lecture05.ppt 1132KB Jun 23 2011 12:09:44 PM

Introduction to

Computer Graphics

CS 445 / 645

Lecture 5

Lecture 5

Transformations

Transformations Lecture 5

Lecture 5

Transformations Transformations

Modeling Transformations

Specify transformations for objects

Specify transformations for objects

• _{Allows definitions of objects in own coordinate systemsAllows definitions of objects in own coordinate systems} • Allows use of object definition multiple times in a scene_{Allows use of object definition multiple times in a scene}

–Remember how OpenGL provides a transformation stack _{Remember how OpenGL provides a transformation stack}

because they are so frequently reused

because they are so frequently reused

Chapter 5 from Hearn and Baker

Chapter 5 from Hearn and Baker

Specify transformations for objects Specify transformations for objects

• Allows definitions of objects in own coordinate systems_{Allows definitions of objects in own coordinate systems}

• Allows use of object definition multiple times in a scene_{Allows use of object definition multiple times in a scene}

–_{Remember how OpenGL provides a transformation stack Remember how OpenGL provides a transformation stack}

because they are so frequently reused

because they are so frequently reused

Chapter 5 from Hearn and Baker Chapter 5 from Hearn and Baker

Overview

2D Transformations

2D Transformations

• _{Basic 2D transformationsBasic 2D transformations} • Matrix representation_{Matrix representation}

• Matrix composition_{Matrix composition}

3D Transformations

3D Transformations

• Basic 3D transformations_{Basic 3D transformations} • Same as 2D_{Same as 2D}

2D Transformations 2D Transformations

• Basic 2D transformations_{Basic 2D transformations}

• Matrix representation_{Matrix representation}

• _{Matrix compositionMatrix composition} 3D Transformations 3D Transformations

• _{Basic 3D transformationsBasic 3D transformations} • Same as 2D_{Same as 2D}

2D Modeling Transformations

Scale Rotate Translate

Scale Translate

x y

World Coordinates Modeling

2D Modeling Transformations

x y

World Coordinates Modeling

Coordinates

Let’s look at this in detail…

2D Modeling Transformations

x y

Modeling Coordinates

Initial location at (0, 0) with x- and y-axes aligned

2D Modeling Transformations

x y

Modeling Coordinates

Scale .3, .3

Rotate -90 Translate 5, 3

2D Modeling Transformations

x y

Modeling Coordinates

Scale .3, .3

Rotate -90

2D Modeling Transformations

x y

Modeling Coordinates

Scale .3, .3 Rotate -90

Translate 5, 3

Scaling

Scaling

Scaling a coordinate means multiplying each of its a coordinate means multiplying each of its components by a scalar

components by a scalar

Uniform scaling

Uniform scaling means this scalar is the same for all means this scalar is the same for all components:

components:

Scaling

Scaling a coordinate means multiplying each of its a coordinate means multiplying each of its components by a scalar

components by a scalar

Uniform scaling

Uniform scaling means this scalar is the same for all means this scalar is the same for all components:

components:

Non-uniform scaling

Non-uniform scaling: different scalars per component:: different scalars per component:

How can we represent this in matrix form?

How can we represent this in matrix form? Non-uniform scaling

Non-uniform scaling: different scalars per component:: different scalars per component:

How can we represent this in matrix form?

How can we represent this in matrix form?

Scaling

X  2,

Scaling

Scaling operation:

Scaling operation:

Or, in matrix form:

Or, in matrix form: Scaling operation: Scaling operation:

Or, in matrix form: Or, in matrix form:

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

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by

ax

y

x

'

'

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

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

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y

x

b

a

y

x

0

0

'

'

2-D Rotation



(x, y)

(x’, y’)

x’ = x

cos

(



) - y

sin

(



)

y’ = x

sin

(



) + y

cos

(



)

2-D Rotation

x = r cos ()

y = r sin ()

x’ = r cos ( + )

y’ = r sin ( + )

Trig Identity…

x’ = r cos() cos() – r sin() sin()

y’ = r sin() sin() + r cos() cos()

Substitute…

x’ = x cos() - y sin()

y’ = x sin() + y cos()



(x, y)

(x’, y’)

2-D Rotation

This is easy to capture in matrix form: This is easy to capture in matrix form:

Even though sin(

Even though sin() and cos() and cos() are nonlinear functions ) are nonlinear functions of

of ,,

• x’ is a linear combination of x and y_{x’ is a linear combination of x and y} • y’ is a linear combination of x and y_{y’ is a linear combination of x and y}

This is easy to capture in matrix form:

This is easy to capture in matrix form:

