TNM046: Datorgrafik Transformations

TNM046: Datorgrafik Transformations

Sasan Gooran

VT 2014

2

Linear Algebra

Transposition

x =

(

)

Scalar (dot) product:

x^t =

x₁ x₂ x₃

!

"

##

##

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&

&

&&

y^t =

y₁ y₂ y₃

!

"

##

##

$

%

&

&

&&

y =

(

)

x • 

y = 

x ∗ 

y^t = x₁y₁ + x₂y₂ + x₃y₃

Length (Norm):

Normalization:

x x

Linear Algebra

x = 1, 2, 2

( )

(

)

Example: Find the angle between the two vectors below!

(answer: 1.68 radians or 96.38 degrees)

x • 

y = 

x ⋅ 

y ⋅ cos(θ⁾

x • 

y > 0 →θ < 90^ x • 

y = 0 →θ = 90^ x • 

y < 0 →θ > 90^

"

#$$

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$$

4

Linear Algebra

Cross (vector) product:

x × 

y = ( x₂y₃ − x₃y₂ x₃y₁ − x₁y₃ x₁y₂ − x₂y₁ ) x × 

y =

ˆx ˆy ˆz

x₁ x₂ x₃ y₁ y₂ y₃

x y x × 

y

x y

y ×  x

x × 

y = 

x ⋅ 

y ⋅ sin(θ⁾ (

x × 

y)• 

x = ? (

x × 

y)• 

y = ?

Basic Transformations (2D)

Translation

P₁

P₂ T

P₁ = x₁ y₁

!

"

##

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&

& P₂ = x₂ y₂

!

"

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&

T = ( t _x, t_y )

O

x₂ = x₁ + t_x y₂ = y₁ + t_y

!

"

#

$#

OP→₂ = →OP₁ + →T

x y

à

Basic Transformations (2D)

Translation

What is the translation vector, from A to B?

from B to A?

15 5

3 6

A

12 2

5 8

B

6

Basic Transformations (2D)

Rotation

P₁ = x₁ y₁

!

"

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&

P₂ = x₂ y₂

!

"

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ϕ

θ r

r

x₂ = r cos(ϕ +θ) = r cosϕ ^cosθ − r sinϕ ^sinθ y₂ = r sin(ϕ +θ) = r sinϕ ^cosθ + r cosϕ ^sinθ

"

#$

%$

x₁ = r cosϕ y₁ = r sinϕ

!

"

#

$#

x₂ = x₁ cosθ − y₁sinθ

"

$

Basic Transformations (2D)

Rotation

x₂ y₂

!

"

##

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&

& = cosθ −sinθ

sinθ ^cosθ

!

"

# $

%& x₁ y₁

!

"

##

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&

&

P₂ = R ∗ P₁

Counter-clockwise rotation by angle θ about the coordinate origin

à

8

Basic Transformations (2D)

Rotation

P₁ = x₁ y₁

!

"

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&

P₂ = x₂ y₂

!

"

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θ r

r

Rotation about an arbitrary point M

x₂ = x_M + (x₁ − x_M )cosθ − (y₁ − y_M )sinθ

"

$

M = x_M y_M

!

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10

Basic Transformations (2D)

Scaling

Fixed point: Coordinate origin

x₂ = s_x ⋅ x₁ y₂ = s_y ⋅ y₁

"

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P₂ = S ∗ P₁ S = s_x 0

0 s_y

!

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à where

s_x and s_y > 0

Scaling values less than 1 reduces the size of the object Scaling values greater than 1 produce an enlargement

s_x=s_y: Uniform scaling, which maintains the relative object properties

Basic Transformations (2D)

Scaling

Arbitrary fixed point: (x_f, y_f) x₂ = x_f + s_x ⋅ (x₁ − x_f ) y₂ = y_f + s_y ⋅ (y₁ − y_f )

#

$%

&%

1 2 4

Object

Scaled sx=sy=2

Scaled sx=sy=0.5

fixed point: (0, 0)

1 2

Object Scaled sx=sy=2

Scaled sx=sy=0.5

fixed point: (2, 1) 3

Matrix representation

Homogenous coordinates (translation)

x₂ y₂

!

