Linear and Affine Transformations

Linear and affine transformations

• Linear Algebra Review ▪ Matrices ▪ Transformations • Affine transformations in Euclidean space

Tricky examples of nonlinear transformations (Youtube)1

Geometric transformations

• Geometric transformations map points in one space to points in another: (x',y',z') = f(x,y,z), i.e. in vector form X’ = f(X) • These transformations can be very simple, such as scaling each coordinate, or complex, such as non-linear twists and bends. • We'll focus on transformations that can be represented easily with matrix operations. • We'll start in 2D...

1 2D Affine Transformations

• An affine transformation is any transformation that preserves co-linearity (i.e., all points lying on a line initially still lie on a line after transformation) and ratios of distances (e.g., the midpoint of a line segment remains the midpoint after transformation).

)

LENGTHS

ONGRUENCE

RESERVES

SOMETRY I

After any of those transformations (turn, flip or slide), the shape still has the same size, area, angles and line lengths.

2 ) When you resize a shape it gets bigger or smaller. ... but it still looks similar:

all angles are the same

ESIZING

ANGLES (R

The face and body are still in proportion

RESERVES

IMILARITY

P S

Affine transformation preserves parallelism,

dividing proportion, linearity and incidence. LINE Similarity and congruence can be viewed as a

special case of the affine transformation.

PROPORTION

PARALLEL

FFINITY

DIVIDING

RESERVES

AND P

3 )

ROJECTIVE The projective transformation does not

(P preserve parallelism, length, and angle. But it still preserves collinearity

and incidence.

OMOGRAPHY H

4 Linearity Parallel Angle Length

X X X

X X

Matrix Multiplication if A is an n × m matrix and B is an m × p matrix, their matrix product AB is an n × p matrix, in which the m entries across a row of A are multiplied with the m entries down a column of B and summed to produce an entry of AB.

A . B

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5 Matrix Multiplication is not commutative

122044  = 3412108

201224  = 1234710

Expand vector notation for linear equations

Solution x = Ax x = Ax xx    xx== ,    xx  21   yy     =    yy  01−   21 A =  x =+21 x y 01− y'0=− x y

6 ◼ The linear transformation given by a matrix

Let A be an 2  2 matrix. The function T defined by T(v) = Av is a linear transformation from R2 into R2.

◼ Note: xAx = xabx = ycdy xaxby =+ ycxdy' =+

◼ The linear transformation given by a matrix

Let A be an mn matrix. The function T defined by

is a linear transformation from Rn into Rm. Rn vector Rm vector ◼ Note:

a 11  aava 121111  v 1 a 12nn va 21 n v ++ + a   aava  v a va v ++ + Av == 21  222221  1 22nn 22 n         am 121  aava12 mmn 2 v nmmmn a van v ++ +

T(v) = Av T : Rn ⎯⎯ → Rm

7 ◼ Two representations of the linear transformation T:R3→R3 :

(1)T(x1, x2 , x3 ) = (2x1 + x2 − x3,−x1 + 3x2 − 2x3,3x2 + 4x3 )

 2 1 −1x1  (2)T(x) = Ax = −1 3 − 2x    2   0 3 4 x3 

◼ Three reasons for matrix representation of a linear transformation:

◼ It is simpler to write.

◼ It is simpler to read.

◼ It is more easily adapted for computer use.

Representation of 2D linear map

• We can represent a 2-D transformation M by a matrix a b M =   c d • If x is a column vector, M goes on the left: x' = Mx x   a b   x   =      y   c d   y  • If x is a row vector, MT goes on the right: xxM = T ac xyxy =    bd • We will use column vectors.

8 Property of Linear Transforms

• Basis vectors map to columns of matrix. • Origin (0,0) is always fixed point. • Composition of M and M-1 gives identity. • Determinant det(M) is scaling factor of the linear transformation described by the matrix M. xMx' = xMx = xabx = abxaxby  + ycdy Mx==   cdycxdy  +

abaabb  10      ==  cdccdd  01   (1, 0)

Scaling by 0.5 xMx =

M = .50  0.5

(1, 0) (0.5, 0)

(0, 1) (0, 0.5)

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9 Scaling by 0.5 x M = x 1 y d e t( )M = y 4 0.5 0 −1 M =  d e t( )M 4 = 0 0.5

−1 20 M =  x 02 x

Inverse mapping = scaling by 2

Composition of M and M-1 gives identity. Determinant is scaling factor of the linear transformation described by the matrix.

Homothety - Scaling

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10 Homothety - Scaling

Describe the transformation represented by matrix S = {{2,0},{0,2}}. Find all fixed points and directions. List all invariants. 20 S =  02 Fixed points: ';'XXXSX== xx= 2   xy==0,0 XSX= yy= 2  Only one fixed point (0, 0).