Even though sin(

Even though sin() and cos() and cos() are nonlinear functions ) are nonlinear functions

of

of ,,

• x’ is a linear combination of x and y_{x’ is a linear combination of x and y}

• y’ is a linear combination of x and y_{y’ is a linear combination of x and y}

 

 

 

 

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       

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 

 

  

 

y x y

x

 

 

cos sin

sin cos

' '

Basic 2D Transformations

Translation:

Translation:

• x’ = x + tx’ = x + tx_x • y’ = y + ty’ = y + ty_y

Scale:

Scale:

• x’ = x * sx’ = x * s_xx • y’ = y * sy’ = y * sy_y

Shear:

Shear:

• x’ = x + hx’ = x + hx_x*y*y • y’ = y + h_{y’ = y + h}y_y*x*x

Rotation:

Rotation:

• x’ = x*cos_{x’ = x*cos} - y*sin - y*sin

• y’ = x*sin_{y’ = x*sin} + y*cos + y*cos

Translation:

Translation:

• x’ = x + tx’ = x + tx_x

• y’ = y + ty’ = y + tyy

Scale:

Scale:

• x’ = x * sx’ = x * sx_x

• y’ = y * sy’ = y * syy

Shear:

Shear:

• x’ = x + hx’ = x + h_xx*y*y

• y’ = y + hy’ = y + h_yy*x*x Rotation:

Rotation:

• x’ = x*cos_{x’ = x*cos} - y*sin - y*sin

• y’ = x*sin_{y’ = x*sin} + y*cos + y*cos

Transformations can be combined (with simple algebra)

Basic 2D Transformations

Translation:

Translation:

• x’ = x + tx’ = x + tx_x • y’ = y + ty’ = y + ty_y

Scale:

Scale:

• x’ = x * sx’ = x * s_xx • y’ = y * sy’ = y * sy_y

Shear:

Shear:

• x’ = x + hx’ = x + hx_x*y*y • y’ = y + h_{y’ = y + h}y_y*x*x

Rotation:

Rotation:

• x’ = x*cos_{x’ = x*cos} - y*sin - y*sin

• y’ = x*sin_{y’ = x*sin} + y*cos + y*cos

Translation:

Translation:

• x’ = x + tx’ = x + tx_x

• y’ = y + ty’ = y + tyy

Scale:

Scale:

• x’ = x * sx’ = x * sx_x

• y’ = y * sy’ = y * syy

Shear:

Shear:

• x’ = x + hx’ = x + h_xx*y*y

• y’ = y + hy’ = y + h_yy*x*x Rotation:

Rotation:

• x’ = x*cos_{x’ = x*cos} - y*sin - y*sin

Basic 2D Transformations

Translation:

Translation:

• x’ = x + tx’ = x + tx_x • y’ = y + ty’ = y + ty_y

Scale:

Scale:

• x’ = x x’ = x * s* s_xx • y’ = y y’ = y * s* s_yy

Shear:

Shear:

• x’ = x + hx’ = x + hx_x*y*y • y’ = y + h_{y’ = y + h}y_y*x*x

Rotation:

Rotation:

• x’ = x*cos_{x’ = x*cos} - y*sin - y*sin

• y’ = x*sin_{y’ = x*sin} + y*cos + y*cos

Translation:

Translation:

• x’ = x + tx’ = x + tx_x

• y’ = y + ty’ = y + tyy

Scale:

Scale:

• x’ = x x’ = x * s* sx_x

• y’ = y y’ = y * s* syy

Shear:

Shear:

• x’ = x + hx’ = x + h_xx*y*y

• y’ = y + hy’ = y + h_yy*x*x Rotation:

Rotation:

• x’ = x*cos_{x’ = x*cos} - y*sin - y*sin

• y’ = x*sin_{y’ = x*sin} + y*cos + y*cos

x’ = x*s_x y’ = y*s_y

x’ = x*s_x

y’ = y*s_y

(x,y)

Basic 2D Transformations

x’ = (x*s_x)*cos - (y*s_y)*sin

y’ = (x*s_x)*sin + (y*s_y)*cos x’ = (x*s_x)*cos - (y*s_y)*sin

y’ = (x*s_x)*sin + (y*s_y)*cos

(x’,y’)

Translation:

Translation: • x’ = x + tx’ = x + tx_x

• y’ = y + ty’ = y + ty_y

Scale:

Scale:

• x’ = x * sx’ = x * s_xx

• y’ = y * sy’ = y * s_yy

Shear:

Shear:

• x’ = x + hx’ = x + hx_x*y*y

• y’ = y + hy’ = y + h_yy*x*x

Rotation:

