Basic Geometry in CG

Transcription

1 Bic Geomer in CG CSE 87 Fll 9 Inro 3D geomer in liner lgebr Rigid bod rnformion Deformion in liner lgebr One mri o repreen cmer Quernion CSE 87 Fll 9

2 Bic Rendering Model Model for objec nd cmer? [Slullek 5] Reriion: Projec geomer forwrd CSE 87 Fll 9 R rcing: Projec imge mple bckwrd 3 Projecie Plne* Projecie Spce PV Sud line Euilence cl / / Impornce Cmer rnformion ner CSE 87 Fll 9 4

3 Repreenion How do we repreen n objec? Poin p [ ] Mhemicl Funcion X Y R Polgon mo commonl ued Poin CSE 87 Fll 9 Connecii 5 Poin nd Vecor A 3D poin p [ ] Repreen locion wih repec o ome coordine em ffine A 3D ecor [ ] Repreen diplcemen from poiion liner CSE 87 Fll 9 6

4 Vecor Spce Coni of e of elemen clled ecor nd wo operion h re defined on hem ddiion nd clr muliplicion CSE 87 Fll 9 7 Vecor Addiion Gien V [X Y Z] nd W [A B C] VW [XA YB ZC] Properie of Vecor ddiion Commuie: VWWV Aociie UVW UVW Addiie Ideni: V V Addiie Inere: VW W-V CSE 87 Fll 9 8

5 Prllelogrm Rule o iulie wh ecor ddiion i doing here i D emple: V VW W CSE 87 Fll 9 9 Sclr Produc Gien V [X Y Z] nd Sclr nd V [X Y Z] Properie of Vecor muliplicion Aociie: V V Muliplicie Ideni: V V Sclr Diribuion: V VV Vecor Diribuion: VW VW CSE 87 Fll 9

6 Do Produc nd Dince Gien u [ ] nd [ b c] u bc he Eucliden dince of u from he origin i r nd i denoed b u Noice h u ru u he Eucliden dince beween u nd i r - -b -c nd i denoed b u- CSE 87 Fll 9 Properie of he Do Produc Gien ecor u w nd clr he reul of do produc i SCALAR lue Commuie: w w Non-degenere: onl when Biliner: uw u w CSE 87 Fll 9

7 Angle nd Projecion Alernie iew of he do produc w w co where i he ngle beween nd w If i uni ecor hen if we perpendiculrl projec w ono cn cll hi newl projeced ecor u hen u w w CSE 87 Fll 9 u 3 Mrice Liner operion on ecor pce 33 Mri A look like ij refer o he elemen of mri A CSE 87 Fll 9 4

8 Mri Muliplicion If A i n n k mri nd B i k p hen AB i n p mri wih enrie cijwhere cij ibj Alerniel if we ook he row of A nd column of B indiidul ecor hen cija i B j where he ubcrip refer o he row nd column repeciel CSE 87 Fll 9 5 Mri Muliplicion Properie Aociie: ABC ABC Diribuie: ABC ABAC Muliplicie Ideni: I dig ure mri NO commuie: AB BA CSE 87 Fll 9 6

9 Deerminn Defined on ure mri nn de A A n i A i i Where A i deerminn of n-n- ubmri A goen b deleing he fir row nd he ih column CSE 87 Fll 9 7 Recurie Definiion!! he bi ce de of mri i defined o be d-bc where b c d CSE 87 Fll 9 8

10 Ue of he Deerminn? Liner Independence of column in mri Cro Produc Gien ecor [ 3 ] w[w w w 3 ] he cro produc i defined o be he deerminn of i w j w k 3 w 3 w3 3w 3w w w w 3 CSE 87 Fll 9 9 Cro Produc Properie he Cro Produc of nd w i denoed b w I VECOR perpendiculr o he plne defined b nd w w w in i he ngle beween nd w w-w w w CSE 87 Fll 9

11 Mri rnpoe nd Inere he rnpoe of mri A denoed b A i defined ij ji echnging he row nd column If A nd B re nn mrice nd ABBAI hen B i he inere of A denoed b A - AB - B - A - me pplie for rnpoe M M CSE 87 Fll 9 Mehod for finding he Inere Eplici Mehod Guin-Jordn Eliminion Cree he Augmened mri [A I] nd reduce he lef ide o he ideni uing elemenr row operion nd he righ hnd ide will be he inere. ie. [I A - ] Crmer Rule Sole for A where ijdeubmria ij A - /deaa CSE 87 Fll 9

