Computer Graphics and Animation 3-Viewing

Transcription

1 Compute Gaphics and Animation 3-Viewing Pof. D. Chales A. Wüthich, Fakultät Medien, Medieninfomatik Bauhaus-Univesität Weima caw AT medien.uni-weima.de Ma 5 Chales A. Wüthich

2 Viewing Hee: Viewing in 3D Plana Pojections Camea and Pojection View tansfomation

3 Pipeline 3D Wold Coodinates 2D Device Coodinates Clipping against the view volume Pojection onto pojection plane Tansfomation into Viewpot fo displa 2D Device (Raste coodinates)

4 Pojections Maps Points of a coodinate sstem in the n-dimensional space into a space of smalle dimension. In compute gaphics :3D -> 2D Idea: Compute intesections of pojection as p with a pojection plane π The as pass though point to be pojected and the cente of pojection NOTE: ou can t invet this ~ loss of infomation

5 Pojections Pespective Pojection Pojection plane A B A B Paallel Pojection Pojection plane A B A B Cente of pojection Cente of pojection at infinit

6 Pojections Paallel (othogaphic) Pojection Pespective Pojection ( vanishing pt) Pespective Pojection (2 vanishing pts) hoizon

7 Pojections Pespective pojection models human view sstem (o photogaph) Realistic but: Scales not peseved Angles not peseved paallel pojection less ealistic but peseve scales and angles Peseve paallel lines

8 Plana pojections paallel pespective othogonal Non-othogonal Point House map aonometic Ach. View... Cavalie 2 Points 3 Points isometic...

9 Camea metapho Goal: use camea to tansfom wold coodinates into sceen coodinates Requiement: desciption of the camea

10 Desciption of the camea Position and oientation in Wold Coodinates (WCS) Pojection point (pojection efeence point, PRP) Nomal to the pojection plane (view plane nomal, VPN) Up-vecto (view up vecto, VUP)

11 Camea desciption image plane = view efeence plane Field of view: angle suttended b the viewing window Pojection point Window in image plane Fa plane nea plane Focal length Wold coodinates Up vecto clipping planes View plane nomal

12 Camea desciption Clipping Window on pojection plane (e.g., 35mm film) Detemines also the view diection (von PRP to the mid point of the Window) Field of View Distance of the view plane fom the oigin (focal length). Altenativel, Opening angle (field of view) (FOV) Mapping to aste coodinates Resolution Aspect atio Font and back clipping-planes Limits view to inteesting pat of the scene. Avoids singulaities in computations (b looking back) Limits objects that ae too fa awa (backgound)

13 Pojection with Matices Pojective tansfomations can be epesented though Matices Eas eample: Paallel pojection onto - plane = = = p p p z " # % & = ot M P M P ot ot =

14 Pespective pojection P(,,z) P p ( p, p,) X p d negative z-ais p d = d z p = d d z = z d

15 Pespective pojection The tansfomation P(,,z) -> P p ( p, p,) is pefomed b multipling with the mati M pe : " # % & = " # % & " # % & = = d z z d P M P pe p

16 View Tansfomation Poblem: This woks well if coodinate sstems ae alead unified and aligned with wold coodinates, but not fo the geneal case. Thus we tansfom the wold to whee we need it. Goal: VRP is at oigin View diection is -Z, Y ist Up vecto Sceen Cente VRP=Cente of Pojection Y=UP UP -z View diection

17 Nomalization Move the VRP to the oigin though a tanslation Rotate coodinate sstem, so that Up vecto points UP and the view diection is Z This is done though a otation The elements ae the pojections of the vectos onto the new aes " # % & = z z z R [ ] z T R = [ ] z T R = [ ] z T z R =

18 Recapping Camea VRP T(-VRP) R M Pojection Wold Coodinate Sstem (WCS)

19 Recapping Tansfomation of the WCS into 2D sceen coodinates though mati multiplication Paamete of the vitual camea detemine the composing tansfomation steps Of couse, if I descibe othewise the camea and viewing sstem -> diffeent matices Note: Some camea paametes ae missing, e.g. CW and the aspect atio of the window. Such paamete can be integated though simple tansfomations in the viewing tansfomations.

20 End Copight (c) 988 ILM +++ Ende - The end - Finis - Fin - Fine +++ Ende - The end - Finis - Fin - Fine +++

21 Thank ou Web pages Ma 5 Chales A. Wüthich 2