Computer Graphics. 2D transformations. Transforma3ons in computer graphics. Overview. Basic classes of geometric transforma3ons

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1 Transforma3ons in computer graphics omputer Graphics Transforma3ons leksandra Piurica Goal: introduce methodolog to hange coordinate sstem Move and deform objects Principle: transforma3ons are applied to object ver3ces In 2D, point P(X,Y) is transformed to P (X,Y ); in 3D, P(X,Y,Z) à P (X,Y,Z ) L4_2 Overview 2D transforma3ons 3D transforma3ons Quaternions Transforma3ons in OpenGL 2D transformations L4_3 2D affine transforma3ons Basic classes of geometric transforma3ons General linear (preserve lines) Translate Rotate Scale Shear ffine (preserve paralelism) rbitrar shearing General scaling onformal (preserve angles) Uniform scaling Rigid (preserve lengths) Transla3on Rota3on L4_5 L4_6

2 Elementar 2D transforma3ons Transla3on T(T,T ) Matri representa3on Suppose we represent a 2D transforma3on b a matri X = X + T X Y = Y + T Scaling S(S,S ) a b = c d original point transformed point transforma3on matri = a + b = c + d X = X * S X Y = Y * S Y Wh is this useful? - sequence of transforma3ons à matri mul3plica3on Rota3on R(θ) X = X * θ Y*θ Y = X * θ + Y*θ L4_7 an we do it for an affine transform? L4_8 22 Matrices 22 Matrices Which transforma3ons can be represented b 22 matrices? - Lets look at some eamples: Which transforma3ons can be represented b 22 matrices? - Lets look at some eamples: 2D iden3t = = = 2D mirror over Y ais = - = = 2D rota3on around (,) = θ θ = θ + θ θ = θ θ θ 2D scale around (,) = S S = S = S 2D mirror over (,) = - = = - 2D Shear = + H X = + H Y = H H L4_9 L4_ 22 Matrices Which transforma3ons can be represented b 22 matrices? - an we represent transla3on b a 22 matri? 2D Transla3on = + T = + T NO 22 matri! Matri representa3on for affine 2D transforms We want a representa3on where 2D transla3on is also represented b a matri (so that we can easil combine different transforma3ons b mul3plica3on) 2D matri representa3on of transla3on does not eist! What do we do? Solu3on: use a 33 matri Onl linear 2D transforma3ons can be represented b a 22 matri Linear transforma3ons sa3sf: T (s P +s 2 P 2 )=s T (P )+s 2 T (P 2 ) are combina3ons of scale, rota3on, shear and mirror 2D Transla3on = + T = + T T = T Homogeneous coordinates L4_ L4_2

3 Homogeneous coordinates n equivalent formula3on: adding a 3 rd coordinate to 2D points such that (, ) à (W, W, W) (/W, /W) are artesian coordinates of the homogeneous point (,, W) (,, ) represents a point at infinit (,, ) not allowed 3 2 (2,3,) or (4,6,2) or (6,9,3) or (3,,) or (6,2,2) or W Homogeneous point is a line in 3D space L4_3 Homogeneous coordinates Even points infinitel far have a representa3on in homogeneous coordinates Points at infinit have their last coordinate equal to ero Eamples: P =(,,); P 2 =(,,); P 3 =(,,), P 4 =(2,,) P P P 2 P 3 L4_4 Matri form of elementar 2D transforma3ons 2D Shear transforma3on T = T Translation T(T,T ) S = S Scaling S(S,S ) General shear opera3on SH (H,H) = H H θ θ = θ θ Rotation R(θ) L4_5 Shear in the X- direc3on: H = a SH = Shear in the Y- direc3on: H = SH = b Shear can be represented as a combina3on of rota3ons and non- uniform scaling opera3ons. L4_6 Translate Rotate Eample: comple transforma3on as a sequence of elementar transforma3ons L4_7 Inverse elementar transforms T - (T X, T Y ) = T (-T X, -T Y ) S - (S X, S Y ) = S (/S X, /S Y ) R - (θ) = R (-θ) omple transforma3ons can be described as a combina3on (composi3on) of elementar transforma3ons (X, Y,) T = S (-) R (-) T (-) R (-) S (-) (X, Y, ) T Each rota3on brings an etra parameter, scaling and transla3ons two etra parameters an this be simplified? L4_8