"

##

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&

& = x₁ y₁

!

"

##

$

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&

&+ t_x t_y

!

"

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&

In order to represent this with a matrix multiplication we add a third coordinate, 1.

x₂ y₂ 1

!

"

##

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&& =

1 0 t_x

0 1 t_y

0 0 1

!

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##

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&&

x₁ y₁ 1

!

"

##

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&

&&

P₂ = T ∗ P₁ à

12

Matrix representation

Homogenous coordinates (translation)

T is the translation matrix. What is the inverse matrix?

T =

1 0 t_x

0 1 t_y

0 0 1

!

"

##

##

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&

&

&&

T⁻¹ = ?

Matrix representation

Homogenous coordinates (Rotation)

x₂ y₂ 1

!

"

##

##

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&

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&& =

cosθ −sinθ ⁰

sinθ ^cosθ ⁰

0 0 1

!

"

##

#

$

%

&

&

&

x₁ y₁ 1

!

"

##

##

$

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&

&&

P₂ = R ∗ P₁

R is the rotation matrix. What is the inverse matrix?

R⁻¹ = ?

14

Matrix representation

Homogenous coordinates (Scaling)

x₂ y₂ 1

!

"

##

##

$

%

&

&

&& =

s_x 0 0

0 s_y 0

0 0 1

!

"

##

##

$

%

&

&

&&

x₁ y₁ 1

!

"

##

##

$

%

&

&

&&

P₂ = S ∗ P₁

S is the scaling matrix. What is the inverse matrix?

S⁻¹ = ?

Composite Transformation

P₂ = T (t_{x 2}, t_y2)∗ T (t

{

}

{

}

If two successive translation vectors (t_x1, t_y1) and (t_x2, t_y2)

are applied to a coordinate position P₁, the final transformed location P₂ is calculated:

1 0 t_{x 2}

0 1 t_y2

0 0 1

!

"

##

##

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&

&&

1 0 t_x1

0 1 t_y1

0 0 1

!

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##

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&& =

1 0 t_x1 + t_{x 2} 0 1 t_y1 + t_y2

0 0 1

!

"

##

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$

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&&

16

Composite Transformation

P₂ = R(θ₂)∗ R(

{

}

{

}

Two successive Rotations by θ₁ and θ₂:

s_{x 2} 0 0

0 s_y2 0

0 0 1

!

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##

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&&

s_x1 0 0

0 s_y1 0

0 0 1

!

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##

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&& =

s_x1 ⋅ s_{x 2} 0 0 0 s_y1 ⋅ s_y2 0

0 0 1

!

"

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&&

Two successive Scaling-s:

General Pivot-Point Rotation

Rotation about an arbitrary point (x_M, y_M):

x₂ y₂ 1

!

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&& =

1 0 x_M

0 1 y_M

0 0 1

!

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&&

cosθ −sinθ ⁰

sinθ ^cosθ ⁰

0 0 1

!

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&

1 0 −x_M

0 1 −y_M

0 0 1

!

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&&

x₁ y₁ 1

!

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&&

Step 1: Translate (x_M, y_M) to the coordinate origin (0,0) Step 2: Rotate about (0, 0)

Step 3: Translate back to (x_M, y_M)

x₂ y₂ 1

!

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##

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&& =

cosθ −sinθ ^x_M (1− cosθ) + y_M sinθ sinθ ^cosθ ^y_M (1− cosθ) − x_M sinθ

0 0 1

!

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##

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&&

x₁ y₁ 1

!

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&& ₁₈

General Fixed-Point Scaling

Scaling about an arbitrary point (x_f, y_f):

x₂ y₂ 1

!

"

##

##

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&

&& =

1 0 x_f

0 1 y_f

0 0 1

!

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##

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&

s_x 0 0

0 s_y 0

0 0 1

!

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##

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&&

1 0 −x_f

0 1 −y_f

0 0 1

!

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##

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x₁ y₁ 1

!

