[PDF] - 1 Translation and Scalling Matrice Translation o The translation PDF Document

SLIDE 1

1 3D Coordinate Systems

3D computer graphics involves the additional dimension of

depth, allowing more realistic representations of 3D objects in the real world

There are two possible ways of “attaching” the Z-axis, which

gives rise to a left-handed or a right-handed system

3D Transformation

The translation, scaling and rotation transformations used for 2D

can be extended to three dimensions

In 3D, each transformation is represented by a 4x4 matrix
Using homogeneous coordinates it is possible to represent each

type of transformation in a matrix form and integrate transformations into one matrix

To apply transformations, simply multiply matrices, also easier in

hardware and software implementation

Homogeneous coordinates can represent directions
Homogeneous coordinates also allow for non-affine

transformations, e.g., perspective projection

Homogeneous Coordinates

In 2D, use three numbers to represent a point
(x,y) = (wx,wy,w) for any constant w≠0
To go backwards, divide by w, (x,y) becomes (x,y,1)
Transformation can now be done with matrix multiplication

                    =           ′ ′ 1 1 1 y x b a a b a a y x

y yy yx x xy xx

Basic 2D Transformations

Translation:
Scaling:
Rotation:

          1 1 1

y x

b b           1

y x

s s           − 1 cos sin sin cos θ θ θ θ

SLIDE 2

2 Translation and Scalling Matrice

The translation and scaling transformations may

be represented in 3D as follows: 1 0 0 trX 0 1 0 trY 0 0 1 trZ 0 0 1 SX 0 0 0 SY 0 0 0 SZ 0 0 1

Translation matrix Scaling matrix

Translation

translation along y,

r V = (0, k, 0)
riginal

V=(ai + bj +ck) == (a, b, c)

Scaling

Original scale all axes scale Y axis

ffset from origin

                        =             ′ ′ ′ 1 1 1 z y x c b a z y x

3D Shearing

                        =             ′ ′ ′ 1 1 1 1 1 1 z y x f e d c b a z y x

Shearing: The change in each coordinate is a linear combination of all three Transforms a cube into a general parallelepiped

SLIDE 3

3 Rotation

In 2D, rotation is about a point
In 3D, rotation is about a vector, which can be done through

rotations about x, y or z axes

Positive rotations are anti-clockwise, negative rotations are

clockwise, when looking down a positive axis towards the

rigin

x y z x y z x y z

Major Axis Rotation Matrices

about X axis
about Y axis
about Z axis

            − = 1 1 θ θ θ θ cos sin sin cos

x

R             − = 1 1 θ θ θ θ cos sin sin cos

z

R             − = 1 1 θ θ θ θ cos sin sin cos

y

R Rotations are

rthogonal matrices,

preserving distances and angles.

Rotation

rotation of 45o about the Z axis

ffset from origin translation then rotate about Z axis

Rotation Axis

In general rotation vector does not pass through
rigin

SLIDE 4

4 Rotation about an Arbitrary Axis

Rotation about an Arbitrary Axis

Basic Idea

1. Translate (x1, y1, z1) to the origin 2. Rotate (x’2, y’2, z’2) on to the z axis 3. Rotate the object around the z-axis 4. Rotate the axis to the original

rientation

5. Translate the rotation axis to the

riginal position

Basic Idea

1. Translate (x1, y1, z1) to the origin 2. Rotate (x’2, y’2, z’2) on to the z axis 3. Rotate the object around the z-axis 4. Rotate the axis to the original

rientation

5. Translate the rotation axis to the

riginal position

(x2,y2,z2) (x1,y1,z1) x z y

R-1 R-1 T-1 T-1 R R T T

( ) ( ) ( ) ( ) ( )

] ][ ][ ][ [ ] [ ] [ ] [ ] [

1 1 1

R x y z y x R arb

T R R R R R T R

T T T T T T T T

α φ θ φ α − − −

=

Rotation about an Arbitrary Axis

Step 1. Translation

            − − − = 1 1 1 1

1 1 1

z y x T

R

T

(x2,y2,z2) (x1,y1,z1) x z y

Rotation about an Arbitrary Axis

Step 2. Establish

[ ]

            − =             − = 1 / / / / 1 1 cos sin sin cos 1

) (

d c d b d b d c T

x

R

α α α α

α

(a,b,c) (0,b,c)

