This lecture. Transformations in 2D. Where are we at? Why do we need transformations?

Transcription

1 Thi lectue Tanfomation in 2D Thoma Sheme Richa (Hao) Zhang Geomet baic Affine pace an affine tanfomation Ue of homogeneou cooinate Concatenation of tanfomation Intouction to Compute Gaphic CMT 36 Lectue 8 Febua 3, 26 2 Wh o we nee tanfomation? Whee ae we at? Nee to tanfom object fom object cooinate tem (OCS) to wol CS Nee to tanfom object fom wol CS to the cooinate tem of the camea o ee (VCS) WCS OCS Geometic tanfomation. Nee to tanfom object fom VCS into an OpenGL winow/ceen (SCS) VCS Object ma move (tanlate o otate) o efom (cale, hea, o geneal non-igi efomation) SCS Febua 3, 26 3 Febua 3, 26 4

2 Object in OCS iel = SCS (2D) Rateization The eie of CS Moeling tanfom WinCS (3D) WCS (4D) Viewpot tanfom OCS: object cooinate tem WCS: wol cooinate tem VCS: viewing cooinate tem Viewing tanfom NDCS (3D) Moel-view tanfom VCS (4D) ojection (nomalization) clipping then pepective / CCS (4D) CCS: clip cooinate tem NDCS: nomalize evice CS WinCS: winow cooinate tem with epth infomation SCS: Sceen cooinate tem Febua 3, 26 5 Geomet baic Scala, point, an vecto Vecto pace an affine pace Baic point an vecto opeation Sie-ne tet Line, plane, an tiangle Linea inepenence Cooinate tem an fame Febua 3, 26 6 Scala, point, an vecto Vecto pace oint: a location in pace Specifie b an k-tuple fo k- point Alwa given with epect to ome cooinate tem Scala: a quantit, e.g., ege length Vecto: a iecte line egment between point Space: vecto pace, affine pace, Eucliean pace, etc. = z A et of vecto with cala multiplication an vecto aition Scala-vecto multiplication u = v Vecto-vecto aition: w = u + v Epeion uch a v = u + 2w 3 make ene in a vecto pace But vecto lack poition Inaequate fo epeenting geomet we nee poition, which ae given b point Febua 3, 26 7 Febua 3,

3 Affine pace Baic point an vecto opeation A vecto pace + point = affine pace Opeation Vecto-vecto aition Scala-vecto multiplication oint-vecto aition Affine um of point an conve um A vecto pace + itance/nom = metic pace (e.g. Eucliean pace) point point = vecto point + vecto = point vecto opeation: cala * vecto = vecto vecto + vecto = vecto vecto vecto = cala, the ot pouct vecto vecto = vecto, the co pouct u v v u Right-han ule Febua 3, 26 9 Febua 3, 26 Moe on ot pouct u v = u * v + u * v + u z * v z u u = u 2 i alwa non-negative u v i commutative an itibutive ove aition u v = u * v * Two vecto u an v ae othogonal (pepenicula) if an onl if u v = If v i nomalize, i.e., v =, then u v give the pojection of u in the iection of v Febua 3, 26 Moe on co pouct Co pouct u v i a vecto pepenicula to u an v fequentl ue to compute the nomal to a tiangle Diection of the co pouct i etemine b the ight han ule u v = (v u) u v = u * v * = aea of the paallelogam How to compute? ue eteminant i j k u u u z v v v z u v Febua 3, 26 2 v u 3

4 Affine an conve um Aition of two abita point i not efine in an affine pace But conie two point an Q with Q = + av, we can alwa fin a point R uch that v = R o now we have Q = + a(r ) o Q = ar + ( a) Thu, affine um (combination) of point can be efine t + t t n n, t + t t n = Conve um (combination) of point t + + t n n, whee t + + t n = an t i fo all i v R Q Conve hull Conve hull of a et of point: et of all conve combination of thee point Altenativel, the conve hull i the mallet conve object containing the et of point Fome b hink wapping point Ueful in man application, e.g., fat colliion etection, a tacing, hape egmentation, etc. Febua 3, 26 3 Febua 3, 26 4 Sie-ne tet On which ie oe a point V lie with epect to a line, pecifie b a point an a vecto u? Solution : ue implicit line o plane equation; plug in the point cooinate an check the ign Solution 2: in 2D, let z cooinate be zeo, compute the co pouct u (V-) an check the ign of the z component Solution 3: fin a vecto u pepenicula to u an check ign of u v u = ( u, u ) Febua 3, 26 5 u v v u V u = (u, u ) V u Line epeentation Conie all point of the fom (a) = + a thi i the et of all point ling on a line that pae though in the iection of the vecto Known a the paametic fom of the line Given two point R an Q on the line, we have Othe epeentation Eplicit: = m + h Implicit: a + b + c = (a) = a R + ( a) Q (a) = a R + ( a) Q Febua 3,

