3D Geometrical TransformationsFoley & Van Dam, Chapter 5

3D Geometrical Transformations

• 3D point representation• Translation• Scaling, reflection• Shearing• Rotations about x, y and z axis• Composition of rotations• Rotation about an arbitrary axis• Transforming planes

3D Coordinate SystemsRight-handed coordinate system:

Left-handed coordinate system:

y

z

x

x

y

z

Reminder: Cross Product

U

V

UxV

θ

VxU

θsinˆ VUnVU =×

−−−

==×

xyyx

zxxz

yzzy

zyx

zyx

vuvuvuvuvuvu

vvvuuuzyx

VUˆˆˆ

3D Point Representation• A 3D point P is represented in homogeneous coordinates by a 4-dimensional vector:

•As for 2D points:

=

1zyx

P

≡

=

αααα

zyx

zyx

p

1

3D Geometrical Transformations• In homogeneous coordinates, 3D affine transformations are represented by 4x4 matrices:

•A point transformation is performed:

1000z

y

x

tihgtfedtcba

=

110001'''

zyx

tihgtfedtcba

zyx

z

y

x

3D TranslationP in translated to P' by:

Or:

Inverse:

+++

=

=

11'''

11000100010001

z

y

x

z

y

x

tztytx

zyx

zyx

ttt

'PPT =

y

z

x

PPT =− '1

3D Scaling

=

=

11'''

11000000000000

czbyax

zyx

zyx

cb

a

'PPS =Or

PPS =− '1

y

z

x y

z

x

3D Reflection• A reflection through the xy plane:

•Reflections through the xz and the yz planes are defined similarly

−=

−111000

010000100001

zyx

zyx

3D Shearing• Shearing:

• Change in each coordinate is a linear combination of all three• Transforms a cube into a general parallelepiped

++++++

=

111000010101

zfyexdzycxbzayx

zyx

fedcba

3D Rotation• To generate a rotation in 3D we have to specify:

– axis of rotation (2 d.o.f.)– amount of rotation (1 d.o.f.)

•Note, the axis passes through the origin

x

y

z

3D Rotation

• Counterclockwise rotation about x-axis

−

=

110000cossin00sincos00001

1'''

zyx

zyx

θθθθ

pRp x )(' θ=

y

z

x

3D Rotation

• Counterclockwise rotation about y-axis

−

=

110000cos0sin00100sin0cos

1'''

zyx

zyx

θθ

θθ

pRp y )(' θ=

y

z

x

3D Rotation

• Counterclockwise rotation about z-axis

−=

1zyx

1000010000θcosθsin00θsinθcos

1'z'y'x

pRp z )(' θ=

yx

z

Composite Rotation• Rx, Ry, and Rz, can perform any rotation about an axis passing through the origin

•Inverse rotation:

)(')(')(1 θθθ TRpRpRp =−== −

Change of Coordinates• Problem: Given the XYZ orthonormal coordinate system, find a transformation M, that maps a representation in XYZ into a representation in the orthonormal system UVW, with the same origin•The matrix M transforms the UVW vectors to the XYZ vectors

y

z

x

u=(ux,uy,uz)

v=(vx,vy,vz)

Change of Coordinates• Solution: M is rotation matrix whose rows are U,V, and W:

• Note: the inverse transformation is the transpose:

000

0 0 0 1

x y z

x y z

x y z

u u uv v v

Mw w w

=

==−

10000wvu0wvu0wvu

MMzzz

yyy

xxx

T1

Change of Coordinates• Let's check the transformation of U under M:

• Similarly, V goes into Y, and W goes into Z

X1001

100

uuu

1uuu

10000www0vvv0uuu

MU

2z

2y

2x

z

y

x

zyx

zyx

zyx

=

=

++=

=

Change of Coordinates• Let's check the transformation of the X axis under M-1:

• Similarly, Y goes into V, and Z goes into W

U

1uuu

1001

10000wvu0wvu0wvu

1001

Mz

y

x

zzz

yyy

xxx

1 =

=

=

−

Rotation About an Arbitrary Axis• Axis of rotation can be located at any point: 6 d.o.f. (we must specify 2 points p1 and p2)• The idea: make the axis coincident with one of the coordinate axes (z axis), rotate by θ, and then transform back

y

z

x

p1p2

Rotation About an Arbitrary Axis

z xp1

y

x

p1

y

xp1

z

z

step 1

Initial Position

Translate p1 to the Origin

step 2

Rotate p2 onto the z Axis

p2

p2

p2

xp1

z

step 3

Rotate the Object Around the z Axis

p2

x

p1

z

step 4

Rotate the Axis to the original Orientation

p2

y

z xp1

step 5

Translate to the Original Position

p2

y y

y

Rotation About an Arbitrary Axis• Step1:

• Step 2:

• Step 3:

• Step 4:

• Step 5:

• Composition:

−−−

=

1000z100y010x001

T1

1

1

=

10000www0vvv0uuu

Mzyx

zyx

zyx

−=

1000010000θcosθsin00θsinθcos

R

=−

10000wvu0wvu0wvu

Mzzz

yyy

xxx

1

=−

1000z100y010x001

T1

1

11

PTMRMTP 11 −−=′

Rotation About Arbitrary Axis• Constructing an orthonormal system along the rotation axis:

– A vector W parallel to the rotation axis:

– A vector V perpendicular to W:

– A vector U forming a right-handed orthogonal system with W and V:

ssw;pps 12 =−=

aavwa =

×= ;

100

wvu ×=

Transforming Planes• Plane representation:

– By three non-collinear points– By implicit equation:

[ ] 0

1

xy

A x B y C z D A B C Dz

+ + + = =

P0

P1P2

[ ]1zyx

1

DC

DB

DA

Transforming Planes• One way to transform a plane is by transforming any three non-collinear points on the plane

• Another way is to transform the plane equation: Given a transformation T such that

T [x, y, z, 1] = [x', y', z', 1]find [A',B',C',D'], such that:

[ ]

''

' ' ' ' 0'

1

xy

A B C Dz

=

Transforming Planes• Note that:

• Thus, the transformation that we should apply to the plane equation is:

[ ] 1 0

1

xy

A B C D T Tz

−

=

[ ] [ ] 1A B C D A B C D T −′ ′ ′ ′ =