May 3, 20231

y

X

Z

Translations

Objects are usually defined relative to their own coordinate system. We can translate points in space to new positions by adding offsets to their coordinates.

The following figure shows the effect of translating a teapot.

May 3, 20233

4

Properties of Translation

v=v)0,0,0(T=v),,(),,( zyxzyx tttTsssT

=

=v),,(1zyx tttT

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

v),,( zzyyxx tststsT

v),,( zyx tttT

May 3, 2023

5

Translation Revisited

=

1

'

'

'

zyx

1000100010001

ttt

z

y

x

1zyx

=T(tx, ty, tz )

May 3, 2023

6

Scaling

zsysxs

zyx

z

y

x

'''

z

y

x

zyx

ss

ssssS

000000

),,(

Uniform scaling iff zyx sss

May 3, 2023

3D Rotations

May 3, 20237

May 3, 20239

May 3, 202310

11

An Alternative View

• We can view the rotation around an arbitrary axis as a set of simpler steps

• We know how to rotate and translate around the world coordinate system

• Can we use this knowledge to perform the rotation?

May 3, 2023

12

Rotations about an arbitrary axis

Rotate by around a unit axis r

r

May 3, 2023

13

Rotation about an arbitrary axisA rotation matrix for any axis that does not coincide with a coordinate axis can be set up as a composite transformation involving combination of translations and the coordinate-axes rotations.

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

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

3. Perform the specified rotation about that coordinate axis

4. Apply inverse rotation axis back to its original orientation

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

May 3, 2023

1010000100001

zyx

T=

1010000100001

zyx

T-1=

•Translate origin to rotation axis

May 3, 202314

A, B, C

V

)0 ,B, C(

θ

CBV 22

Sin θ = B / V

Cos θ = C / V

May 3, 202315

10000cossin00sincos00001

Rx=

10000//00//00001

VCVBVBVC

Rx=

Sin θ = B / V

Cos θ = C / V

10000//00//00001

VCVBVBVC Rx

-1 =

• Rotation about X-axis

May 3, 202316

Lß

A

V

CBAL 222

May 3, 202317

• Rotation about Y-axis

10000cos0sin00100sin0cos

Ry=

sin = V / L

cos = A / L

10000/0/00100/0/

LVLA

LALV

Ry=

Ry-1=

10000/0/00100/0/

LVLA

LALV

May 3, 202318

• Finally Rotate about Z-axis with angle

1000010000cossin00sincos

Rz =

Rotation about an arbitrary axis with a angle

R = T Rx Ry Rz Ry-1 Rx

-1 T-1

May 3, 202319

Shear (Kxy, Kxz, Kyz)=

1000010001001

KKK

yz

xzxy

Shear

May 3, 202320