Rotation:

• x’ = x*x’ = x*coscos - y* - y*sinsin

• y’ = x*_{y’ = x*}sinsin + y* + y*coscos

Translation:

Translation:

• x’ = x + tx’ = x + t_xx

• y’ = y + ty’ = y + t_yy Scale:

Scale:

• x’ = x * sx’ = x * sx_x

• y’ = y * sy’ = y * sy_y

Shear:

Shear:

• x’ = x + hx’ = x + hx_x*y*y

• y’ = y + hy’ = y + h_yy*x*x

Rotation:

Rotation:

• x’ = x*_{x’ = x*}coscos - y* - y*sinsin

Basic 2D Transformations

Translation:

Translation:

• x’ = x x’ = x + t+ tx_x • y’ = y y’ = y + t+ ty_y

Scale:

Scale:

• x’ = x * sx’ = x * s_xx • y’ = y * sy’ = y * sy_y

Shear:

Shear:

• x’ = x + hx’ = x + hx_x*y*y • y’ = y + h_{y’ = y + h}y_y*x*x

Rotation:

Rotation:

• x’ = x*cos_{x’ = x*cos} - y*sin - y*sin

• y’ = x*sin_{y’ = x*sin} + y*cos + y*cos

Translation:

Translation:

• x’ = x x’ = x + t+ tx_x

• y’ = y y’ = y + t+ tyy

Scale:

Scale:

• x’ = x * sx’ = x * sx_x

• y’ = y * sy’ = y * syy

Shear:

Shear:

• x’ = x + hx’ = x + h_xx*y*y

• y’ = y + hy’ = y + h_yy*x*x Rotation:

Rotation:

• x’ = x*cos_{x’ = x*cos} - y*sin - y*sin

• y’ = x*sin_{y’ = x*sin} + y*cos + y*cos

x’ = ((x*s_x)*cos - (y*s_y)*sin) + t_x

y’ = ((x*s_x)*sin + (y*s_y)*cos) + t_y x’ = ((x*s_x)*cos - (y*s_y)*sin) + t_x

y’ = ((x*s_x)*sin + (y*s_y)*cos) + t_y

Basic 2D Transformations

Translation:

Translation:

• x’ = x + tx’ = x + tx_x • y’ = y + ty’ = y + ty_y

Scale:

Scale:

• x’ = x * sx’ = x * s_xx • y’ = y * sy’ = y * sy_y

Shear:

Shear:

• x’ = x + hx’ = x + hx_x*y*y • y’ = y + h_{y’ = y + h}y_y*x*x

Rotation:

Rotation:

• x’ = x*cos_{x’ = x*cos} - y*sin - y*sin

• y’ = x*sin_{y’ = x*sin} + y*cos + y*cos

Translation:

Translation:

• x’ = x + tx’ = x + tx_x

• y’ = y + ty’ = y + tyy

Scale:

Scale:

• x’ = x * sx’ = x * sx_x

• y’ = y * sy’ = y * syy

Shear:

Shear:

• x’ = x + hx’ = x + h_xx*y*y

• y’ = y + hy’ = y + h_yy*x*x Rotation:

Rotation:

• x’ = x*cos_{x’ = x*cos} - y*sin - y*sin

• y’ = x*sin_{y’ = x*sin} + y*cos + y*cos

x’ = ((x*s_x)*cos - (y*s_y)*sin) + t_x

y’ = ((x*s_x)*sin + (y*s_y)*cos) + t_y x’ = ((x*s_x)*cos - (y*s_y)*sin) + t_x

Overview

2D Transformations

2D Transformations

• _{Basic 2D transformationsBasic 2D transformations} • Matrix representation_{Matrix representation}

• Matrix composition_{Matrix composition}

3D Transformations

3D Transformations

• Basic 3D transformations_{Basic 3D transformations} • Same as 2D_{Same as 2D}

2D Transformations 2D Transformations

• Basic 2D transformations_{Basic 2D transformations}

• Matrix representation_{Matrix representation}

• _{Matrix compositionMatrix composition} 3D Transformations 3D Transformations

• _{Basic 3D transformationsBasic 3D transformations} • Same as 2D_{Same as 2D}

Matrix Representation

Represent 2D transformation by a matrix

Represent 2D transformation by a matrix

Multiply matrix by column vector

Multiply matrix by column vector





apply transformation to point _{apply transformation to point}

Represent 2D transformation by a matrix Represent 2D transformation by a matrix

Multiply matrix by column vector Multiply matrix by column vector





apply transformation to point apply transformation to point

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y x d

c b a

y x

' '

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

d

c b

a

dy cx

y

by ax

x

 

 

' '

Matrix Representation

Transformations combined by multiplication

Transformations combined by multiplication Transformations combined by multiplication Transformations combined by multiplication

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   

   

  

 



y x l

k

j i

h g

f e

d c b a

y x

' '

Matrices are a convenient and efficient way to represent a sequence of transformations!