Fixed directions: v';'==vvSv xxx=−=2(2)0   vSv= yyy=−=2(2)0 = 2,,xRyR All directions are fixed.

General Scaling

xx = scaless(,)xy y y

1 scaless( xy, ) = sy

sx 0  0 sy

x x 1 sx

11 General Scaling

General Scaling

Describe the transformation represented by matrix S = {{2,0},{0,1}}. Find all fixed points and directions. List all invariants. 20 S =  01 Fixed points: X '== X ; X ' SX xx= 2   xyR=0, XSX= yy=  FP= (0,ttR ),.  

Fixed directions: v';== 'v vSv x=2 x x ( − 2) = 0 vS= v   y= y y( − 1) = 0  =2,x  R , y = 0; fd = ( t ,0)  =1,x = 0, y  R ; fd = (0, t ) 2 fixed directions: [(1,0)] and [(0,1)].

12 Scaling of a circle

x x = xy22+=1 xx = M a 22 −1  y xy xx= M y =  +=1 b ab

y a 0 y M =  0 b

b 1 0 −1 a x M =  a 1 1 x 0 b

Real image

? Driver’s eye 1m

5 m

13 Shear-x x M = x

y y

1 s  01

x x

Rotation xx = Rt()

cossin(tt) − ( ) rot (t) =  sincos(tt) ( )

t sin(t) cos(t)

-sin(t)

t cos(t)

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14 Trajectory of point A = (r, 0) in revolution xx = R

cossin() − ( ) RR==;det()1 sincos() ( ) y

xr cossin() − ( )  =  y sincos() ( ) 0 xr =cos A’ yr =sin 

A x GeoGebra book 2.1 Rotace

Exercise: Rotation xx = R()

Estimate parameter a so that matrix B represents revolution about origin. Find all fixed points and directions.

2 a − 2 B =  cos() − sin ( ) 2 R( ) =  a sin cos 2 ( ) ( )

1. method: comparing elements R and B. 22 sin= a = cos = 22 2. method: 2 det(Ba )= 1 2 + = 1 4 Matrix Representation of rotation

15 22 − 22 Rotation B =  22 Find all fixed points.  22 xxxx==B , xxx=−=BBEo ()

GeoGebra tool ReducedRowEchelonForm(M)eliminates non diagonal elements by row operations (= Gaussian elimination). 10 ()BE−  01 Using back-substitution, unknowns x, y can be solved for. Solution x = 0 and y = 0 gives only one fixed point FP = (0,0).

22 − 22 Rotation B =  22 Find all fixed directions.  22 x=B  x, x =  x x=B  x () B − E x = o

Matrix(B-E)must be singular for non trivial solutions x, but Det(B-E)=0 has no real solution.

22 −− 22 BE−== 0 22 −  22

General rotation hasn’t fixed directions.

16 Reflection in y-axis xx = r e f y

refy = −10  01

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Reflection in y-axis

y y

refy = −10  01

x x

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17 Line reflection

Reflection in the line y= x

18 Composing Linear Transformations

TM11()vv= • If T1 and T2 are transformations TM22()vv= ▪ T2 T1(v) =def T2( T1(v))

• If T1 and T2 are represented by matrices M1 and M2

▪ T2 T1 is represented by M2 M1

▪ T2 T1(v) = T2( T1(v)) = (M2 M1)(v)

• Order is important! reflect(x) (rot(O,훂)): A → A’ → A’’ rot(O,훂) (reflect(x)): A → A → A’

AA'Rot=

AA''Ref'= 37

Composing Linear Transformations

• Order is important! reflect(x) (rot(O,훂)): A → A’ → A’’ rot(O,훂) (reflect(x)): A → A → A’

AAAA'= Rot  '' = Ref  ' cos− sin   1 0  Rot==  ; Ref   sin cos   0− 1  cos sin Rot*Ref =  sin− cos cos− sin Ref*Rot =  −−sin cos 38

19 Composition of Linear Transformations

Composition of linear transformations 39

*Decomposing Linear Transformations • Any 2D Linear Transformation can be decomposed into the product of a rotation, a scale (or line reflection), and a rotation

M = R1SR2 .

• Any 2D congruence can be decomposed into the product of 3 line reflection at the most.

Isometry (congruent transformation) • Isometry preserves length, whereas direct isometry preserves orientation and opposite does not preserve orientation • Direct Isometry |R| = 1 (Rotation) • Opposite Isometry |R| = -1 (Line Reflection)

20 Linear Transformations

• Scale, Reflection, Rotation, and Shear are all linear transformations • They satisfy: T(au + bv) = aT(u) + bT(v) ▪ u and v are vectors ▪ a and b are scalars • If T is a linear transformation ▪ T((0, 0)) = (0, 0) • What important operation does that leave out?

