9/22 /2011 Transformations. What is a transformation? • P′=T(P) What does it do? Computer Graphics Transform the coordinates / normal vectors of objects Why use them? • Modelling Lecture 2 -Moving the objects to the desired location in the environment -Multiple instances of a prototype shape Transformations -Kinematics of linkages/skeletons – character animation • Viewing – Virtual camera: parallel and perspective projections Lecture 2 1 2 Types of Transformations Geometric Transformation – Once the models are prepared, we need to place them in the environment – Geometric Transformations – Objects are defined in their own local coordinate • Translation system • Rotation • scaling – We need to translate, rotate and scale them to put them •• Linear (preserves parallel lines) into the world coordinate system • Non-uniform scales, shears or skews – Projection (preserves lines) • Perspective projection • Parallel projection – Non-linear (lines become curves) • Twists, bends, warps, morphs, 3 22/09/2011 Lecture 1 4 2D Translations. 2D Scaling from the origin. Point P defined as P( x , y ), Point P defined as P(x, y), ′′′ ′′′ Perform a scale (stretch) to Point P( x , y ) by a factor sx along the x axis, translate to Point P( x , y ) a distance dx parallel to x axis, dy parallel to y axis. and s along the y axis. ′ = + ′ = + y x x dx y y d y x′ = s. x , y′ = s. y Define the column vec tors x y Define the matrix P’ ′ P x x d x P = , P′ = ,T = s 0 ′ = x y y dy S P’ 0 sy Now P Now P′ = P +T x′ s 0 x ′ = ⋅ = x PSP or ′ . y 0 sy y 5 6 1 9/22/2011 2D Rotation about the origin. 2D Rotation about the origin. y y θ P’(x’,y’) P’(x’,y’) P(x,y) P(x,y) r r x= r.cosφ y= r.sinφ θ y r φ r x x x 7 8 2D Rotation about the origin. 2D Rotation about the origin. x′ = r.cos(θ +φ)= r .cosφ.cosθ − r.sinφ.sinθ x′ = r.cos(θ +φ)= r .cosφ.cosθ − r.sinφ.sinθ ′ = θ +φ = φ θ + φ θ ′ = θ +φ = φ θ + φ θ y y r.sin( )r .cos .sin r.sin .cos y r.sin( )r .cos .sin r.sin .cos Substituting for r : P’(x’,y’) x= r.cosφ P(x,y) y= r.sinφ r x= r.cosφ y= r.sinφ Gives us : θ y x′ = x.cosθ − y.sinθ φ r x y′ = x.sinθ + y.cosθ x 9 10 2D Rotation about the origin. Transformations. • Translation. x′ = x.cosθ − y.sinθ – P′=T + P y′ = x.sinθ + y.cosθ • Scale P′=S ⋅ P Rewriting in matrix form gives us : – • Rotation x′ cosθ −sinθx ′ θ − θ = . x cos sin x P′=R ⋅ P ′ θ θ = . – y sin cos y y′ sinθ cosθ y • We would like all transformations to be multiplications so we can concatenate them cosθ − sinθ • ⇒ express points in homogenous coordinates. Define the matrix R = , P′ =RP ⋅ sinθ cosθ 11 12 2 9/22/2011 Homogeneous coordinates Translations in homogenised coordinates • Add an extra coordinate, W, to a point. – P(x,y,W). • Transformation matrices for 2D translation • Two sets of homogeneous coordinates represent the are now 3x3. same point if they are a multiple of each other. – (2,5,3) and (4,10,6) represent the same point. • If W≠ 0 , divide by it to get Cartesian coordinates of point ′ ′ x 1 0 d x x x = x+ d (x/W,y/W,1). x y′ = 0 1 d . y y′ = y+ d • If W=0, point is said to be at infinity. y y 1 0 0 1 1 1 =1 13 14 Concatenation. Concatenation. ⋅ • When we perform 2 translations on the same point The matrix product T( d x1,d y1) T( d x2 ,d y2 ) is : P′ = T(,) d d ⋅ P x1 y 1 1 0 d x1 1 0 d x2 ′′ = ⋅ ′ P T(,) d x2d y 2 P = 0 1 d y1 .0 1 d y2 ? P′′ = T(,)(,) d d ⋅T d d ⋅P = T( d + d ,d + d )⋅ P x1 y 1 x2 y 2 x1 x2 y 1 y2 0 0 1 0 0 1 So we expect : Matrix product is variously referred to as compounding, ⋅ = + + concatenation, or composition T(,)(,) dx1 d y 1 T d x2d y 2 T( d x1 d x2,d y 1 d y2 ) 15 16 Concatenation. Properties of translations. ⋅ = The matrix product T( d x1,d y1) T( d x2 ,d y2 ) is : 1. T (0,0) I ⋅ = + + + 2. T( sx , s y )T ( tx , t y ) T( sx tx, s y t y ) 1 0 d x1 1 0 dx2 1 0 d x1 d x2 ⋅ = ⋅ = + 3. T( sx , s y )T ( tx , t y ) T(,)(,) tx t y T sx s y 0 1 d y1 .0 1 d y2 0 1 d y1 d y2 4. T-1 (s , s ) = T (−s ,−s ) 0 0 1 0 0 1 0 0 1 x y x y Matrix product is variously referred to as compounding, concatenation, or composition. Note : 3. translation matrices are commutative. 17 18 3 9/22 /2011 Homogeneous form of scale. Concatenation of scales. Recall the (x,y) form of Scale : sx 0 ⋅ S(,) s s = The matrix product S(, sx1 s y1) S( sx2 , sy2 ) is : x y 0 s y ⋅ sx1 0 0 sx2 0 0 sx1 sx2 0 0 = ⋅ In homogeneous coordinates : 0 sy1 0 . 