4 We tr to simplif such epressions b ombining transforma3ons, if possible hanging the order of transforma3ons in order to make the combina3ons possible Transla3on and rota3on pseudo- commute T (T X,T Y ) R (θ) = R (θ) T (T X2,T Y2 ) Sequen3al elementar transforms of the same tpe absorb each other T (T X, T Y ) T (T X2, T Y2 ) = T (T X +T X2, T Y +T Y2 ) S (S X, S Y ) S (S X2, S Y2 ) = S (S X. S X2, S Y. S Y2 ) R (θ ) R (θ 2 ) = R (θ +θ 2 ) hanging transforma3on order is not alwas possible! Transforma3ons that commute are onl Two elementar transforma3ons of the same tpe Scaling with S =S and Rota3on T (T X2,T Y2 ) R (θ) R (θ) T (T X,T Y ) L4_9 L4_2 Transla3on and scaling pseudo- commute S (S X, S Y ) T (T X,T Y ) = T (T X2,T Y2 ) S (S X, S Y ) Hpothesis: Most general affine transform can alwas be represented as R (θ ) S (S X, S Y ) R (θ 2 ) T (T X,T Y ) T (T X,T Y ) S (.5,3.) This means: a unit square in the centre is reshaped to an arbitrar parallelogram, brought to an arbitrar posi3on and rotated b an arbitrar angle S (.5,3.) T (T X2,T Y2 ) No similar propert for rota3on and scaling X = ax + by + c Y = dx + ey + f Prove the hpothesis formall Set c=t, f=t Determine the parameters θ, θ 2, S X, S Y as a func3on of a, b, d, e L4_2 L4_22 Efficienc: matri calcula3ons Efficienc: crea3ng successive views The most general affine transforma3on a = d b e c f Mul3plica3on of a 33 matri with column vector requires 9 mul3plies and 6 addi3ons ctuall we need onl 4 mul3plies and 4 addi3ons X = ax + by + c Y = dx + ey + f Even though matri representa3on is useful, prac3cal applica3on should make use of the special structure of the matri, for efficienc L4_23 To produce an impression of a dnamicall rota3ng object, man successive views are needed = θ θ = θ + θ The angle difference between the successive views is ver small (a few degrees). an we simplif the calcula3on? Solu3on : Use approima3on θ : = θ = θ + What is wrong with this solu3on? beser solu3on = θ = θ + heck the determinant of the matri in both cases! L4_24

5 3- D Transforma3ons Generalia3on of 2D transforma3ons Same principle: appl to ver3ces (which are now 3D points) P(X,Y,Z) is transformed to P (X,Y,Z ) 3D transformations We consider general 3D affine transforma3on X = ax + by + cz + d Y = ex + fy + gz + h Z = ix + jy + kz + l Proper3es analogous to 2D case Lines map to lines (plane segments map to plane segments) Parallelism preserved unit cube centered in the origin is transformed into an arbitrar parallelepiped, arbitraril posi3oned in space L4_26 3D transforma3ons 3D rota3on onven3on that we will adopt: Right- hand side sstem 3D rota3on of a rigid object is described b three parameters, such as three angles of Euler In Euler angle formula3on an arbitrar rota3on is represented as a composi3on of three elemental rota3ons, each around a gle coordinate ais L4_27 L4_28 3D rota3on The rota3ons around, and ais are also known as Roll, Pitch and Yaw ( heading ); terms coming from flight dnamics Pitch Yaw 3D transforma3ons Three elementar transforms Elementar transla3on Three elementar rota3ons round the X- ais, Y- ais and Z- ais Elementar scaling In X-, Y-, en Z- direc3on Roll Ver3cal Lateral Longitudinal L4_29 Matri representa3on 4 4 matrices for 3- D, analogous to 3 3 matrices for 2- D ug homogeneous coordinates W W ; W W usual notation W = : Each point in 3D space à a line throught the origin in 4D space L4_3

