a

0 0

0 1 0 0

0 0

0 0 0 1

y

d a

a d

R

29

cos sin 0 0

sin cos 0 0

0 0 1 0

0 0 0 1

z

R

1 1 1x y z y x R T R R R R R T

2

2

2

1 cos cos 1 cos sin 1 cos sin

1 cos sin 1 cos cos 1 cos sin

1 cos sin 1 cos sin 1 cos cos

R

a ab c ac b

ba c b bc a

ca b cb a c

M