0 sy2 0 0 sy1 sy2 0 sx 0 0 0 0 1 0 0 1 0 0 1 = S(,) sx s y 0 sy 0 Only diagonal elements in the matrix - easy to multiply! 0 0 1 19 20 Homogeneous form of rotation. Orthogonality of rotation matrices. x′ cos θ − sin θ 0 x y′ = sin θ cos θ 0 . y cos θ − sin θ 0 cos θ sin θ 0 1 0 0 1 1 θ = θ θ T θ = − θ θ R() sin cos 0 , R () sin cos 0 For rotation matrices, 0 0 1 0 0 1 −1 θ = −θ R () R( ). cos −θ − sin −θ 0 cos θ sin θ 0 −θ = −θ −θ = − θ θ Rotation matrices are orthogonal , i.e : R() sin cos 0 sin cos 0 R−1()θ = RT ()θ 0 0 1 0 0 1 21 22 How are transforms combined? Other properties of rotation. Scale then Translate R )0( = I (5,3) Scale(2,2) (2,2) Translate(3,1) R()()θ ⋅ R φ = R(θ +φ) (1,1) (3,1) (0,0) (0,0) and Use matrix multiplication: p' = T ( S p ) = TS p θ ⋅ φ = φ ⋅ θ R()()R R()()R 1 0 3 2 0 0 2 0 3 But this is only because the axis of rotation TS = 0 1 1 0 2 0 = 0 2 1 is the same 0 0 1 0 0 1 0 0 1 For 3D rotations , need to be more careful Caution: matrix multiplication is NOT commutative! 23 02/10/09 Lecture 4 24 4 9/22 /2011 Non-commutative Composition Non-commutative Composition Scale then Translate: p' = T ( S p ) = TS p Scale then Translate: p' = T ( S p ) = TS p (5,3) 1 0 3 2 0 0 2 0 3 Scale(2,2) (2,2) Translate(3,1) (1,1) (3,1) TS = 0 1 1 0 2 0 = 0 2 1 (0,0) (0,0) 0 0 1 0 0 1 0 0 1 Translate then Scale: p' = S ( T p ) = ST p Translate then Scale: p' = S ( T p ) = ST p (8,4) 2 0 0 1 0 3 2 0 6 (4,2) Translate(3,1) Scale(2,2) (6,2) (1,1) (3,1) ST = 0 2 0 0 1 1 = 0 2 2 (0,0) 0 0 1 0 0 1 0 0 1 02/10/09 Lecture 4 25 02/10/09 Lecture 4 26 How are transforms combined? Non-commutative Composition Rotate then Translate Rotate then Translate: p' = T ( R p ) = TR p (3,sqrt(2)) 1 0 3 0 -1 3 (0,sqrt(2)) 0 -1 0 (1,1) Rotate 45 deg Translate(3,0) TR = 0 1 1 1 0 0 0 0 1 (0,0) (0,0) (3,0) 0 0 1 0 0 1 0 0 1 Translate then Rotate = Translate then Rotate: p' = R ( T p ) = RT p (1,1) Translate(3,0) (3/sqrt(2),3/sqrt(2)) Rotate 45 deg 0 -1 0 1 0 3 0 -1 -1 (0,0) (0,0) (3,0) RT = 1 0 0 0 1 1 = 1 0 3 0 0 1 0 0 1 0 0 1 Caution: matrix multiplication is NOT commutative! 02/10/09 Lecture 4 27 02/10/09 Lecture 4 28 Types of transformations. Transformations of coordinate systems. • Rotation and translation • Have been discussing transformations as – Angles and distances are preserved transforming points. – Unit cube is always unit cube • Always need to think the transformation in the – Rigid-Body transformations. world coordinate system • Rotation, translation and scale. • Sometimes this might not be so convenient – Angles & distances not preserved. – i.e. rotating objects at its location – But parallel lines are. 29 30 5 9/22 /2011 Transformations of coordinate Transformations of coordinate systems - Example systems - example • Concatenate local transformation matrices from left to right • Can obtain the local – world transformation matrix • p’,p’’,p’’’ are the world coordinates of p after • is the world coordinate of point p after n each transformation transformations 31 32 Quiz Solution • I sat in the car, and find the side mirror is 0.4m on • The side mirror position is locally my right and 0.3m in my front (0,4,0.3) • I started my car and drove 5m forward, turned 30 • The matrix of first driving degrees to right, moved 5m forward again, and forward 5m is turned 45 degrees to the right, and stopped 1 0 0 • What is the position of the side mirror now, T = 0 1 5 relative to where I was sitting in the beginning? 1 0 0 1 33 34 Solution Solution • The matrix to turn to the right The local-to-global transformation matrix at the last configuration of the car is 30 and 45 degrees (rotating -30 3 1 1 1 0 0 and -45 degrees around the 1 0 0 2 2 1 0 0 2 2 1 3 1 1 TR TR = − 0 − 0 1 2 0 1 5 2 2 0 1 5 2 2 origin) are 0 0 1 0 0 10 0 1 0 0 1 3 1 1 1 0 0 2 2 2 2 1 3 1 1 The final position of the side mirror can be R = − 0 , R = − 0 , respective ly 1 2 2 2 2 2 computed by TR 1TR 2 p which is around (2.89331, 9.0214) 0 0 1 0 0 1 35 36 6 9/22 /2011 This is convenient for character animation / robotics Transformations of coordinate • In robotics / animation, we systems.