Projected Point

α α

Rotated Point

d c c b c d b c b b = + = α = + = α

2 2 2 2

cos sin

x y z

[ ]

) (α

x

R

T

Rotation about an Arbitrary Axis

Step 3. Rotate about y axis by φ

(a,b,c) (a,0,d) φ l d

2 2 2 2 2 2 2 2

cos , sin c b d d a c b a l l d l a + = + = + + = = φ = φ

( )

[ ]

            − =             − = 1 / / 1 / / 1 cos sin 1 sin cos l d l a l a l d T

y

R

φ φ φ φ

φ

x y

Projected Point

z

Rotated Point

SLIDE 5

5 Rotation about an Arbitrary Axis

Step 4. Rotate about z axis by the desired angle θ

θ l

( )

[ ]

            − = 1 1 cos sin sin cos θ θ θ θ

θ

z

R

T

y x z

Rotation about an Arbitrary Axis

Step 5. Apply the reverse transformation to place the axis back in

its initial position

x y l l z

( ) ( )

            −             −             =

− − −

1 cos sin 1 sin cos 1 cos sin sin cos 1 1 1 1 1 ] [ ] [ ] [

1 1 1 1 1 1

φ φ φ φ α α α α

φ α

z y x T T T

y x R

R R T

( ) ( ) ( ) ( ) ( )

] ][ ][ ][ [ ] [ ] [ ] [ ] [

1 1 1

R x y z y x R arb

T R R R R R T R

T T T T T T T T

α φ θ φ α − − −

=

Rotation about an Arbitrary Axis

Find the new coordinates of a unit cube 90º-rotated about an axis defined by its endpoints A(2,1,0) and B(3,3,1).

Rotation about an Arbitrary Axis

Step1. Translate point A (2,1,0) to the origin

A’(0,0,0)

x z y

B’(1,2,1)

            − − = 1 1 1 1 2 1 ] [

R

T

SLIDE 6

6 Rotation about an Arbitrary Axis

x z y l

( )

[ ]

                − = 1 5 5 5 5 2 5 5 2 5 5 1

α

x

R

T 6 1 2 1 5 5 5 1 cos 5 5 2 5 2 1 2 2 sin

2 2 2 2 2

= + + = = = α = = + = α l

B’(1,2,1)

α

Projected point (0,2,1) B”(1,0,√5)

Step 2. Rotate axis A’B’ about the x axis by and angle α, until it lies
n the xz plane.

Rotation about an Arbitrary Axis

x z y l

( )

[ ]

                − = 1 6 30 6 6 1 6 6 6 30

φ

y

R

T 6 30 6 5 cos 6 6 6 1 sin = = φ = = φ φ

B”(1,0, √ 5) (0,0,√6)

Step 3. Rotate axis A’B’’ about the y axis by and angle φ,

until it coincides with the z axis.

Rotation about an Arbitrary Axis

( )

[ ]

            − =

°

1 1 1 1

90

z

R

T

Finally, the concatenated rotation matrix about the arbitrary axis AB becomes,

Step 4. Rotate the cube 90° about the z axis

( ) ( )

( )

( ) ( )

] ][ ][ ][ [ ] [ ] [ ] [ ] [

90 1 1 1

R x y z y x R arb

T R R R R R T R

T T T T T T T T

α φ φ α

−

− −

=

Rotation about an Arbitrary Axis

            − − − =             − −                 −                 −             −                 −                 −             = 1 560 . 167 . 741 . 650 . 151 . 1 075 . 667 . 742 . 742 . 1 983 . 075 . 166 . 1 1 1 1 2 1 1 5 5 5 5 2 5 5 2 5 5 1 1 6 30 6 6 1 6 6 6 30 1 1 1 1 1 6 30 6 6 1 6 6 6 30 1 5 5 5 5 2 5 5 2 5 5 1 1 1 1 1 2 1 ] [

arb

R

T

SLIDE 7

7 Rotation about an Arbitrary Axis

[ ] [ ] [ ]

P T P

arb

R ⋅

=

*

[ ]

            − − − − − − − =                         − − − = 1 1 1 1 1 1 1 1 076 . 091 . 560 . 726 . 817 . 650 . 301 . 1 467 . 1 483 . 409 . 151 . 1 225 . 1 184 . 258 . 484 . 558 . 891 . 2 909 . 1 742 . 1 725 . 2 816 . 2 834 . 1 667 . 1 650 . 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 560 . 167 . 741 . 650 . 151 . 1 075 . 667 . 742 . 742 . 1 983 . 075 . 166 .