"

##

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&&

Step 1: Translate (x_f, y_f) to the coordinate origin (0,0) Step 2: Scale about (0, 0)

Step 3: Translate back to (x_f, y_f)

x₂ y₂

!#

#

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& =

s_x 0 x_f (1− s_x) 0 s_y y_f (1− s_y)

!#

#

$&

&

x₁ y₁

!#

#

$&

&

Ex: General Scaling Directions

We can scale an object in other directions by rotating the object to align the desired scaling directions

x y

S₁ S₂

θ Step 1: Rotate the object (by θ) to align

Step 2: Scale in the direction of (x, y) Step 3: Rotate back

20

Ex: General Scaling Directions

x₂ y₂ 1

!

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&&=

cosθ ^sinθ ⁰

−sinθ ^cosθ ⁰

0 0 1

!

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##

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$

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&

&

&

s₁ 0 0

0 s₂ 0

0 0 1

!

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&&

cosθ −sinθ ⁰

sinθ ^cosθ ⁰

0 0 1

!

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&

x₁ y₁ 1

!

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##

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&&

x₂ y₂ 1

!

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&& =

s₁ cos²θ + s₂ sin²θ ^(s₂ − s₁)sinθ ^cosθ ⁰ (s₂ − s₁)sinθ ^cosθ ^s₁^sin2θ + s₂ cos²θ ⁰

0 0 1

!

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x₁ y₁ 1

!

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&&

Reflection

1 0 0

0 −1 0

0 0 1

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'' '

Reflection matrix across y=0 (x-axis)

−1 0 0

0 1 0

0 0 1

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'' '

Reflection matrix across x=0 (y-axis)

−1 0 0

0 −1 0

0 0 1

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'' '

Reflection relative to the coordinate origin

22

Ex: Reflection in y=x

Step 1: Rotate counter clockwise (45 degrees) (y=x) à y-axis

Step 2: Reflect across the y-axis Step 3: Rotate back

x₂ y₂ 1

!

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##

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&&=

cosθ ^sinθ ⁰

−sinθ ^cosθ ⁰

0 0 1

!

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##

#

$

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&

&

&

−1 0 0

0 1 0

0 0 1

!

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cosθ −sinθ ⁰

sinθ ^cosθ ⁰

0 0 1

!

"

##

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&

x₁ y₁ 1

!

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##

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&&

θ = π ^{/ 4} x₂

y₂

!#

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& =

0 1 0 1 0 0

!#

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x₁ y₁

!#

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Shear

x₂ y₂ 1

!

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##

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&& =

1 sh_x 0

0 1 0

0 0 1

!

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&&

x₁ y₁ 1

!

"

##

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&&

x₂ = x₁ + sh_x ⋅ y₁ y₂ = y₁

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1 1

1 1

2 3

sh_x = 2

24

3D transformations, Translation

x₂ y₂ z₂ 1

!

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=

1 0 0 t_x

0 1 0 t_y

0 0 1 t_z

0 0 0 1

!

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&

&

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x₁ y₁ z₁ 1

!

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3D Rotation

x

P₂ =

x₂ = a y₂ z₂

!

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&&

θ r r

y z

About x-axis

Counter clockwise

P₁ =

x₁ = a y₁ z₁

!

"

##

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&&

When rotating around x-axis the x-coordinate is unchanged

26

3D Rotation

x₂ y₂ z₂ 1

!

"

##

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&

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=

1 0 0 0

0 cosθ −sinθ ⁰

0 sinθ ^cosθ ⁰

0 0 0 1

!

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##

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&

&

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&

x₁ y₁ z₁ 1

!

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Rotation about x-axis:

x₂ y₂ z₂ 1

!

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=

cosθ ^{0 sin}θ ⁰

0 1 0 0

−sinθ ^{0 cos}θ ⁰

0 0 0 1

!

"

##

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&

&

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&

x₁ y₁ z₁ 1

!

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Rotation about y-axis:

Why changing the signs in front of sinθ?

General 3D Rotation

Rotation about an axis parallel to one of the coordinate axes Step 1: Translate so that the rotation axis coincides with the parallel coordinate axis

Step 2: Perform the specified rotation about that axis Step 3: Translate back

R(α) = T⁻¹ ∗ R_co−axis(α^)*T

28

General 3D Rotation

Example: Counterclockwise rotation by 30 degrees about the axis passing through M1:(2, 1, 3) and M2:(2, 2, 3)

(vector from M1 to M2)

The vector going from M1 to M2 is (2 2 3)-(2 1 3)=(0 1 0), which is obviously parallel to y-axis

R(π 6 ) =

1 0 0 2 0 1 0 1 0 0 1 3 0 0 0 1

!

