3D GeometricTransformation

(Chapt. 5 in FVD, Chapt. 11 in Hearn & Baker)

3D Coordinate Systems

• Right-handed coordinate system:

• Left-handed coordinate system:

y

z

x

x

y

z

Reminder: Vector Product

U

V

UxV

VxU

θsinˆ VUnVU =×

θ

−−−

==×

xyyx

zxxz

yzzy

zyx

zyx

vuvu

vuvu

vuvu

vvv

uuu

zyx

VU

ˆˆˆ

3D Point

• A 3D point P is represented inhomogeneous coordinates by a4-dim. vector:

• Note, that

=

1

z

y

x

P

≡

=

αααα

z

y

x

z

y

x

p

1

3D Transformations

• In homogeneous coordinates, 3Dtransformations are represented by4x4 matrices:

• A point transformation isperformed:

1000z

y

x

tihg

tfed

tcba

=

110001

'

'

'

z

y

x

tihg

tfed

tcba

z

y

x

z

y

x

3D Translation

• P in translated to P' by:

• Inverse translation:

+++

=

=

11

'

'

'

11000

100

010

001

z

y

x

z

y

x

tz

ty

tx

z

y

x

z

y

x

t

t

t

'PPT =Or

y

z

x

PPT =− '1

Scaling

=

=

11

'

'

'

11000

000

000

000

cz

by

ax

z

y

x

z

y

x

c

b

a

'PPS =Or

y

z

x

PPS =− '1

y

z

x

3D Shearing

• Shearing:

• The change in each coordinateis a linear combination of allthree.

• Transforms a cube into ageneral parallelepiped.

++++++

=

111000

01

01

01

zfyex

dzycx

bzayx

z

y

x

fe

dc

ba

3D Rotation

• To generate a rotation in 3D wehave to specify:– axis of rotation (2 d.o.f)

– amount of rotation (1 d.o.f)

• Note, the axis passes through theorigin.

x

y

z

• A counter-clockwise rotationabout the z-axis:

−

=

11000

0100

00cossin

00sincos

1

'

'

'

z

y

x

z

y

x

θθθθ

y

z

x

pRp z )(' θ=

• A counter-clockwise rotationabout the x-axis:

−

=

11000

0cossin0

0sincos0

0001

1

'

'

'

z

y

x

z

y

x

θθθθ

y

z

x

pRp x )(' θ=

• A counter-clockwise rotationabout the y-axis:

−=

11000

0cos0sin

0010

0sin0cos

1

'

'

'

z

y

x

z

y

x

θθ

θθ

y

z

x

pRp y )(' θ=

Inverse Rotation

')(')(1 pRpRp θθ −== −

Composite Rotations

• Rx, Ry, and Rz, can perform anyrotation about an axis passingthrough the origin.

Rotation About anArbitrary Axis

• Axis of rotation can be located atany point: 6 d.o.f.

• The idea: make the axiscoincident with one of thecoordinate axes (z axis), rotate,and then transform back.

• Assume that the axis passesthrough the point p0.

y

z

x

p0

• Transformations:– Translate P0 to the origin.

– Make the axis coincident with thez-axis (for example):

• Rotate about the x-axis into thexz plane.

• Rotate about the y-axis ontothe z-axis.

• Rotate as needed about the z-axis.

• Apply inverse rotations about yand x.

• Apply inverse translation.

y

zx

p0

y

x

p0

y

x

p0

y

x

p0

z

z

z

y

x

p0

z

y

zx

p0

A

B

C

D

E

F

3D Reflection

• A reflection through the xy plane:

• Reflections through the xz and theyz planes are defined similarly.

• How can we reflect through somearbitrary plane?

−=

−110000

0100

0010

0001

z

y

x

z

y

x

Transforming Planes

• Plane representation:– By three non-collinear points

– By implicit equation:

[ ] 0

1

=

=+++z

y

x

DCBADCzByAx

P0

P1P2

[ ]1zyx

1D

C

D

B

D

A

• One way to transform a plane isby transforming any three non-collinear points on the plane.

• Another way is to transform theplane equation: Given atransformation T thattransforms [x,y,z,1] to [x',y',z',1]find [A',B',C',D'], such that:

[ ] 0

1

'

'

'

'''' =

z

y

x

DCBA

• Note that

• Thus, the transformation thatwe should apply to the planeequation is:

[ ] 0

1

1 =

−

z

y

x

TTDCBA

( )

=

−

D

C

B

A

T

D

C

B

A

T1

'

'

'

'

Changing CoordinateSystems

• Problem: Given the XYZorthogonal coordinate system,find a transformation, M, thatmaps XYZ to an arbitraryorthogonal system UVW.

• This transformation changes arepresentation from the UVWsystem to the XYZ system.

y

z

x

u=(ux,uy,uz)

v=(vx,vy,vz)

w=(wx,wy,wz)

(x0,y0,z0)

• Solution: M=RT where T is atranslation matrix by (x0,y0,z0), andR is rotation matrix whose columnsare U,V, and W:

because

=

1000

0

0

0

zzz

yyy

xxx

wvu

wvu

wvu

R

Uu

u

u

wvu

wvu

wvu

RXz

y

x

zzz

yyy

xxx

=

=

=

11

0

0

1

1000

0

0

0

• Similarly, Y goes into V, and Zgoes into W.

• The inverse transform, T-1R-1,

provides the mapping fromUVW back to XYZ. For therotation matrix R-1=RT:

X

uuu

u

u

u

www

vvv

uuu

UR

zyx

z

y

x

zyx

zyx

zyx

T

=

=

++

=

=

1

0

0

1

1

0

0

11000

0

0

0

222

• Comment: Very useful if anarbitrary plane is to be mappedto the XY plane or vice versa.

• Possible to apply if an arbitraryvector is to be mapped to anaxis (How?).

Transformation Pipe-Line

ViewingTransformation

ModelingTransformation

WorkstationTransformation

ModelingCoordinates

World Coordinates

Viewing Coordinates

Device Coordinates

ProjectionTransformation

Projection Coordinates

Modeling CoordinateHierarchy

yw

zw

xw

world

TractorSystem

Front-WheelSystem

World

Tractor

Front-wheel

transformation

transformation