Computer graphics

3D Rotation transformation

3D Rotation 2

z

x

y

Rotation around axis:

- Counterclockwise, viewed from rotation axis

z

x

y z

x

y

1

3D Rotation around arbitrary axis 1

z

P

x

yP

Q

2

3

1) Translate object such that the rotation axis coincides with X

axis.

2) Perform Rotation around X axis

3) Translate the object such that the rotation axis is moved back

to its original position

z x

y

1P

2P

3D Rotation around arbitrary axis 2 Rotation around axis through two points

P1 and P1 .

1.Translate such that axis passes through

origin;

2.Rotate the object such the axis of

rotation coincides with any one axes

3.Perform rotation about that axes

4.Apply inverse rotation to bring the axis

back to its original position

5.Translate back the axis to its original

position

3D Rotation around arbitrary axis 3

z x

y

1P

2P

Initial

z x

y

'

1P

1. translate axis

'P2

zx

y

2. rotate axis

'

1P

''P2

zx

y

3. rotate around

z-axis

'

1P

''P2

4. rotate back

z x

y

'

1P

'P2

z x

y

1P

2P

5. translate back

3D Rotation around arbitrary axis 3

z x

y

1P

2P

Initial

z x

y

'

1P

1. translate axis

'P2

zx

y

2. rotate axis

'

1P

''P2

zx

y

3. rotate around

z-axis

'

1P

''P2

4. rotate back

z x

y

'

1P

'P2

z x

y

1P

2P

5. translate back

T(P1)R

Rz() R1 T(P1)

3D Rotation around arbitrary axis 3

z x

y

1P

2P

Initial

z x

y

'

1P

1. translate axis

'P2

zx

y

2. rotate axis

'

1P

''P2

zx

y

3. rotate around

z-axis

'

1P

''P2

4. rotate back

z x

y

'

1P

'P2

z x

y

1P

2P

5. translate back

T(P1)R

Rz() R1 T(P1)

M = T(P1) R1Rz() RT(P1)

Rotation around an axis

Projections

Use to get a view of a solid object

Parallel

Perspective

Parallel Projection

Perspective Projection