5 Ra an line egment If a, then (a) i the a leaving in the iection of the vecto Fo the two-point epeentation (a) = a R + ( a) Q (a) = a R + ( a) Q if a, then we get all the point on the line egment joining R an Q (thi i leping) Febua 3, 26 7 aametic plane epeentation A plane can be efine b a point an two vecto o b thee point v Q R u R (a, b) = R + au + bv (a, b) = R + a(q R) + b( R) An affine um Febua 3, 26 8 Tiangle an bacentic cooinate Baic tanfomation conve um of an Q conve um of S() & R The mot baic one Tanlation, caling, otation, an hea An othe, e.g., pepective tanfom, pojection, etc. Baic tpe of tanfomation Rigi-bo tanfom: peeve length an angle Similait tanfom: angle peeving, e.g., unifom caling T i a conve um of, Q, an R. The weight ae calle the bacentic cooinate of T Febua 3, 26 9 Affine (linea) tanfom: peeve paallel line, not angle o length Fee-fom efomation: anthing goe Febua 3,

6 6 Febua 3, 26 2 Tanlation in 2D T Febua 3, Scaling in 2D Unifom: = Nonunifom: S Febua 3, Rotation about oigin oitive angle: counteclockwie Fo negative angle R Febua 3, Deivation of otation mati Make ue of pola cooinate: (, ) (, ) otate

7 Sheaing in 2D SH Tanfomation chain O SH b O O O a a O = S /3 (O) O = R -2 (O ) O = R -2 (S /3 (O)) O = T,3 (O ) O = T,3 (R -2 (O )) O = T,3 (R -2 (S /3 (O))) Febua 3, Febua 3, Tanfomation chain Tanfomation chain O O (4) O (5) Q O (6) Q O (6) = T,7 (R 5 (R 9 (S 2 (T,4 (R -3 (T,3 (R -2 (S /3 (O))))))))) Thi mean: fo eve vete v O O = T,3 (R -2 (S /3 (O))) O (5) = T,4 (R -3 (T,3 (R -2 (S /3 (O))))) O (4) = R -3 (T,3 (R -2 (S /3 (O)))) v (6) = T,7 (R 5 (R 9 (S 2 (T,4 (R -3 (T,3 (R -2 (S /3 (v))))))))) v (6) = T,7 (R 5 (R 9 (S 2 (T,4 (R -3 ( 2 2 +( ( /3 /3 v)))))))) Febua 3, Febua 3,

8 v (6) = 7 Tanfomation chain +( ( ( v (6) = U 3 +( M 6 (M 5 (M 4 ( U 2 +( M 3 ( ( ( 2 2 ( 2 2 +( ( /3 /3 v)))))))) U +(M 2 ( M v)))))))) v (6) = U 3 +M 6 M 5 M 4 (U 2 +M 3 (U +M 2 M v)) Febua 3, Tanfomation chain v (6) = U 3 + M 6 M 5 M 4 (U 2 + M 3 (U + M 2 M v)) M 7 = M 6 M 5 M 4 v (6) = U 3 + M 7 (U 2 + M 3 (U + M 8 v)) M 8 = M 2 M Altenating mati multipl an vecto aition. Thi can be acceleate with hawae! We can o even bette. We can epe tanlation a a mati multipl, but we have to ue an epane cooinate tem. (It jut the common enominato tick with ome math theo aoun it.) Febua 3, 26 3 Homogeneou cooinate Onl tanlation not epee a mati-vecto pouct Unifom teatment neee in a pipeline famewok We a a thi cooinate w, an obtain Homogeneou cooinate fo 2D point (, ) tun into (,, ) if (,, w) an (', ', w ) ae multiple of one anothe, the epeent the ame point multiple epeentation fo (, ) tpicall, w. point with w = ae point at infinit One fo each iection in [, 8): pojective plane If intea a gle point at infinit: teeogaphic pojection Febua 3, 26 3 Homogeneou cooinate w i the common enominato fo an. (, ) tun into (,, ) (, ) = (, ) If (,, w) an (', ', w ) ae multiple of one anothe, the epeent the ame point multiple epeentation fo (, ) (,, w ) (,, w) ( w, w ) = ( w, w ) = ( w, w ) tpicall, w. ep! point with w = ae point at infinit (, 4, 2) 3(, 4, 2) = (3, 2, 6) ( 2, 4 2 ) = (3 6, 2 6 ) (, 4, ) (, 4 ) = (3, 2 ) = (, 4 ) Febua 3,