2x2 Matrices

What types of transformations can be

What types of transformations can be

represented with a 2x2 matrix?

represented with a 2x2 matrix?

What types of transformations can be What types of transformations can be

represented with a 2x2 matrix? represented with a 2x2 matrix?

2D Identity? y y x x   ' '              y x y x 1 0 0 1 ' '

2D Scale around (0,0)?

y s y x s x y x * ' * '                      y x s s y x y x 0 0 ' '

2x2 Matrices

What types of transformations can be

What types of transformations can be

represented with a 2x2 matrix?

represented with a 2x2 matrix?

What types of transformations can be What types of transformations can be

represented with a 2x2 matrix? represented with a 2x2 matrix?

2D Rotate around (0,0)?

y x

y x y

x

* cos

* sin

' cos * sin * '                                 y x y x cos sin sin cos ' ' 2D Shear? y x sh y y sh x x y x     * ' * '                    y x sh sh y x y x 1 1 ' '

2x2 Matrices

What types of transformations can be

What types of transformations can be

represented with a 2x2 matrix?

represented with a 2x2 matrix?

What types of transformations can be What types of transformations can be

represented with a 2x2 matrix? represented with a 2x2 matrix?

2D Mirror about Y axis?

y y x x   ' '              y x y x 1 01 0 '

'

2D Mirror over (0,0)?

y y x x    ' '                y x y x 1 01 0 '

2x2 Matrices

What types of transformations can be

What types of transformations can be

represented with a 2x2 matrix?

represented with a 2x2 matrix?

What types of transformations can be What types of transformations can be

represented with a 2x2 matrix? represented with a 2x2 matrix?

2D Translation? y

x

t y

y

t x

x

 

 

' '

Only linear 2D transformations

can be represented with a 2x2 matrix NO!

Linear Transformations

Linear transformations are combinations of …

Linear transformations are combinations of … • Scale,Scale,

• Rotation,_Rotation, • Shear, andShear, and

• MirrorMirror

Properties of linear transformations:

Properties of linear transformations: • Satisfies:Satisfies:

• Origin maps to originOrigin maps to origin

• Lines map to lines_{Lines map to lines}

• Parallel lines remain parallel_{Parallel lines remain parallel} • Ratios are preservedRatios are preserved

• Closed under composition_{Closed under composition}

Linear transformations are combinations of …

Linear transformations are combinations of …

• Scale,_Scale,

• Rotation,Rotation,

• Shear, and_{Shear, and}

• Mirror_Mirror

Properties of linear transformations:

Properties of linear transformations:

• Satisfies:Satisfies:

• Origin maps to origin_{Origin maps to origin}

• Lines map to linesLines map to lines

• Parallel lines remain parallelParallel lines remain parallel

• Ratios are preserved_{Ratios are preserved}

• Closed under composition_{Closed under composition}

) (

) (

)

(s₁p₁ s₂p₂ s₁T p₁ s₂T p₂

T   

         

        

y x d

c

b a

y x ' '

Homogeneous Coordinates

Q: How can we represent translation as a 3x3 matrix?

Q: How can we represent translation as a 3x3 matrix? Q: How can we represent translation as a 3x3 matrix? Q: How can we represent translation as a 3x3 matrix?

y x

t y

y

t x

x

 

 

' '

Homogeneous Coordinates

Homogeneous coordinates

Homogeneous coordinates

• represent coordinates in 2 _{represent coordinates in 2}

dimensions with a 3-vector

dimensions with a 3-vector

Homogeneous coordinates

Homogeneous coordinates

• represent coordinates in 2 _{represent coordinates in 2}

dimensions with a 3-vector

dimensions with a 3-vector

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1

coords s

homogeneou

_y

x

y

x

Homogeneous coordinates seem unintuitive, but they Homogeneous coordinates seem unintuitive, but they

make graphics operations

make graphics operations muchmuch easier easier

Homogeneous coordinates seem unintuitive, but they

Homogeneous coordinates seem unintuitive, but they

make graphics operations

Homogeneous Coordinates

Q: How can we represent translation as a 3x3 matrix?

Q: How can we represent translation as a 3x3 matrix?

A: Using the rightmost column:

A: Using the rightmost column:

Q: How can we represent translation as a 3x3 matrix? Q: How can we represent translation as a 3x3 matrix?

