geometric transformations

Transcription

1 geomeric ranformaion comuer grahic ranform 28 fabio ellacini linear algebra review marice noaion baic oeraion mari-vecor mulilicaion comuer grahic ranform 28 fabio ellacini 2

2 marice noaion for marice and vecor ue column form for vecor m m 2 m ij m 2 m 22 v v2 [ ] v [ v v ] T 2 comuer grahic ranform 28 fabio ellacini 3 mari oeraion addiion T N [ ij ] [ mij nij ] calar mulil T [ ij ] [ mij ] comuer grahic ranform 28 fabio ellacini 4

3 mari oeraion mari-mari mulil row-column mulilicaion no commuaive aociaive T N [ ij ] m k m m 2 m m 2 22 ik n n n kj 2 n n 2 22 comuer grahic ranform 28 fabio ellacini 5 mari oeraion mari-vecor mulil row-column mulilicaion u v [ ui ] m k u m u2 m 2 m m 2 22 v v 2 ik v k comuer grahic ranform 28 fabio ellacini 6

4 ecial marice ideni I for i j [ i ] ij for i j I I I zero O [ o ij ] O O comuer grahic ranform 28 fabio ellacini 7 mari oeraion ranoe fli along he diagonal T T [ ij ] [ m ji ] invere will no comue elicil in hi coure T T I comuer grahic ranform 28 fabio ellacini 8

5 mari oeraion oroerie lineari of mulilicaion A B A B A B A B aociaivi of mulilicaion A BC AB C ranoe and invere of mari mulil AB AB T B B T A T A comuer grahic ranform 28 fabio ellacini 9 geomeric ranformaion funcion ha ma oin o oin X differen ranformaion have rericion on he form of X we will look a linear, affine and rojecion comuer grahic ranform 28 fabio ellacini

6 2D ranformaion comuer grahic ranform 28 fabio ellacini ranlaion imle form invere T T T comuer grahic ranform 28 fabio ellacini 2

7 linear ranformaion fundamenal roer X q X X q can be rereened in mari form X oher roerie ma origin o origin ma line o line arallel line remain arallel lengh raio are reerved cloed under comoiion comuer grahic ranform 28 fabio ellacini 3 uniform cale S S S/ comuer grahic ranform 28 fabio ellacini 4

8 comuer grahic ranform 28 fabio ellacini 5 non-uniform cale S,/ / S S comuer grahic ranform 28 fabio ellacini 6 roaion co in in co co in in co R R R

9 comuer grahic ranform 28 fabio ellacini 7 hear Sh comuer grahic ranform 28 fabio ellacini 8 reflecion Rl o Rl Rl...

10 combining ranlaion and linear ranform rereen linear ogeher wih ranlaion rigid bod ranformaion are a ube of hi X, goal: unified forma for all ranformaion comuer grahic ranform 28 fabio ellacini 9 homogeneou coordinae rereen oin wih addiional coordinae w e i o for oin w comuer grahic ranform 28 fabio ellacini 2

11 comuer grahic ranform 28 fabio ellacini 2 homogeneou coordinae rereen ranlaion wih a 33 mari add one row and column o linear ranform T m m m m m m m m comuer grahic ranform 28 fabio ellacini 22 affine ranformaion combining linear and ranlaion in one mari roerie doe no ma origin o origin ma line o line arallel line remain arallel lengh raio are reerved cloed under comoiion X

12 affine ranformaion ranlaion cale roaion R T S co in in co comuer grahic ranform 28 fabio ellacini 23 comoiing ranformaion aling one ranformaion afer anoher ereed b funcion comoiion X 2 X X 2 o X for he ranform reened before, comued b mari mulilicaion X 2 o X X 2 X 2 2 comuer grahic ranform 28 fabio ellacini 24

13 comuer grahic ranform 28 fabio ellacini 25 comoiing ranformaion ranlaion linear ranformaion affine ranformaion 2 2 I I I comuer grahic ranform 28 fabio ellacini 26 comoiing ranformaion comoiion i no commuaive roae, hen ranlae ranlae, hen roae

14 comoiing ranformaion comoiion i no commuaive roae, hen ranlae ranlae, hen roae comuer grahic ranform 28 fabio ellacini 27 comoiing ranformaion comoiion i no commuaive roae, hen ranlae ranlae, hen roae comuer grahic ranform 28 fabio ellacini 28

15 comle ranformaion ofen rereened a combinaion of imler one inuiive geomeric inerreaion roaion around arbirar ai ranlae o ai cener roae ranlae back R a, Ta R T a comuer grahic ranform 28 fabio ellacini 29 roaion around arbirar ai comuer grahic ranform 28 fabio ellacini 3

16 roaion around arbirar ai comuer grahic ranform 28 fabio ellacini 3 roaion around arbirar ai comuer grahic ranform 28 fabio ellacini 32

