Specifying the view transformation

Transcription

1 3D to D Pojectio Po. Aao Latema Based o sides b Po. Hsie-Hsi Sea Lee Schoo o Eectica ad Compute Egieeig Geogia Istitute o Techoog

2 Speciig the view tasomatio Most commo paameteied b: Positio o camea Positio o poit to ook at Vecto idicatig up diectio o camea I Diect3D: D3DXMatiLookAtLH! D3D uses a LHS, but aso have D3DXMatiLookAtRH I XNA: Mati.CeateLookAt RHS I OpeGL: gulookat RHS Ca aso buid a otatiotasatio mati as i the camea was a object i scee, the take the ivese o that mati msd.micosot.com/e-us/iba/bb534vs.85.asp msd.micosot.com/e-us/iba/bb5343vs.85.asp

3 Pojectio om 3D space Much discussio adapted om Joe Fae s atice / 3

4 Caoica view voume Pojectio tasoms ou geomet ito a caoica view voume i omaied device coodiates O X- ad Y-coodiates wi be mapped oto the scee Z wi be amost useess, but used o depth test,,,, -, -, Caoica view voume LHS -, -, to,, used b Diect3D 4

5 Stage covetios,,,,,,,, -, -, Caoica view voume LHS -, -, to,, used b D3D/XNA emembe ee-space coodiates i XNA ae i a RHS!!! -, -,- Caoica view voume LHS -, -, to,, used b OpeGL emembe ee-space coodiates i OpeGL ae i a RHS!!! 5

6 Othogaphic o paae pojectio View pae Viewe s positio Poject om 3D space to the viewe s D space 6

7 Ste o othogaphic pojectio Same sie i D ad 3D No sese o distace Paae ies emai paae Good o tie-based games whee camea is i ied ocatio e.g., Mahjog o 3D Tetis See /Deivig-Pojectio-Matices.htm 7

8 Othogaphic pojectio, t,,,, b, View Voume a ais-aiged bo -, -, Caoica view voume D3D XNA 8

9 9 Othogaphic pojectio math Deive ad,] [ ' ], [ ' b t b t b t ' See /Deivig-Pojectio-Matices.htm

10 Othogaphic pojectio math Deive sight dieet o the age i D3D [,] ' ], [ ' OpeGL tasom o ooks moe ike tasoms See /Deivig-Pojectio-Matices.htm

11 Otho pojectio mati LHS Put it a togethe # % % % % [', ', ',] [,,,] P whee P % % % % % % $ t b t b t b ' See /Deivig-Pojectio-Matices.htm

12 Otho poj LHS Micosot ste Reaagig to ook ike Micosot documetatio [', ',',] [,,,] P whee P $ % ' t b t b b t I Diect3D: D3DXMatiOthoOCeteLH*o,,,b,t,, LHS is deaut sstem i Diect3D

13 Othogaphic pojectio RHS Math the same, but cippig pae iputs i most API cas ae egated so $ iput paametes ae positive t b [', ',',] [,,,] P whee P I Diect3D: D3DXMatiOthoOCeteRH*o,,,b,t,, I XNA: Mati.CeateOthogaphicOCete,,b,t,, I OpeGL: gotho,,b,t,, mati is dieet OpeGL maps to [-,] uses coum vectos ' t b % b t 3

14 Simpe otho pojectio LHS I most othogaphic pojectio setups Z-ais passes though the cete o ou view voume Fied o view FOV eteds equa a To the et as to the ight i.e., - To the top as to the beow i.e., t-b [',',',] [,,,] P whee P $ w ' h % I Diect3D: D3DXMatiOthoLH*o,w,h,, See /Deivig-Pojectio-Matices.htm 4

15 Simpe otho pojectio RHS Math the same, but cippig pae iputs i most API cas ae egated so iput paametes ae positive [', ',',] [,,,] P whee P $ w ' h % I Diect3D: D3DXMatiOthoRH*o,w,h,, I XNA: Mati.CeateOthogaphicw,h,, See /Deivig-Pojectio-Matices.htm 5

16 Pespective pojectio View pae Viewe s positio 6

17 Viewig ustum Fa pae Nea pae Viewe s positio Thik about ookig though a widow i a dak oom 7

18 Viewig ustum with uitue Nea pae Fa pae Viewe s positio 8

19 Pespective pojectio, t,,,, b, View Fustum a tucated pamid -, -, Caoica view voume D3D XNA 9

20 Pespective pojectio mappig Give a poit,, withi the view ustum, poject it oto the ea pae [, ] ad [b, t] We wi map om [,] to [-,] ad om [b,t] to [-,], t,,, -, -,, b, View Fustum Caoica view voume See 3/Deivig-Pojectio-Matices.htm

21 Pespective pojectio math,,,, ' ',,,, To cacuate ew coodiates o ad ' ' ' ' Net app ou othogaphic pojectio omuas See 3/Deivig-Pojectio-Matices.htm

22 Pespective pojectio math,,,,,,,, ' ' t b t b t b ' ' t b t t b b Now et s tacke the compoet See 3/Deivig-Pojectio-Matices.htm

23 3 Pespective pojectio math 3 We kow depth tasomatio has othig to do with ad b t b t b t ' ' ' p q whee p ad q ae costats See 3/Deivig-Pojectio-Matices.htm

24 Pespective pojectio math 4 ' p q whee p ad q ae costats p q p q p ad q ' We kow boed equatios above whe ea pae whe a pae See 3/Deivig-Pojectio-Matices.htm 4

25 5 Pespective pojectio math 5 b t b t b t ' ' ' w' whee,],, [ ] ', ', ', ' [ b t b t b t P P w See 3/Deivig-Pojectio-Matices.htm