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##

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&

cos(π

6 ) 0 sin(π

6 ) 0

0 1 0 0

−sin(π

6 ) 0 cos(π

6 ) 0

0 0 0 1

!

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##

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&

1 0 0 −2 0 1 0 −1 0 0 1 −3

0 0 0 1

!

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General 3D Rotation

Rotation about an axis parallel to one of the coordinate planes Step 1: Translate the object so that the rotation axis passes through the coordinate origin. This vector is now on one of the coordinate planes.

Step 2: Rotate the object so that the axis of rotation coincides with one of the coordinate axes (one rotation is needed)

Step 3: Perform the specified rotation about that axis Step 4: Rotate back

Step 5: Translate back

30

R(α) = T⁻¹ ∗ Raxis −>co−axis

−1 ∗ R_co−axis(α)∗ Raxis−>co−axis ∗T

General 3D Rotation

Example: Counterclockwise rotation by 30 degrees about the axis passing through M1:(2, 1, 3) and M2:(2, 2, 4)

(vector from M1 to M2)

The vector going from M1 to M2 is (2 2 4)-(2 1 3)=(0 1 1), which is obviously parallel to YZ-plane (The translation is as before) It is easily seen that if we rotate (0, 1, 1) by 45 degrees and counterclockwise around X-axis, it will coincide with the Z-axis.

Raxis−>co _ axis =

1 0 0 0

0 cos(π

4) −sin(π

4) 0

0 sin(π

4) cos(π

4) 0

0 0 0 1

"

#

$$

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'' '' '' '

General 3D Rotation

Example: Continuation

R = T⁻¹

1 0 0 0

0 cos(π

4) sin(π

4) 0

0 −sin(π

4) cos(π

4) 0

0 0 0 1

"

#

$$

$$

$$

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'' '' '' '

cos(π

6) −sin(π

6) 0 0

sin(π

6) cos(π

6) 0 0

0 0 1 0

0 0 0 1

"

#

$$

$$

$$

$

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&

'' '' '' '

1 0 0 0

0 cos(π

4) −sin(π

4) 0

0 sin(π

4) cos(π

4) 0

0 0 0 1

"

#

$$

$$

$$

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'' '' '' '

T

32

Where,

T =

1 0 0 −2 0 1 0 −1 0 0 1 −3

0 0 0 1

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$$

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'' ''

General 3D Rotation

Rotation about an arbitrary axis that is not parallel to any coordinate axes, or coordinate planes.

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

Step 2: Rotate the object so that the axis of rotation coincides with one of the coordinate axes (two rotations are needed)

Step 3: Perform the specified rotation about the coordinate axis Step 4: Rotate back

Step 5: Translate back

R(α) = T⁻¹ ∗ Raxis −>co−axis

−1 ∗ R_co−axis(α)∗ Raxis−>co−axis ∗T

General 3D Rotation

M₂ =

x_m2 y_m2 z_m2

!

"

##

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&

&

&&

x

y z

M₁ =

x_m1 y_m1 z_m1

!

"

##

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&

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&

Eye

→U = →

M₁M₂ = (x_m2 − x_m1, y_m2 − y_m1, z_m2 − z_m1)

ˆu = U

U = (a, b, c) The vector is normalized

a² + b² + c² = ? ³⁴

General 3D Rotation

x

y z

After translation

(Step 1) ˆu = (a, b, c)

T =

1 0 0 −x_m1

0 1 0 −y_m1

0 0 1 −z_m1

0 0 0 1

"

#

$$

$$

$

%

&

'' '' '

General 3D Rotation

x

y z

ˆu = (a, b, c)

Two rotations are needed to make u coincide with one of the coordinate axes. (step 2)

Here we find the rotations to coincide it with z-axis

v = (a, b, 0) ϕ

The first rotation

is about the z-axis by φ to rotate u to the xz-plane (clockwise)

θ

The second rotation is about the y-axis by θ to align that with z-axis (clockwise)