17 comuer grahic ranform 28 fabio ellacini 33 roaion around arbirar ai comuer grahic ranform 28 fabio ellacini 34 ranforming oin and vecor oin and vecor are differen eniie vecor: encode direcion difference of oin oin: encode oiion origin lu a vecor oin and vecor ranform differenl we have hown how oin ranform revioul vecor iml ignore he ranlaion q q q v q v X X X X

18 comuer grahic ranform 28 fabio ellacini 35 ranforming oin and vecor ue homogeneou coordinae wih w everhing i conien bu wha i ha w anwa? v v comuer grahic ranform 28 fabio ellacini 36 homogeneou coordinae oin will become ueful laer on vecor w w w,, v v v v

19 comuer grahic ranform 28 fabio ellacini 37 coordinae em review oin are rereened wr a coordinae em careian coordinae in he canonical coord. em canonical coordinae em o o,,,, o comuer grahic ranform 28 fabio ellacini 38 coordinae em review wrie a oin in a new coordinae em can be rereened a an affine mari mulil, o o o o o

20 comuer grahic ranform 28 fabio ellacini 39 coordinae em review an affine ranform i a change of coord. em anoher inerreaion for combinaion of ranform wha i he mari I hould ue o change coord? ju inver reviou definiion i.e. inver combinaion of ranlaion and orhonormal o o comuer grahic ranform 28 fabio ellacini 4 3D ranformaion

21 comuer grahic ranform 28 fabio ellacini 4 2D o 3D ranformaion ado homogeneou formulaion in 3d oin have 4 coordinae ue 44 marice for ranformaion mo conce generalize ver eail roaion much more comle comuer grahic ranform 28 fabio ellacini 42 affine ranformaion z T ranlaion cale z S

22 comuer grahic ranform 28 fabio ellacini 43 affine ranformaion co in in co z R roaion around z comuer grahic ranform 28 fabio ellacini 44 affine ranformaion co in in co R roaion around co in in co R roaion around

23 roaion around arbirar ai in 2d, roaion are around a oin change coordinae frame ranlaion roae around he origin change coordinae frame back imle geomeric conrucion in 3d, he are around an ai change coordinae frame align z wih ai roae around z ai change coordinae frame back comle geomeric conrucion R a, Ta R T a R a, Fa R F a comuer grahic ranform 28 fabio ellacini 45 roaion around arbirar ai ue a change of coordinae em define new coordinae em wih z arallel o ai and origin on he ai build ranform mari a een revioul F z o comuer grahic ranform 28 fabio ellacini 46

24 conruc 3d frame from vecor given wo non-arallel vecor a and b i.e. a lane e, arallel o a, b a / a z b; z z / z z given he vecor a e z arallel o a, chooe arbirar, coninue a above comuer grahic ranform 28 fabio ellacini 47 rereening roaion 3 roaion around major ai remember o chooe order imle bu ha quirk when combining roaion will ue hi for imlici ai and angle combinaion of roaion can be rereened hi wa wih more formalim become elegan and conien quaernion comuer grahic ranform 28 fabio ellacini 48

25 ranforming normal oin and vecor work angen, i.e. difference of oin, work oo normal work differenl defined a orhogonal o he ranformed urface i.e. orhogonal o all angen comuer grahic ranform 28 fabio ellacini 49 ranforming normal b definiion afer ranform for all, we have which give T n n T Xn T T Xn T T T n normal are ranformed b he invere ranoe comuer grahic ranform 28 fabio ellacini 5

26 ranformaion hierarchie ofen need o ranform an objec wr anoher e.g. he comuer on he able when he able move, he comuer move naurall build a hierarch of ranformaion o ranform he able, al i ranform o ranform he comuer, al he able and he comuer ranform comuer grahic ranform 28 fabio ellacini 5 ranformaion hierarchie rereened a a ree daa rucure ranformaion node objec node - leave walk he ree when drawing ver convenien rereenaion for objec all objec can be defined in heir imle form e.g. ever here can be rereened b a ranformaion alied o he uni here comuer grahic ranform 28 fabio ellacini 52

27 ranformaion hierarchie roo ranform ranform here riangle ranform here riangle here comuer grahic ranform 28 fabio ellacini 53 imlemening ranformaion hierarchie ranformaion funcion for each node ge he aren mari mulil he aren and curren marice a he combined mari when calling children ack of ranform uh/o when walking down/u ued b grahic librarie OenGL more fleible generalized mechanim for all aribue comuer grahic ranform 28 fabio ellacini 54

28 raracing and ranformaion ranform he objec imle for riangle ince he ranform o riangle bu mo objec require comle inerecion e here do no ranform o here, bu ellioid ranform he ra much more elegan work on an urface allow for much imler inerecion e onl worr abou uni here, all oher are ranformed comuer grahic ranform 28 fabio ellacini 55 raracing and ranformaion ranforming ra ranform origin/direcion a oin/vecor noe ha direcion i no normalized now i.e. ra arameer i no he diance inerec a ranformed objec ranform he ra b mari invere inerec urface ranform hi oin and normal b mari comuer grahic ranform 28 fabio ellacini 56