A: Using the rightmost column: A: Using the rightmost column:

     

    

1 0

0

1 0

0 1

y x

t t ranslation

T

y x

t y

y

t x

x

 

 

' '

Translation

Example of translation

Example of translation





Example of translation

Example of translation

                                              1 1 1 0 0 1 0 0 1 1 ' ' y x y x t y t x y x t t y x

t_x = 2 t_y = 1

Homogeneous Coordinates

Homogeneous Coordinates Homogeneous Coordinates

Homogeneous Coordinates

Add a 3rd coordinate to every 2D point

Add a 3rd coordinate to every 2D point

• (x, y, w) represents a point at location (x/w, y/w)_{(x, y, w) represents a point at location (x/w, y/w)} • (x, y, 0) represents a point at infinity_{(x, y, 0) represents a point at infinity}

• (0, 0, 0) is not allowed_{(0, 0, 0) is not allowed}

Add a 3rd coordinate to every 2D point

Add a 3rd coordinate to every 2D point • (x, y, w) represents a point at location (x/w, y/w)_{(x, y, w) represents a point at location (x/w, y/w)}

• (x, y, 0) represents a point at infinity_{(x, y, 0) represents a point at infinity}

• (0, 0, 0) is not allowed_{(0, 0, 0) is not allowed}

Convenient coordinate system to

represent many useful transformations

1 2 1

2

(2,1,1) or (4,2,2) or (6,3,3)

x y

Basic 2D Transformations

Basic 2D transformations as 3x3 matrices

Basic 2D transformations as 3x3 matrices Basic 2D transformations as 3x3 matrices Basic 2D transformations as 3x3 matrices

                                    1 1 0 0 0 cos sin 0 sin cos 1 ' ' y x y x                                1 1 0 0 1 0 0 1 1 ' ' y x t t y x y x                                1 1 0 0 0 1 0 1 1 ' ' y x sh sh y x y x Translate Rotate Shear                                1 1 0 0 0 0 0 0 1 ' ' y x s s y x y x Scale

Affine Transformations

Affine transformations are combinations of …

Affine transformations are combinations of …

• Linear transformations, and_{Linear transformations, and} • Translations_Translations

Properties of affine transformations:

Properties of affine transformations:

• Origin does not necessarily map to origin_{Origin does not necessarily map to origin} • Lines map to lines_{Lines map to lines}

• Parallel lines remain parallel_{Parallel lines remain parallel} • Ratios are preserved_{Ratios are preserved}

• Closed under composition_{Closed under composition}

Affine transformations are combinations of …

Affine transformations are combinations of … • Linear transformations, and_{Linear transformations, and}

• Translations_Translations

Properties of affine transformations:

Properties of affine transformations: • Origin does not necessarily map to origin_{Origin does not necessarily map to origin}

• Lines map to lines_{Lines map to lines}

• Parallel lines remain parallel_{Parallel lines remain parallel}

• Ratios are preserved_{Ratios are preserved}

• Closed under composition_{Closed under composition}

            

           

w y x f

e

da b c w

y x

1 0

0 '

Projective Transformations

Projective transformations …

Projective transformations …

• Affine transformations, and_{Affine transformations, and} • Projective warps_{Projective warps}

Properties of projective transformations:

Properties of projective transformations:

• Origin does not necessarily map to origin_{Origin does not necessarily map to origin} • Lines map to lines_{Lines map to lines}

• Parallel lines do not necessarily remain parallel_{Parallel lines do not necessarily remain parallel} • Ratios are not preserved_{Ratios are not preserved}

• Closed under composition_{Closed under composition}

Projective transformations …

Projective transformations … • Affine transformations, and_{Affine transformations, and}

• Projective warps_{Projective warps}

Properties of projective transformations:

Properties of projective transformations: • Origin does not necessarily map to origin_{Origin does not necessarily map to origin}

• Lines map to lines_{Lines map to lines}

• Parallel lines do not necessarily remain parallel_{Parallel lines do not necessarily remain parallel}

• Ratios are not preserved_{Ratios are not preserved}

• Closed under composition_{Closed under composition}

            

        

  

w y x i

h g

f e

da b c w

y x ' ' '

Overview

2D Transformations

2D Transformations

• _{Basic 2D transformationsBasic 2D transformations} • Matrix representation_{Matrix representation}