12 Implici Mehod Ined of eplicil clculing A - here re echniue h ole euion of he form Ab em of liner euion. Clerl A - b bu we do no need o eplicil clcule A - o clcule. LU Decompoiion QR Fcoriion Singulr Vlue Decompoiion SVD Conjuge Grdien if pre CSE 87 Fll 9 3 rnformion Wh ue rnformion? Cree objec in conenien coordine Reue bic hpe muliple ime Hierrchicl modeling Sem independen Virul cmer CSE 87 Fll 9 4

13 CSE 87 Fll 9 5 rnlion CSE 87 Fll 9 6 Properie of rnlion

14 CSE 87 Fll 9 7 Roion D φ φ in co r r φ φ φ φ co in in co in in co co r r r r in co φ φ r r φ φ φ φ φ φ co in in co in in in co co co φ co in in co CSE 87 Fll 9 8 Roion D So in mri noion co in in co

15 CSE 87 Fll 9 9 Roion 3D co in in co co in in co co in in co R R R CSE 87 Fll 9 3 Properie of Roion R I φ φ R R R R φ φ R R R R R R φ φ b b R R R R order mer!

16 Combining rnlion & Roion R 45 R 45 CSE 87 Fll 9 3 Combining rnlion & Roion R R R R R R CSE 87 Fll 9 3

17 CSE 87 Fll 9 33 Scling S Uniform cling iff CSE 87 Fll 9 34 Affine operion rnlion c f Scling d Roion CCW bou co -in in co Skew or Sher

18 CSE 87 Fll 9 35 Homogeneou Coordine w Z Y X cn be repreened where w Z w Y w X 3D Projecie Spce: ech poin een line in 4D CSE 87 Fll 9 36 rnlion Reiied

19 CSE 87 Fll 9 37 Roion & Scling Reiied S co in in co R CSE 87 Fll 9 38 Combining rnformion M RS R RS R S where RS M

20 CSE 87 Fll 9 39 rnforming ngen p p p p M M M M CSE 87 Fll 9 4 rnforming Norml n n n n n n n n n n n n M M M M M M

21 Roion bou n rbirr i Roe b round uni i r r CSE 87 Fll 9 4 Roion bou n rbirr i Roion i liner ro b ro ro b ro α α ro b CSE 87 Fll 9 b ro b rob ro 4

22 CSE 87 Fll 9 43 Roion bou n rbirr i Projecion operor: b b b b b α α \ b \ b CSE 87 Fll 9 44 Roion bou n rbirr i Cree n orhonorml bi: 3 3 \ \ e e e r r e r e

23 Roion bou n rbirr i Decompoe before roing: \ r α r ro ro \ r α r ro \ r ro α r CSE 87 Fll 9 ro \ r \ r \ r e α r ro e α r co e in e α r co \ r in r r r 45 An Alernie View We cn iew he roion round n rbirr i e of impler ep We know how o roe nd rnle round he world coordine em Cn we ue hi knowledge o perform he roion? CSE 87 Fll 9 46

24 Roion bou n rbirr i rnle he pce o h he origin of he uni ecor i on he world origin Roe uch h he eremi of he ecor now lie in he plne -i roion Roe uch h he poin lie in he -i -i roion Perform he roion round he -i Undo he preiou rnformion CSE 87 Fll 9 47 Roion bou n rbirr i Sep Roe -i bc CSE 87 Fll 9 48

25 Cloer Look Y-Z Plne Need o roe α degree round he -i α CSE 87 Fll 9 49 Euion for α b c in α b c b c co α b c CSE 87 Fll 9 5

26 Roion bou he Y-i Uing he me nli before we need o roe β degree round he Y-i CSE 87 Fll 9 5 Roion bou he Z-i Now i i ligned wih he Z-i hu we cn impl roe degree round he Z-i. hen undo ll he rnformion we ju did CSE 87 Fll 9 5