9 9 Febua 3, D Homogeneou Cooinate Cateian cooinate of the homogenou point (,, w): /w, /w (ivie though b w) Ou tpical homogenize point: (,, ) Connection to 3D? (,, ) epeent a 3D point on the plane w = A homogeneou point i a line (minu oigin) in 3D, though the oigin Nee at leat one non-zeo cooinate, o oigin (,, ) i not coniee Febua 3, 26 = + + Tanfomation in homogeneou cooinate Geneal fom of affine (linea) tanfomation E.g., 2D tanlation in homogeneou cooinate q c p b a ' ' q c p b a ' ' Febua 3, Baic 2D tanfomation ), ( Tan ), ( Scale ) ( a a Shea ) ( Rot Febua 3, Invee of tanfomation Invee of Tan(, ) = Tan(, ) Rot() = Rot() Scale(, ) = Scale(/, / ) Shea(a ) = Shea(a ) Tan(, ) = Tan(, )

10 Compoun tanfomation Often a eie of tanfom to move an object Tanfom ma not be the tana one e.g., otation about an abita point/line? Concatenate baic tanfom equentiall Coepon to (onl) multiplication of tanfom matice, thank to homogeneou cooinate Compoun tanlation What happen when a point goe though Tan(, ) an then Tan( 2, 2 )? Combine tanlation: Tan( + 2, + 2 ) Concatenation of tanfomation: mati multiplication Febua 3, Febua 3, Compoun otation Commutativit What happen when a point goe though Rot() an then Rot()? Combine otation: Rot( + ) ( ) ( ) ( ) ( ) Scaling o heaing (in one iection) i imila Thee concatenation ae commutative Tanfomation that o commute: Tanlate tanlate Rotate otate Scale cale Unifom cale otate Shea in () hea in (), etc. In geneal, the oe in which tanfomation ae compoe i impotant Afte all, mati mult ae not commutative in geneal Febua 3, Febua 3, 26 4

11 Rotation about an abita point Rotation about an abita point Beak it own into tana tanfom: Tanlate to the oigin: Tan(, ) Rotate: Rot(q) An tanlate back: Tan(, ) Compoun tanfomation (non-commutative): Tan(, ) Rot(q) Tan(, ) = Tan(, )Rot(q)Tan(, ) = T Wh combine thee matice into a gle T? Febua 3, 26 4 Febua 3, Anothe eample Rigi-bo tanfomation A tanfomation mati Compoun tanfomation: Tan( 2, 2 ) Rot(9 o ) Scale(, ) Tan(, ) whee the uppe 2 2 ubmati i othonomal, peeve angle an length calle igi-bo tanfomation an equence of tanlation an otation give a igi-bo tanfomation Febua 3, Febua 3, 26 44

12 Affine tanfomation eeve line, plane, an paallelim, but not length o angle 2D Line Recall Tom favoite line equation: A + B + C = F(z) = i calle a homogeneou equation. Unit cube Rot(45 o ) Scale in ouct of a equence of tanlation, caling, otation, an hea i affine; thee ae othe, e.g., eflection. Febua 3, We can wite thi a a mati (vecto) equation: [A B C] = Becaue it i homogeneou, we can cale the left-han ie without affecting the tuth of the equation: w[a B C] = w =, o [A B C] w w w Febua 3, = Equation hol fo all epeentation of (, ) in homogenou cooinate! 2D Line We coul have abobe the cala into the fit vecto intea: [wa wb wc] = If [A B C] epeent a line, then [wa wb wc] epeent the ame line. 2D Line oint p i on line l iff lp =. oint p i on line l iff l p =. oint p i on line l iff l p. Thee i a line that contain all of the point at infinit. What i it epeentation a a ow vecto? We let the line l be the ow vecto A B C, an the point p be the column vecto, o in geneal. The line equation become: w lp = The homogeneou line equation. Febua 3, What i the line l between point p an p 2? What i the line l uch that l p an l p 2? l = p p 2 What i the point p at the inteection of line l an l 2? What i the point p uch that l p an l 2 p? p = l l 2 Febua 3,