• Matrix composition_{Matrix composition}

3D Transformations

3D Transformations

• Basic 3D transformations_{Basic 3D transformations} • Same as 2D_{Same as 2D}

2D Transformations 2D Transformations

• Basic 2D transformations_{Basic 2D transformations}

• Matrix representation_{Matrix representation}

• _{Matrix compositionMatrix composition} 3D Transformations 3D Transformations

• _{Basic 3D transformationsBasic 3D transformations} • Same as 2D_{Same as 2D}

Matrix Composition

Transformations can be combined by

Transformations can be combined by

matrix multiplication

matrix multiplication

Transformations can be combined by Transformations can be combined by

matrix multiplication matrix multiplication                                                       w y x sy sx tytx w y x 1 0 0 0 0 0 0 1

0

0 cos 0

sin sin 0

cos 1 0 0 1 0 0 1 ' ' '

Matrix Composition

Matrices are a convenient and efficient way to

Matrices are a convenient and efficient way to

represent a sequence of transformations

represent a sequence of transformations

• General purpose representation_{General purpose representation} • Hardware matrix multiply_{Hardware matrix multiply}

Matrices are a convenient and efficient way to Matrices are a convenient and efficient way to

represent a sequence of transformations represent a sequence of transformations

• General purpose representation_{General purpose representation}

• Hardware matrix multiply_{Hardware matrix multiply}

p

’ = (T * (R * (S*

p

) ) )

Matrix Composition

Be aware: order of transformations matters

Be aware: order of transformations matters

–_{Matrix multiplication is not commutativeMatrix multiplication is not commutative}

Be aware: order of transformations matters Be aware: order of transformations matters

–Matrix multiplication is not commutative_{Matrix multiplication is not commutative}

p

’ = T * R * S *

p

Matrix Composition

What if we want to rotate

What if we want to rotate andand translate? translate?

• Ex: _Ex: Rotate line segment by 45 degrees about endpoint Rotate line segment by 45 degrees about endpoint aa and lengthen

and lengthen

What if we want to rotate

What if we want to rotate andand translate? translate?

• _{Ex: Ex:} Rotate line segment by 45 degrees about endpoint _{Rotate line segment by 45 degrees about endpoint} aa and lengthen

and lengthen

Multiplication Order – Wrong Way

Our line is defined by two endpoints

Our line is defined by two endpoints

• Applying a rotation of 45 degrees, R(45), affects both points_{Applying a rotation of 45 degrees, R(45), affects both points}

• We could try to translate both endpoints to return endpoint _{We could try to translate both endpoints to return endpoint} aa to its to its original position, but by how much?

original position, but by how much?

Our line is defined by two endpoints

Our line is defined by two endpoints

• Applying a rotation of 45 degrees, R(45), affects both points_{Applying a rotation of 45 degrees, R(45), affects both points}

• We could try to translate both endpoints to return endpoint _{We could try to translate both endpoints to return endpoint} aa to its to its original position, but by how much?

original position, but by how much?

Wrong Correct

T(-3) R(45) T(3)

R(45)

Multiplication Order - Correct

Isolate endpoint

Isolate endpoint aa from rotation effects from rotation effects

• First translate line so _{First translate line so} aa is at origin: T (-3) is at origin: T (-3)

• Then rotate line 45 degrees: R(45)_{Then rotate line 45 degrees: R(45)}

• Then translate back so _{Then translate back so} aa is where it was: T(3) is where it was: T(3)

Isolate endpoint

Isolate endpoint aa from rotation effects from rotation effects • First translate line so _{First translate line so} aa is at origin: T (-3) is at origin: T (-3)

• Then rotate line 45 degrees: R(45)_{Then rotate line 45 degrees: R(45)}

• _{Then translate back so Then translate back so aa is where it was: T(3) is where it was: T(3)}

a

a

a

Will this sequence of operations work?

Will this sequence of operations work?

Will this sequence of operations work?

Will this sequence of operations work?

Matrix Composition

                                                     1 ' ' 1 1 0 0 0 1 0 3 0 1 1 0 0 0 ) 45 cos( ) 45 sin( 0 ) 45 sin( ) 45 cos( 1 0 0 0 1 0 3 0 1 y x y x a a a a

Matrix Composition

After correctly ordering the matrices

After correctly ordering the matrices

Multiply matrices together

Multiply matrices together

What results is one matrix –

What results is one matrix – store it (on stack)!store it (on stack)! Multiply this matrix by the vector of each vertex

Multiply this matrix by the vector of each vertex

All vertices easily transformed with one matrix

All vertices easily transformed with one matrix

multiply

multiply

After correctly ordering the matrices

After correctly ordering the matrices

Multiply matrices together

Multiply matrices together

What results is one matrix –

What results is one matrix – store it (on stack)!store it (on stack)! Multiply this matrix by the vector of each vertex