Linear transformation

Affine transformation

21 Rotation about an Arbitrary Point y y

  x x

This is not a linear transformation. The origin moves.

Translation

y (x, y)→(x+a,y+b) y

(a, b)

x x

This is not a linear transformation. The origin moves.

22 Homogeneous Coordinates y Embed the xy-plane in R3 at z = 1. y (x, y)  (x, y, 1)

x x X A' X= z xaaxa'0  xay11121112 + yaayaxay'0 ==+   21222122 z '00111 

2D Linear Transformations as 3D Matrices Any 2D linear transformation can be represented by a 2x2 matrix

aaa11121112 xa yx +  = aaa21222122 xa yy + or a 3x3 matrix

a11 a 120   x   a 11 x+ a 12 y  a a0   y =+  a x a y  21 22     21 22  0 0 1   1   1 

23 2D afinne transformation

Image of a point (x, y, 1)T

xefmxe'  xfym1111 ++ yefnyexfyn' ==++   2222 100111 

Image of a vector (x, y, 0)T

xefmxe'  xfy1111 + yefnyexfy' ==+   2222 000100 

2D afinne transformation

Image of a origin (0, 0, 1)T is 3rd column.

x'0   e11 f m     m  y'0 ==  e f n     n    22      1   0 0 1   1   1 

Image of a basis vectors (1, 0, 0)T, (0, 1, 0)T are 1st and 3rd columns. e f m1 e 1 1     1  e1 f 1 m  0   f 1        e f n0 = e e f n  1 =  f  2 2     2  2 2     2  0 0 1   0   0  0 0 1   0   0 

24 2D Linear Translations as 3D Matrices

Any 2D translation can be represented by a 3x3 matrix.

10  axxa + 01  byyb =+   00111 

With homogeneous coordinates, we can represent all 2D affine transformations as 3D linear transformations. We can then use matrix multiplication to transform objects.

Rotation 180° about an arbitrary point

Z_affine_reflectionPoint.ggb November 10, 2020 50

25 Recall that the column vectors of the matrix M are given by images of the basis vectors and origin O(0, 0).

Affine transformation

Windowing Transforms Windowing is the process of transforming co-ordinates from one space to another. It is used when scaling and transforming the view of a program. For example: when you zoom into an image, the original image data is transformed to fill the current screen. (A,B)

(a,b) translate (A-a,B-b)

scale (C,D) (C-c,D-d)

translate (c,d) November 10, 2020 53

26 Fixed point of the plane isometry

Classify the transformation A. Determine all fixed points and directions.

0 1 6− vv =  01− A =  v= Av 10 A = 1 0 1   0 0 1 ( A−= E) v o

Determinant A = 1, first two orthonormal columns yields the congruent transformation. Vectors could be investigated by linear part of matrix A. Translation has no influence on vectors. AE−=+=2 10

Characteristic polynomial has only complex solution. Isometry without fixed direction is rotation.

Fixed point of the plane isometry

Classify the transformation A. Determine all fixed points.

XX = XAX= ( AEXo−=) * −1 − 1 6   − 1 − 1 6   1 0 − 2.5        ( AE−) =1 − 1 1   0 − 2 7   0 1 − 3.5        0 0 0   0 0 0   0 0 0 

GeoGebra tool ReducedRowEchelonForm(M)provides Gaussian elimination with echelon form *. Using back- substitution, unknowns x, y can be solved for. Transformation has only one fixed point FP = ( 2.5, 3.5).

27 Fixed point of the plane isometry

Classify the transformation A. Determine all fixed points. * 102.50− x  013.50−= y  00010 x −=2.50 y −=3.50

3D Transformations

x Remember: x y y    z z  1 A 3D linear transformation can be represented by a 3x3 matrix. aaa 0 aaa 111213 111213 aaa 0 aaa  212222 212223 aaa313233 0 aaa313233  0001

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28 3D Affine Transformations

sx 000 000s y scale,,(sssxyz ) = 000 sz  0001

100 tx 010 t y translate,,(tttxyz ) = 001 tz  0001  58

3D Rotations

1000 0 cossin0() − ( ) rotate ( ) =  x 0 sincos0() ( )  0001 cos0() sin0 ( )  0100 rotate ( ) =  y −sin0() cos0 ( )  0001 cos() − sin( ) 0 0  sin() cos( ) 0 0 rotate ( ) =  z 0 0 1 0  0 0 0 1 59