*

P

Multiplying [TR]AB by the point matrix of the original cube

Rotation about an Arbitrary Axis

Reflection Relative to the xy Plane
Z-axis Shear

x z y x z y

                        − =             1 1 1 1 1 1 z y x z y x ' ' '                         =             1 1 1 1 1 1 ' ' ' z y x b a z y x

Q1 - Translate by <1, 1, 1>

A translation by an offset (tx, ty, tz) is achieved

using the following matrix:

So to translate by a vector

(1, 1, 1), the matrix is simply:             = 1 1 1 1 ) , , (

z y x z y x T

t t t t t t M

            = 1 1 1 1 1 1 1 ) 1 , 1 , 1 (

T

M

Q2- Rotate by 45 degrees about x axis

So to rotate by 45 degrees about the x-axis, we

use the following matrix:

                − =             − = 1 2 1 2 1 2 1 2 1 1 1 45 cos 45 sin 45 sin 45 cos 1 ) 45 (

x

R

SLIDE 8

8 Q3 - Rotate by 45 about axis <1, 1, 1>

So a rotation by 45 degrees about <1, 1, 1> can

be achieved by a few succesive rotations about the major axes. Which can be represented as a single composite transformation 414 . 1 2

2 2

= = + =

z x

n n d 45 1 1 tan tan

1 1

= = =

− − z x

n n α 264 . 35 2 1 tan tan

1 1

= = =

− −

d n y β nx = 1 ny = 1 nz = 1 SO

Q3 - Arbitrary Axis Rotation

The composite transformation can then be
btained as follows:

) ( ) ( ) ( ) ( ) ( ) 1 , 1 , 1 (

1 1

α β θ β α

y x z R

R R R R R M

x y

=

− −             − 1 1 ) 45 cos( ) 45 sin( ) 45 sin( ) 45 cos(             − − − − − 1 ) 45 cos( ) 45 sin( 1 ) 45 sin( ) 45 cos(             − 1 ) 2 . 35 cos( ) 2 . 35 sin( ) 2 . 35 sin( ) 2 . 35 cos( 1             − − − − − 1 ) 2 . 35 cos( ) 2 . 35 sin( ) 2 . 35 sin( ) 2 . 35 cos( 1

            − 1 ) 45 cos( ) 45 sin( 1 ) 45 sin( ) 45 cos(

=

?? x y z

Directions vs. Points

We have looked at transforming points
Directions are also important in graphics
Viewing directions
Normal vectors
Ray directions
Directions are represented by vectors, like points, and can

be transformed, but not like points

Say we define a direction as the difference of two points:

d=a–b. This represents the direction of the line between two points

Now we translate the points by the same amount:

a’=a+t, b’=b+t

Have we transformed d?

(1,1) (-2,-1)

Homogeneous Directions

Translation does not affect directions!
Homogeneous coordinates give us a clear way of handling

this, e.g., direction (x,y) becomes homogeneous direction (x,y,0), and remains the same after translation:

(x, y, 0) is a vector, (x,y,1) is a point.
The same applies to rotation and scaling, e.g., scaling

changes the length of vector, but not direction

Normal vectors are slightly different though (can’t always

use the matrix for points to transform the normal vector)

          =                     1 1 1 y x y x b b

y x

SLIDE 9

9 Alternative Rotations

Specify the rotation axis and the angle (OpenGL method)
Euler angles: Specify how much to rotate about X, then how

much about Y, then how much about

These are hard to think about, and hard to compose
Quaternions
4-vector related to axis and angle, unit magnitude, e.g.,

rotation about axis (nx,ny,nz) by angle θ:

Only normalized quaternions represent rotations, but you can

normalize them just like vectors, so it isn’t a problem

But we don’t want to learn all the maths about quaternions in

this module, because we have to learn how to create a basic application before trying to make rotation faster

( ) ( ) ( ) ( )

( )

2 / sin , 2 / cos , 2 / cos , 2 / cos θ θ θ θ

z y x

n n n

OpenGL Transformations

OpenGL internally stores two matrices that control viewing
f the scene
The GL_MODELVIEW matrix for modelling and world to view

transformations

The GL_PROJECTION matrix captures the view to canonical

conversion

Mapping from canonical view volume into window space is through a

glViewport function call

Matrix calls, such as glRotate, glTranslate, glScale right