Topic 3: 2D Transformations 9/10/2016. Today s Topics. Transformations. Lets start out simple. Points as Homogeneous 2D Point Coords

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1 Tody s Topics 3. Trnsformtions in 2D 4. Coordinte-free geometry 5. (curves & surfces) Topic 3: 2D Trnsformtions 6. Trnsformtions in 3D Simple Trnsformtions Homogeneous coordintes Homogeneous 2D trnsformtions Affine trnsformtions & restrictions Trnsformtions Lets strt out simple Trnsformtion/Deformtion in Grphics: A function f, mpping points to points. simple trnsformtions re usully invertile. [x y] T f [x y ] T f -1 Applictions: Plcing ojects in scene. Composing n oject from prts. Animting ojects. Trnslte point [x y] T y [t x t y ] T : x = x + t x y = y + t y Rotte point [x y] T y n ngle t : x = x cost - y sint y = x sint + y cost Scle point [x y] T y fctor [s x s y ] T x = x s x y = y s y Processing Tree Demo! Representing 2D trnsforms s 2x2 mtrix Points s Homogeneous 2D Point Coords Rotte point [x y] T y n ngle t : x = cost -sint 0 x y sint cost 0 y Scle point [x y] T y fctor [s x s y ] T p= x[1 0 0] T + y[0 1 0] T Trnslte? x = s x 0 0 x y 0 s y 0 y [0 1] T [1 0] T p=[x y 1] T sis vectors +[0 0 1] T 1

2 Crtesin Homogeneous 2D Points Points t in Homogeneous Coordintes Crtesin [x y] T => Homogeneous [x y 1] T [x y w] T with w=0 represent points t infinity, though with direction [x y] T nd thus provide nturl representtion for vectors, distinct from points in Homogeneous coordintes. Homogeneous [x y w] T => Crtesin [x/w y/w 1] T Homogeneous points re eul if they represent the sme Crtesin point. For eg. [4-6 2] T = [-6 9-3] T. Wht out w=0? Line Eutions in Homogeneous Coordintes The Line Pssing Through 2 Points A line given y the eution x+y+c=0 cn e represented in Homogeneous coordintes s: l=[ c], mking the line eution l.p= [ c][x y 1] T =0. For line l tht psses through two points p 0, p 1 we hve l.p 0 = l.p 1 = 0. In other words we cn write l using cross product s: l= p 0 X p 1 Aside: cross product s mtrix [ 0 -c ] [x y 1] T [ c 0 -] [- 0] p 1 p 0 Point of intersection of 2 lines Representing 2D trnsforms s 3x3 mtrix For point tht is the intersection of two lines l 0, l 1 we hve p.l 0 = p.l 1 = 0. In other words we cn write p using cross product s: p= l 0 X l 1 Trnslte point [x y] T y [t x t y ] T : x = 1 0 t x x y 0 1 t y y Rotte point [x y] T y n ngle t : l 0 p x = cost -sint 0 x y sint cost 0 y Scle point [x y] T y fctor [s x s y ] T Wht hppens when the lines re prllel? l 1 x = s x 0 0 x y 0 s y 0 y 2