36

General 3D Rotation

y z

ˆu = (a, b, c)

v = (a, b, 0) ϕ

The first rotation:

ˆx • 

v = ˆx ⋅ 

v cosϕ = a² + b² cosϕ ˆx • 

v = (1, 0, 0)⋅ (a, b, 0) = a ^à cosϕ = a d set d = a² + b²

sinϕ = ± 1− (a

d)² = ± b d

if b > 0 --> 0 ϕ  π --> sinϕ  0 if b < 0 --> π ϕ  2π --> sinϕ  0

Therefore:

sinϕ = b

General 3D Rotation

The second rotation:

ˆz• ˆu = ˆz ⋅ ˆu cosθ = cosθ

ˆz• ˆu = (0, 0,1)⋅ (a, b, c) = c ^à cosθ = c

sinθ ^{= 1− cos2}θ ^{= 1− c2 = d}

x

y z

ˆu = (a, b, c) θ

Since 0 ≤θ ≤ π ^{then sin}θ ≥ 0

Observe: a² + b² + c² = 1 and d = a² + b²

38

General 3D Rotation

R(α) = T⁻¹ ∗ Raxis −>co−axis

−1 ∗ R_co−axis(α)∗ Raxis−>co−axis ∗T

Raxis−>co−axis =

cosθ 0 −sinθ ⁰

0 1 0 0

sinθ ^{0 cos}θ ⁰

0 0 0 1

"

#

$$

$$

%

&

'' ''

cosϕ ^sinϕ ^{0 0}

−sinϕ ^cosϕ ^{0 0}

0 0 1 0

0 0 0 1

"

#

$$

$$$

%

&

'' '''

Raxis−>co−axis =

c 0 −d 0

0 1 0 0

d 0 c 0

"

$$

$

%' ''

a / d b / d 0 0

−b / d a / d 0 0

0 0 1 0

"

$$

$

%' ''

Notice: Clockwise rotations Step 2

General 3D Rotation

R(α) = T⁻¹ ∗ Raxis −>co−axis

−1 ∗ R_co−axis(α)∗ Raxis−>co−axis ∗T

Raxis−>co−axis

−1 =

a / d −b / d 0 0 b / d a / d 0 0

0 0 1 0

0 0 0 1

"

#

$$

$$

%

&

'' ''

c 0 d 0

0 1 0 0

−d 0 c 0

0 0 0 1

"

#

$$

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'' ''

Why?

T⁻¹ =

1 0 0 x_m1

0 1 0 y_m1

0 0 1 z_m1

0 0 0 1

"

#

$$

$$

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'' '' '

40

Step 4

Step 5

General 3D Rotation

R(α) = T⁻¹ ∗ Raxis −>co−axis

−1 ∗ R_co−axis(α)∗ Raxis−>co−axis ∗T

R_co−axis =

cos(α) −sin(α^{) 0 0}

sin(α^{) cos(}α^{) 0 0}

0 0 1 0

0 0 0 1

"

#

$$

$$$

%

&

'' '''

Notice that in the approach presented here we coincided the axis of rotation with z-axis. Therefore the coordinate axis is z-axis.

Step 3

General 3D Rotation Exercises

42

Exercise 1: Counterclockwise rotation by α about the axis passing through M1:(1, 2, 3) and M2:(1, 1, 5)

(vector from M1 to M2)

Exercise 2: Counterclockwise rotation by α about the axis passing through M1: (0, 0, 0) and M2: (-2, 3, -6) (vector from M1 to M2)

Parallel Projection

Just to make it consistent with the assignments in the labs, here we rearranged the x, y and z-axes. The xy plane is the view plane and the z-axis points towards the viewer.

If the parallel projection on the view plane

(xy-plane or z=0) occurs along the z-direction, then:

x y

P₁ = x₁ y₁

!#

###

$&

&

&&

P₂ =

x₂ y₂ z₂

!

"

##

##

$

%

&

&

&&

x₂ = x₁ y₂ = y₁ z₂ = 0

!