26 Simpe pespective pojectio Simia to othogaphic pojectio, i - ad t-b, we ca simpi to # % % % % [', ', ', w'] [,,,] P whee P % % % % % % $ w h ' I a case, we wi have to divide b to obtai [,,, w ] Impemeted b dividig b the outh w' coodiate See 3/Deivig-Pojectio-Matices.htm 6

27 Deie viewig ustum Fa Nea Viewe s positio ov/ Paametes: FOV: Fied o View Aspect atio Width/Height Nea Fa XNA: Mati.CeatePespectiveFiedOView Height See 3/Deivig-Pojectio-Matices.htm 7

28 8 Repaameteied mati h a cot h w P a h/ h/ Need to epace w ad h with FOV ad aspect atio cot cot a a h w h w See 3/Deivig-Pojectio-Matices.htm

29 9 Fia mati o pespective poj LHS :a pae : ea pa height width :aspect atio View FOV a : Fied o cot cot whee,],, [ ] ', ', ', ' [ a a P P w I Diect3D: D3DXMatiPespectiveFovLH*o,a,,, LHS is deaut sstem i Diect3D

30 Fia mati o pespective poj RHS [', ',',w'] [,,,] P whee P $ cota ' cot a % a : Fied o View FOV : aspect atio width height : ea pa : a pae I Diect3D: D3DXMatiPespectiveFovRH*o,a,,, I XNA: Mati.CeatePespectiveFiedOViewa,,, I OpeGL: gupespectivea,,, 3

31 Uit s coodiate sstems Viewpot space:, is bottom-et, is top-ight Scee space coodiates: is i wod uits om the camea, is bottom-et Camea.pieWidth,Camea.pieHeight is top-ight 3

32 Custom pojectios i Uit Fom Camea.pojectioMati documetatio: Use a custom pojectio o i ou ea eed a ostadad pojectio. This popet is used b Uit's wate edeig to setup a obique pojectio mati. Usig custom pojectios equies good kowedge o tasomatio ad pojectio matices. 3

33 Viewpot tasomatio The actua D pojectio to the viewe Cop to ou back bue ame bue Ca be pogammed, scaed,... 33

34 34 Backace cuig Detemie acig diectio Tiage ode mattes How to compute a oma vecto o give vectos? Usig coss poduct o give vectos c, c, c3 b, b, b3 a, a, a3 k j i k j i k j i V V V V Coss poduct k j i k j i a c a c a c V a b a b a b V Vectos

35 Compute the suace oma o a tiage Cockwise omas, LHS 3, 3, v 3i 3j k,, v 4,, v 4i j k 3, 3, v 4i j k,, v 4,, v 3i 3j k 35

36 Backace cuig method Check i the oma is acig the camea How to detemie that? Use Dot Poduct Suace vectos Ee vectos 36

37 Backace cuig method Check i the oma is acig the camea How to detemie that? Use Dot Poduct Suace vectos Ee vectos 37

38 Dot poduct method A B A B cosθ θ B A B > π < θ < π 38

39 Dot poduct method A B A B cosθ θ B A B > π < θ < π 39

40 Dot poduct method 3 A B > π < θ < π 4

41 Cautio! A B > π < θ < π 4

42 Whe to peom backace cuig? Wod Tasom Backace Cuig View Tasom Vete Lightig Pojectio Tasom Cippig Pespective Divide Viewpot Tasom Rasteiatio Wod Tasom View Tasom Backace Cuig Vete Lightig Pojectio Tasom Cippig Pespective Divide Viewpot Tasom Rasteiatio Make sue camea is i coect coodiates! 4

43 How about beoe ou eve stat? Backace Cuig Wod Tasom View Tasom Vete Lightig Pojectio Tasom Cippig Pespective Divide Viewpot Tasom Rasteiatio Wod Tasom Backace Cuig View Tasom Vete Lightig Pojectio Tasom Cippig Pespective Divide Viewpot Tasom Rasteiatio Tasom camea vectos ito object spaces 43

44 O how about at the ve ed? Wod Tasom View Tasom Vete Lightig Pojectio Tasom Cippig Pespective Divide Backace Cuig Viewpot Tasom Rasteiatio Now ou ca just check widig ode i D Wod Tasom View Tasom Vete Lightig Pojectio Tasom Cippig Pespective Divide Viewpot Tasom Backace Cuig Rasteiatio As oicia doe b OpeGL 44

45 3D cippig Test 6 paes i a tiage is iside, outside, o patia iside the view ustum Cippig ceates ew tiages tiaguatio Itepoate ew vetices io 45

46 Appedi 46

47 Cippig agaist a pae Test each vete o a tiage Outside Iside Patia iside Icued computatio ovehead Save uecessa computatio ad badwidth ate Need to kow how to detemie a pae Need to kow how to detemie a vete is iside o outside a pae 47

48 Speciig a pae V5, 6, 7 Pae equatio K,, P,, 3 5*-6*-7*-3 You eed two thigs to speci a pae A poit o the pae p, p, p A vecto oma pepedicua to the pae a, b, c Pae à a* p b* p c* - p 48

49 49 Distace cacuatio om a pae Give a poit R, cacuate the distace Distace > iside the pae Distace o the pae Distace < outside the pae Noma R υ d d θ υ υ υ υ θ cos d P R P R P R P R P R P

50 Distace cacuatio om a pae υ P 8-θ d θ R d R P cos8 θ 5

51 5 Tiaguatio usig itepoatio a, b, c a, b, c,, d d c c s c b b s b a a s a d d d s