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). 3 ) LENGTHS ONGRUENCE (C RESERVES P 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 5 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 A DIVIDING RESERVES AND P 6 3 ) ROJECTIVE The projective transformation does not (P preserve parallelism, length, and angle. But it still preserves collinearity and incidence. OMOGRAPHY H 7 4 Linearity Parallel Angle Length X X X X X X 9 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 November 10, 2020 10 5 Matrix Multiplication is not commutative 1 2 2 0 4 4 = 3 4 1 2 10 8 2 0 1 2 2 4 = 1 2 3 4 7 10 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: x = Ax x a b x = y c d y x =+ ax by y' =+ cx dy ◼ The linear transformation given by a matrix Let A be an mn matrix. The function T defined by is a linear transformation from Rn into Rm. Rn vector Rm vector ◼ Note: a11 a 12 a 1n v 1 a 11 v 1+ a 12 v 2 + + a 1 n v n a a a v a v+ a v + + a v Av ==21 22 2n 2 21 1 22 2 2 n n am1 a m 2 a mn v n a m 1 v 1+ a m 2 v 2 + + a mn v n 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 −1x1 (2)T(x) = Ax = −1 3 − 2x 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: x = xMT ac x y = x y 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. x = M x a b x ax+ by Mx== c d y cx+ dy a b 10 a a b b == c d 01 c c d d (1, 0) Scaling by 0.5 x = M x M = .5 0 0 .5 x' = Mx (1, 0) x a b x (0.5, 0) = y c d y (0, 1) (0, 0.5) November 10, 2020 18 9 Scaling by 0.5 1 y det(M ) = y 4 0.5 0 −1 M = det(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 x = M x November 10, 2020 20 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: X'== X ; X ' SX xx= 2 xy==0, 0 XSX= yy= 2 Only one fixed point (0, 0). Fixed directions: v'==v ; v ' Sv x=2 x x ( − 2) = 0 v= S v y=2 y y ( − 2) = 0 =2,x R , y R All directions are fixed. General Scaling xx = scale(,) sxy s y y 1 scale( sxy, s ) = sy sx 0 0 sy x x 1 sx 11 General Scaling Fixed directions: v'==v ; v ' Sv v= S v 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 x=0, y R XSX= yy= FP= (0,t ), t R . x=2 x x ( − 2) = 0 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 y y 1 s 01 x x Rotation xx = Rtx() = M x cos(tt) − sin ( ) rot (t) = sin(tt) cos( ) t sin(t) cos(t) -sin(t) t cos(t) November 10, 2020 28 14 Trajectory of point A = (r, 0) in revolution xx = R cos() − sin ( ) RR==; det( ) 1 sin() cos( ) y xr cos() − sin ( ) = y sin() cos( ) 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 x=B x, x = x x=B x () B − E x = o 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 = ref y refy = −10 01 November 10, 2020 33 Reflection in y-axis y y refy = −10 01 x x November 10, 2020 34 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 42 21 Rotation about an Arbitrary Point y y x x This is not a linear transformation.