27 CSE 87 Fll 9 53 Euion ummr c b R R R R R c b ro i α β β α CSE 87 Fll 9 54 Deformion rnformion h do no preere hpe Non-uniform cling Shering pering wiing Bending

28 CSE 87 Fll 9 55 Shering CSE 87 Fll 9 56 pering f f Imge coure of W 3D Compuer Grphic

29 CSE 87 Fll 9 57 wiing co in in co Imge coure of W 3D Compuer Grphic CSE 87 Fll 9 58 Bending k h g f Imge coure of W 3D Compuer Grphic

30 Coordine Sem Objec coordine World coordine Cmer coordine Normlied deice coordine Window coordine CSE 87 Fll 9 59 Wihin OpenGL Objec Coordine CSE 87 Fll 9 Modeliew Mri Ee coordine Projecion Mri Clip coordine glbegingl_polygon; glvere3d; glvere3db; glvere3d c; glend; Homogenie Window o Viewpor Normlied deice coordine Viewpor coordine 6

31 Objec Coordine Conenien plce o model he objec O CSE 87 Fll 9 6 World Coordine Common coordine for he cene O O W CSE 87 Fll 9 6

32 CSE 87 Fll 9 63 Cmer Coordine Coordine em wih he cmer in conenien poe u n n r r u r n n n u u u M CSE 87 Fll 9 64 Normlied Deice Coordine Deice independen coordine Viible coordine uull rnge from:

33 Window Coordine Adjuing he NDC o fi he window i he lower lef of he window w w nd nd CSE 87 Fll 9 widh heigh 65 Perpecie Projecion king he cmer coordine o NDC ner CSE 87 Fll 9 66

34 Perpecie Projecion p p ner ner ner CSE 87 Fll 9 67 Perpecie Projecion Mp lefrigh o - when -ner ner fr CSE 87 Fll 9 ner lef ner lef righ lef ner lef righ lef ner righ lef righ lef righ lef lef righ ner 68

35 Peudodeph Mp -ner-fr o - fr ner fr ner fr ner fr ner Line re preered hrough he rnformion See Newmn nd Sproull 8 for he full deriion CSE 87 Fll 9 69 Perpecie Projecion ner righ lef P ner op boom righ lef righ lef op boom op boom fr ner fr ner fr ner fr ner CSE 87 Fll 9 7

36 CSE 87 Fll 9 7 Perpecie Projecion Preere Srigh Line b b b b F b E b D b C b B b A mp o Compue All he numeror h depend on -. All he he me denominor. herefore he rio re conn. CSE 87 Fll 9 7 Orhogrphic Projecion ner

37 Orhogrphic Projecion p p ner CSE 87 Fll 9 73 Orhogrphic Projecion Mp lefrigh o - lef righ lef righ lef lef lef righ lef righ lef righ lef CSE 87 Fll 9 74

38 Orhogrphic Projecion Mp nerfr o - ner fr ner fr ner CSE 87 Fll 9 ner ner fr ner fr ner fr ner 75 Orhogrphic Projecion righ lef P op boom fr ner righ lef righ lef op boom op boom fr ner fr ner CSE 87 Fll 9 76

39 CSE 87 Fll 9 77 Orhogrphic Projecion Preere Srigh Line b b b F b E D b C B b A mp o CSE 87 Fll 9 78 Wh uernion? he moion of he keleon i underling mo of chrcer nimion Mo keleon re riculed: A figure mde up of erie of link bone conneced join roion

40 CSE 87 Fll 9 79 Roion uing Quernion* Quernion k j i k j i k j i CSE 87 Fll 9 8 Roion uing Quernion* Quernion muliplicion p p p p p p r p r p p p Aociie No Commuie

41 CSE 87 Fll 9 8 Roion uing Quernion* Quernion mgniude 4 3 Quernion inere CSE 87 Fll 9 8 Roion uing Quernion* Quernion roion: ro r / in / co

42 CSE 87 Fll 9 83 Roion uing Quernion* Combining roion: ro ro ro ro ro CSE 87 Fll 9 84 Roion uing Quernion* Mri Form: w w w w w w w w w w k j i w

43 Summr Poin/Vecor repreened b Vecor Decribed in coordine em rnformion Objec Chnge of coord Projecion mri for cmer Quernion for roion CSE 87 Fll 9 85