3 Properties of 2D trnsforms these 3x3 trnsforms hve vriety of properties. most generlly they mp lines to lines. Such invertile Liner trnsforms re lso clled Homogrphies. more restricted set of trnsformtions lso preserve prllelism in lines. These re clled Affine trnsforms. trnsforms tht further preserve the ngle etween lines re clled Conforml. trnsforms tht dditionlly preserve the lengths of line segments re clled Rigid. Where do trnslte, rotte nd scle fit into these? Properties of 2D trnsforms Homogrphy, Liner (preserve lines) Affine (preserve prllelism) sher, scle Conforml (preserve ngles) uniform scle Rigid (preserve lengths) rotte, trnslte Homogrphy: mpping four points Homogrphy: preserving lines Show tht if points p lie on some line l, then their trnsformed points p lso lie on some line l. Proof: We re given tht l.p = 0 nd p =Hp. Since H is invertile, p=h -1 p. Thus l.(h -1 p )=0 => (lh -1 ).p =0, or p lies on line l = lh -1. QED How does the mpping of 4 points uniuely define the 3x3 Homogrphy mtrix? Affine: preserving prllel lines Affine: preserving prllel lines Wht restriction does the Affine property impose on H? If two lines re prllel their intersection point t infinity, is of the form [x y 0] T. If these lines mp to lines tht re still prllel, then [x y 0] T trnsformed must continue to mp to point t infinity or [x y 0] T Wht restriction does the Affine property impose on H? If two lines re prllel their intersection point t infinity, is of the form [x y 0] T. If these lines mp to lines tht re still prllel, then [x y 0] T trnsformed must continue to mp to point t infinity or [x y 0] T i.e. [x y 0] T = * * * [x y 0] T * * *??? i.e. [x y 0] T = A t [x y 0] T In Crtesin co-ordintes Affine trnsforms cn e written s: p = Ap + t 3

4 Affine properties: composition Affine properties: inverse Affine trnsforms re closed under composition. i.e. Applying trnsform (A 1,t 1 ) (A 2,t 2 ) in seuence results in n overll Affine trnsform. p = A 2 (A 1 p+t 1 ) + t 2 => (A 2 A 1 )p+ (A 2 t 1 + t 2 ) The inverse of n Affine trnsform is Affine. - Prove it! Affine trnsform: geometric interprettion Affine trnsform: chnge of reference frme A chnge of sis vectors nd trnsltion of the origin p A 2 t 1 How cn we trnsform point p from one reference frme <1,1,o1>, to nother frme <2,2,o2>? 2 o2 2 1 o1 p 1 point p in the locl coordintes of reference frme defined y <1,2,t> is t p Composing Trnsformtions Rottion out fixed point Any seuence of liner trnsforms cn e collpsed into single 3x3 mtrix y conctenting the trnsforms in the seuence. In generl trnsforms DO NOT commute, however certin comintions of trnsformtions re commuttive try out vrious comintions of trnslte, rotte, scle. The typicl rottion mtrix, rottes points out the origin. To rotte out specific point, use the ility to compose trnsforms T R T - 4

5 Topic 4: Coordinte-Free Geometry (CFG) CFG: dimension free geometric resoning Points p [ 1] Vectors v [ 0] Lines l [.. ] Dot products, Cross products, Length of vectors, Weighted verge of points A rief introduction & sic ides How do you find the ngle etween 2 vectors? 3D prmetric curves p(t)=(f x (t),f y (t),f z (t))) 3D prmetric surfces 3D prmetric plne p(t,s)=(f x (t,s),f y (t,s),f z (t,s))) p(s,t)= + s +t 5

6 Tngent / Norml vectors of 2D curves Explicit: y=f(x). Prmetric: x=f x (t) y=f y (t) Implicit: f(x,y) = 0 Tngent is dy/dx. Tngent is (dx/dt, dy/dt) Norml is grdient(f). direction of mx. chnge Given tngent or norml vector in 2D how do we compute the other? Wht out in 3D? Norml vector of plne Norml vector of plne n p(s,t)= + s +t n=x Norml vector of prmetric surfce Norml vector of prmetric surfce n [f(u 0,v 0 )] [f(u 0,v 0 )] f(u 0,v) f(u,v 0 ) f(u 0,v) f(u,v 0 ) n=f (u 0,v) X f (u,v 0 ) 6

7 Implicit function of plne n f(p) = (p-).n=0 Implicit function: level sets 3D prmetric surfces 3D prmetric surfces: Coons interpoltion Extrude Revolve Loft Sure interpolte( 0, 2) My Live Demo interpolte( 1, 3) iliner interpoltion 7