Multiply this matrix by the vector of each vertex

All vertices easily transformed with one matrix

All vertices easily transformed with one matrix

multiply

Overview

2D Transformations

2D Transformations

• _{Basic 2D transformationsBasic 2D transformations} • Matrix representation_{Matrix representation}

• Matrix composition_{Matrix composition}

3D Transformations

3D Transformations

• Basic 3D transformations_{Basic 3D transformations} • Same as 2D_{Same as 2D}

2D Transformations

2D Transformations

• Basic 2D transformations_{Basic 2D transformations}

• Matrix representation_{Matrix representation}

• _{Matrix compositionMatrix composition} 3D Transformations 3D Transformations

• _{Basic 3D transformationsBasic 3D transformations} • Same as 2D_{Same as 2D}

3D Transformations

Same idea as 2D transformations

Same idea as 2D transformations

• _{Homogeneous coordinates: (x,y,z,w) Homogeneous coordinates: (x,y,z,w)} • 4x4 transformation matrices_{4x4 transformation matrices}

Same idea as 2D transformations Same idea as 2D transformations • Homogeneous coordinates: (x,y,z,w) _{Homogeneous coordinates: (x,y,z,w)}

• 4x4 transformation matrices_{4x4 transformation matrices}

               

          

   

wz y x p

o n

mi j k l

h g

f

e b c d

a wz

y x ' '' '

Basic 3D Transformations

                               wz y x wz y x 1 0 0

0 0 1 0

0 1 0 0

0 0 0 0

1 '' '                                      w z y x t t t w z y x z y x 1 0 0 0 1 0 0 0 1 0 0 0 1 ' ' '                                      w z y x s s s w z y x z y x 1 0 0 0 0 0 0 0 0 0 0 0 0 ' ' '                                wz y x wz y x 1 0 0

0 0 1 0

0 1 0 0

01 0 0 0 '

' '

Identity Scale

Basic 3D Transformations

                                   wz y x wz y x 1 0 0

0 0 1 0

0 cos 0 0 sin sin 0 0 cos

' ' ' Rotate around Z axis:

                                          w z y x w z y x 1 0 0 0 0 cos 0 sin 0 0 1 0 0 sin 0 cos ' ' ' Rotate around Y axis:

                                   wz y x wz y x 1 0 0

0 sin cos 0 0 cos sin 0

0 0 0 0

1 '

' ' Rotate around X axis:

Reverse Rotations

Q: How do you undo a rotation of

Q: How do you undo a rotation of R(R()?)? A: Apply the inverse of the rotation… R

A: Apply the inverse of the rotation… R-1-1(() = R(-) = R(-) )

How to construct R-1(

How to construct R-1(_) = R(-) = R(-_))

• Inside the rotation matrix: cos(_{Inside the rotation matrix: cos(}) = cos(-) = cos(-))

– The cosine elements of the inverse rotation matrix are unchanged_{The cosine elements of the inverse rotation matrix are unchanged}

• The sign of the sine elements will flip_{The sign of the sine elements will flip}

Therefore… R

Therefore… R-1-1(() = R(-) = R(-) = R) = RTT(())

Q: How do you undo a rotation of

Q: How do you undo a rotation of R(R()?)? A: Apply the inverse of the rotation… R

A: Apply the inverse of the rotation… R-1-1(() = R(-) = R(-) )

How to construct R-1(

How to construct R-1() = R(-) = R(-))

• Inside the rotation matrix: cos(_{Inside the rotation matrix: cos(}) = cos(-) = cos(-))

– The cosine elements of the inverse rotation matrix are unchanged_{The cosine elements of the inverse rotation matrix are unchanged}

• The sign of the sine elements will flip_{The sign of the sine elements will flip}

Therefore… R

Summary

Coordinate systems

Coordinate systems

• World vs. modeling coordinates_{World vs. modeling coordinates}

2-D and 3-D transformations

2-D and 3-D transformations

• Trigonometry and geometry_{Trigonometry and geometry} • Matrix representations_{Matrix representations}

• Linear vs. affine transformations_{Linear vs. affine transformations}

Matrix operations

Matrix operations

• Matrix composition_{Matrix composition}

Coordinate systems

Coordinate systems

• World vs. modeling coordinates_{World vs. modeling coordinates}

2-D and 3-D transformations

2-D and 3-D transformations • Trigonometry and geometry_{Trigonometry and geometry}