"

#

$

##

x₂ y₂ z

!#

##

#

$&

&

&

&

=

1 0 0 0 0 1 0 0 0 0 0 0

!#

##

$&

&

&

x₁ y₁ z

!#

##

#

$&

&

&

&

Parallel Projection

z

x y

If the view plane is parallel to xy-plane, z=z_v.

If the parallel projection on the view plane (z=z_v) occurs along the z-direction, then:

P₁ = x₁ y₁ z₁

!

"

##

##

$

%

&

&

&&

x₂ = x₁ y₂ = y₁ z₂ = z_v

!

"

#

$#

#

x₂ y₂ z₂ 1

!

"

##

##

#

$

%

&

&

&

&

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=

1 0 0 0

0 1 0 0

0 0 0 z_v

0 0 0 1

!

"

##

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$

%

&

&

&

&

x₁ y₁ z₁ 1

!

"

##

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$

%

&

&

&

&

&

P₂ =

x₂ y₂ z₂

!

"

##

##

$

%

&

&

&&

44

Coordinate System Transformation

x

y N

U V

The axes of the two coordinate systems are parallel.

O

O' =

x_o' y_o' z_o'

!

"

##

##

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&

&

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&

Assume that a point P has coordinates x

(

)

and coordinates u

(

)

World

View

T =

1 0 0 −x_o'

0 1 0 −y_o'

0 0 1 −z_o'

0 0 0 1

"

#

$$

$$

$

%

&

'' '' '

T transforms (translates) View system to World system.

Coordinate System Transformation

x₁ y₁ z₁ 1

!

"

##

##

#

$

%

&

&

&

&

&

=

1 0 0 x_o'

0 1 0 y_o'

0 0 1 z_o'

0 0 0 1

!

"

##

##

#

$

%

&

&

&

&

&

u₁ v₁ n₁ 1

!

"

##

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$

%

&

&

&

&

&

u₁ v₁ n₁ 1

!

"

##

##

#

$

%

&

&

&

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=

1 0 0 −x_o'

0 1 0 −y_o'

0 0 1 −z_o'

0 0 0 1

!

"

##

##

#

$

%

&

&

&

&

&

x₁ y₁ z₁ 1

!

"

##

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%

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&

In order not to get confused think of the coordinates of the origin of the View

system (O’), which is (0,0,0) in View, but

O' = x

(

)

in World system.

The axes of the two coordinate systems are parallel.

View World

World View ⁴⁶

Coordinate System Transformation

x

y N

U V

One of the axes are parallel (here z and N) but UV plane is rotated by θ in relation to xy plane.

O

O' = x( _o', y_o', z_o')^t World

View The view coordinate

system is transformed to World by a translation followed by a clockwise rotation about z-axis.

θ

Coordinate System Transformation

The z-axes are parallel but xy are rotated by θ.

x₁ y₁ z₁ 1

!

"

##

##

#

$

%

&

&

&

&

&

=

1 0 0 x_o'

0 1 0 y_o'

0 0 1 z_o'

0 0 0 1

!

"

##

##

#

$

%

&

&

&

&

&

cosθ −sinθ ^{0 0}

sinθ ^cosθ ^{0 0}

0 0 1 0

0 0 0 1

!

"

##

##

$

%

&

&

&

&

u₁ v₁ n₁ 1

!

"

##

##

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$

%

&

&

&

&

&

u₁ v₁ n₁ 1

!

"

##

##

#

$

%

&

&

&

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=

cosθ ^sinθ ^{0 0}

−sinθ ^cosθ ^{0 0}

0 0 1 0

0 0 0 1

!

"

##

##

$

%

&

&

&

&

1 0 0 −x_o'

0 1 0 −y_o'

0 0 1 −z_o'

0 0 0 1

!

"

##

##

#

$

%

&

&

&

&

&

x₁ y₁ z₁ 1

!

"

##

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$

%

&

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&

View World

World View ⁴⁸

Coordinate System Transformation

None of the axes are parallel

x

y N

U V

O

O' = x( _o', y_o', z_o')^t World

View First a translation is needed. Assume then that the unit vectors of the

coordinate axes of the View system can be expressed in World

coordinates.

ˆu = (u_x, u_y, u_z) ˆv = (v_x, v_y, v_z) ˆn = (n_x, n_y, n_z)