6 3D Transla3on and Scaling Transla3on L4_3 T (T X, T Y, T Z ) = T T T X = X + T X Y = Y + T Y Z = Z + T Z Elementar scaling S (S X, S Y, S Z ) = S S S Y Y X X = S X X Y = S Y Y Z = S Z Z 3D Rotation Rota3on around - ais L4_32 θ θ θ θ R X (θ ) = R Y (ϕ) = R Z (ψ) = ϕ ϕ ϕ ϕ ψ ψ ψ ψ X = X Y = Y (θ) - Z (θ) Z = Y (θ) + Z (θ Similarl for rota3ons around other two aes: 3D Shear Shear around the Z- ais X = X + H X Z Y = Y + H Y Z Z = Z an be represented in matri form as L4_33 = H H Shear around the X- ais and Y- ais have a similar form 3D Transforma3ons: proper3es Inverse transforms are defined as T - (T X, T Y, T Z ) = T (-T X, -T Y, -T Z ) S - (S X, S Y, S Z ) = S (/S X, /S Y, /S Z ) R X - (θ) = R X (-θ); R Y - (φ) = R Y (-φ); R Z - (ψ) = R X (-ψ) The same absorp3on and pseudo- commuta3on are valid like in the 2D case Transla3on pseudo- commutes with scaling and with three rota3ons Scaling and rota3ons do not pseudo- commute ddi3onal proper3es, specific for 3D (without proof) Two (different) rota3ons do not pseudo- commute (not even with an addi3onal transla3on) Three (different) rota3ons and a transla3on do pseudo commute L4_34 3D Transforma3ons: proper3es general posi3oning transforma3on involves 6 parameters T (T X, T Y, T Z ) R X (θ) R Y (φ) R Z (ψ) Object deforma3on: 6 other parameters (scaling + rota3on) general affine transforma3on involves thus 2 parameters R X (θ) R Y (φ) R Z (ψ) S (S X,S Y,S Z ) R X (θ ) R Y (φ ) R Z (ψ ) T (T X,T Y,T Z ) There are man equivalent forms L4_35. rbitrar displacement in 3D B Ini3al posi3on B Desired posi3on () B B (2) (4) B (3) B L4_36 See eercises

7 omposi3ons of 3D transforma3ons Rota3on about arbitrar point: in this eample about - ais Quaternions and 3D rotation From the book of Edward ngel: Interactive omputer Graphics Top-Down pproach Ug OpenGL L4_37 omple numbers and rota3on Polar representa3on of a comple number iθ 2 2 c = a + ib = re r = a + b, θ = arctan( b/ a) Denote b c the result of rota3ng c about the origin b φ: iθ iφ i( θ + φ) c = re e = re Quaternions (Sir William Rowan Hamilton, 843) are etensions of comple numbers in 3D, ielding elegant rota3ons in 3D a = ( q, q); q = qi + q2j + q3k i = j = k = ijk = L4_39 omple numbers and rota3on point in space p = (, p) Rota3on of the point p b θ about the vector v: where: θ θ r =, v, 2 2 p = rpr θ θ r =, v 2 2 Wh are quaternions interes3ng for 3D rota3ons in computer graphics? Euler angle representa3on suffers from Gimbal lock Simpler representa3on Lower computa3onal compleit, but not necessaril awer conversion to matri form L4_4 Gimbal lock Transformations in OpenGL When the aes of two of the three gimbals align, "locking" into rota3on in a degenerate 2D space. Eample: when the pitch (green) and aw (magenta) gimbals become aligned, changes to roll (blue) and aw appl the same rota3on to the airplane. L4_4

8 OpenGL func3on format func3on name dimensions Transforma3ons in OpenGL We can load matri with the func3on glloadmatri(pointer_to_matri) glverte3f(,,),, are floats belongs to GL librar glverte3fv(p) p is a pointer to an arra Or set a matri to iden3t matri with the func3on glloadidentit() Rota3on, transla3on and scaling are provided through glrotatef(angle,v,v,v) gltranslatef(d,d,d) glscalef(s,s,s) ngel: Interac3ve omputer Graphics 4E ddison- Wesle 25 L4_43 L4_44 Summar 2D transforma3ons arbitrar 2D affine transforma3on 6 parameters comple transforma3ons as composi3ons of the elementar ones 3D transforma3ons arbitrar 3D affine transforma3on 2 parameters comple transforma3ons as composi3ons of the elementar ones Euler angles Quaternions advantage over Euler angle representa3on (Gimbal lock avoided) Transforma3ons in OpenGL L4_45