• Matrix representations_{Matrix representations}

• Linear vs. affine transformations_{Linear vs. affine transformations}

Matrix operations

Matrix operations • Matrix composition_{Matrix composition}

Overview

2D Transformations

2D Transformations

•

_{Basic 2D transformations}

_{Basic 2D transformations}

•

Matrix representation

_{Matrix representation}

•

Matrix composition

_{Matrix composition}

3D Transformations

3D Transformations

•

Basic 3D transformations

_{Basic 3D transformations}

•

Same as 2D

_{Same as 2D}

2D Transformations

2D Transformations

•

Basic 2D transformations

_{Basic 2D transformations}

•

Matrix representation

_{Matrix representation}

•

_{Matrix composition}

_{Matrix composition}

3D Transformations

3D Transformations

•

_{Basic 3D transformations}

_{Basic 3D transformations}

•

Same as 2D

_{Same as 2D}

3D Transformations

Same idea as 2D transformations

Same idea as 2D transformations

•

_{Homogeneous coordinates: (x,y,z,w)}

_{Homogeneous coordinates: (x,y,z,w)}

•

4x4 transformation matrices

_{4x4 transformation matrices}

Same idea as 2D transformations

Same idea as 2D transformations

•

Homogeneous coordinates: (x,y,z,w)

_{Homogeneous coordinates: (x,y,z,w)}

•

4x4 transformation matrices

_{4x4 transformation matrices}

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

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







































w

z

y

x

p

o

n

m

i

j

k

l

h

g

f

e

b

c

d

a

w

z

y

x

'

'

'

'

Basic 3D Transformations

                               wz y x wz y x 1 0 0

0 0 1 0

0 1 0 0

0 0 0 0

1 '' '                                      w z y x t t t w z y x z y x 1 0 0 0 1 0 0 0 1 0 0 0 1 ' ' '                                      w z y x s s s w z y x z y x 1 0 0 0 0 0 0 0 0 0 0 0 0 ' ' '                                wz y x wz y x 1 0 0

0 0 1 0

0 1 0 0

01 0 0 0 '

' '

Identity

Scale

Basic 3D Transformations

                                   wz y x wz y x 1 0 0

0 0 1 0

0 cos 0 0

sin sin 0 0

cos '

' '

Rotate around Z axis:

                                          w z y x w z y x 1 0 0 0 0 cos 0 sin 0 0 1 0 0 sin 0 cos ' ' '

Rotate around Y axis:

                                   wz y x wz y x 1 0 0

0 sin cos 0

0 cos sin 0

0 0 0 0

1 '

' '

Reverse Rotations

Q: How do you undo a rotation of

Q: How do you undo a rotation of





R(

R(





)?

)?

A: Apply the inverse of the rotation… R

A: Apply the inverse of the rotation… R

-1-1

(

(





) = R(-

) = R(-





)

)

How to construct R-1(

How to construct R-1(



_

) = R(-

) = R(-



_

)

)

• Inside the rotation matrix: cos(_{Inside the rotation matrix: cos(}) = cos(-) = cos(-))

– The cosine elements of the inverse rotation matrix are unchanged_{The cosine elements of the inverse rotation matrix are unchanged} • The sign of the sine elements will flip_{The sign of the sine elements will flip}

Therefore… R

Therefore… R

-1-1

(

(





) = R(-

) = R(-





) = R

) = R

TT

(

(





)

)

Q: How do you undo a rotation of

Q: How do you undo a rotation of





R(

R(





)?

)?

A: Apply the inverse of the rotation… R

A: Apply the inverse of the rotation… R

-1-1

(

(





) = R(-

) = R(-





)

)

How to construct R-1(

How to construct R-1(





) = R(-

) = R(-





)

)

• Inside the rotation matrix: cos(_{Inside the rotation matrix: cos(}) = cos(-) = cos(-))

– The cosine elements of the inverse rotation matrix are unchanged_{The cosine elements of the inverse rotation matrix are unchanged}

• The sign of the sine elements will flip_{The sign of the sine elements will flip}

Therefore… R

Summary

Coordinate systems

Coordinate systems

• World vs. modeling coordinates_{World vs. modeling coordinates}

2-D and 3-D transformations

2-D and 3-D transformations

• Trigonometry and geometry_{Trigonometry and geometry} • Matrix representations_{Matrix representations}

• Linear vs. affine transformations_{Linear vs. affine transformations}

Matrix operations

Matrix operations

• Matrix composition_{Matrix composition}

Coordinate systems

Coordinate systems

• World vs. modeling coordinates_{World vs. modeling coordinates}

2-D and 3-D transformations

2-D and 3-D transformations

• Trigonometry and geometry_{Trigonometry and geometry}

• Matrix representations_{Matrix representations}

• Linear vs. affine transformations_{Linear vs. affine transformations}

Matrix operations

Matrix operations

• Matrix composition_{Matrix composition}