Intersection Testing Chapter 16. Department of Computer Engineering Chalmers University of Technology

Transcription

1 Inrion Ting Chapr 6 Dparmn of Compur Enginring Chalmr Uniriy of Thnology

2 Wha for? l A ool n for h graphi popl all h im l Vry imporan omponn: N o mak hm fa! l Fining if an whr a ray hi an obj Piking Ray raing an global illuminaion l For p-up hniqu l Colliion ion ra in a lar lur

3 Exampl Miown Man 3, DICE

4 Som bai gomrial primii l Ray: l Sphr: l Box Axi-align AABB Orin OBB l k-dop

5 Four iffrn hniqu l Analyial l Gomrial l Sparaing axi horm SAT l Dynami l Gin h, on an ri many qui aily Howr, ofn rik ar n o mak hm fa

6 Analyial: Ray/phr l Sphr nr:, an raiu r l Ray: ro+ l Sphr formula: p-r l Rpla p by r, an quar i: r r r + + ο ο ο r + + r o o + + r ο ο ο o r + o + o r

7 Analyial, oninu + + r ο ο ο l B a lil mar? > ο o? < r ο ο l Suh ar all rjion l Ohr hap: r p p y x + / / / + + p b p a p z y x / / + z y x p b p a p

8 Gomrial: Ray/Box Inrion l Box an phr ofn u a bouning olum l A lab i h olum bwn wo parallll plan: l A box i h logial inrion of hr lab in D: BOX

9 Gomrial: Ray/Box Inrion l Inr h plan of ah lab wih h ray max y min x min y max x l Kp max of min an min of max l If min < max hn w go an inrion l Spial a whn ray parallll o lab

10 Sparaing Axi Thorm SAT Pag 563 in book l Two onx polyhron, A an B, ar ijoin if any of h following ax para h obj: An axi orhogonal o a fa of A An axi orhogonal o a fa of B An axi form from h ro prou of on g from ah of A an B axi A an B orlap on hi axi

11 SAT xampl: Triangl/Box l E.g an axi-align box an a riangl l h ax ha ar orhogonal o h fa of h box l Tha i, x,y, an z

12 Triangl/Box wih SAT l Aum ha hy orlapp on x,y,z l Mu oninu ing l Axi orhogonal o fa of riangl axi Triangl n from i

13 Triangl/Box wih SAT 3 l If ill no paraing axi ha bn foun l 3 T axi: box x riangl l Exampl: x-axi from box: box,, riangl - l T all uh ombinaion l If hr i a la on paraing axi, hn h obj o no olli l El hy o orlap

14 Rul of Thumb for Inrion Ting l Apan an rjion Try hm arly on o mak a fa xi l Popon xpni alulaion if poibl l U imnion ruion E.g. 3 on-imnional ina of on omplx 3D, or D ina of 3D l Shar ompuaion bwn obj if poibl l Timing!

15 Anohr analyial xampl: Ray/ Triangl in ail l Ray: ro+ l Triangl ri:,, l A poin in h riangl: l u, +u u- +u + [u,>, u+<] l S u,r, an ol! - - o u

16 Ray/Triangl l Sol wih Cramr rul: o u,,,,,,,, u o # $ & ' u # $ & ' # $ & ' A x b

17 Ray/Triangl l Sol wih Cramr rul: o u,,,,,,,, u o u l Shar faor o p up ompuaion b a b a b a : hi fa U,, # $ & ' u # $ & ' # $ & '

18 Ray/Triangl 3 Implmnaion l B mar! Compu a lil a poibl. Thn l Exampl: u a f a / p p l Compu p f u l Thn ali boun l if u< or u> xi;

19 Ngai half pa Poii half pa Poin/Plan Plan : π : np + l Inr a poin x ino plan quaion: f x n x + f x nx + for x' on h plan f x nx + < for x' on on i of h plan f x nx + > for x' on h ohr i origin x nx x oγ < n nx x oφ > x π

20 Sphr/Plan Box/Plan Plan : π : np + Sphr : r AABB: b min b max l Sphr: ompu f n + l f i h ign ian n normaliz l ab f > r no olliion l ab f r phr ouh h plan l ab f < r phr inr plan l Box: inr all 8 ornr l If all f ha h am ign, hn all poin ar on h am i, an no olliion

21 AABB/plan l Th mar way hown in D Plan : π : np + Sphr : r Box : min b max b l Fin h wo ri ha ha h mo poii an mo ngai alu whn again h plan po N only h r poin OBB almo a ay. Ju fir proj n on OBB ax p: 757 ng n pox n x >?b maxx : b minx poy poz n y >?b maxy : b miny n z >?b max z : b minz ngx n x <?b maxx : b minx ngy ngz n y <?b maxy : b miny n z <?b max z : b minz

22 Ray/Polygon: ry brifly l Inr ray wih polygon plan l Proj from 3D o D l How? l Fin maxn x,n y,n z l Skip ha oorina! l Thn, oun roing in D Toma Aknin-Mőllr 3

23 Volum/Volum l U in olliion ion l Sphr/phr Compu quar ian bwn phr nr, an ompar o r +r l Axi-Align Bouning Box AABB T in D for x,y, an z If A min_x > B max_x or A min_y > B max_y or A min_z > B max_z or B min_x > A max_x or B min_y > A max_y or B min_z > A max_z rurn no_inrion El rurn inrion. x max,y max A x max,y max l Orin Bouning box x min,y min U SAT [ail in book] x min,y min B

24 Viw fruum ing l Viw fruum i 6 plan: l Nar, far, righ, lf, op, l Cra plan from projion marix L all poii half pa b oui fruum No al wih hr -- p , 3r. l Sphr/fruum ommon approah: T phr again ah of h 6 fruum plan: l If oui h plan > no inrion l If inring h plan or ini, oninu If no oui afr all ix plan, hn onraily onir phr a ini or inring l Exampl follow boom

25 Viw fruum ing xampl oui fruum inring fruum l No xa, bu no inorr A phr ha i rpor o b ini, an b oui No i ra l Similarly for box

26 Dynami Inrion Ting [In book: 6-68] l Ting i ofn on ry rnr fram, i.., a ir im inral l Thrfor, you an g quanum ff Fram n Fram n+ l Dynami ing al wih hi l I mor xpni l Dal wih a im inral: im bwn wo fram

27 n Dynami inrion ing Sphr/Plan r l No olliion our: If hy ar on h am i of h plan > l an: >r an >r l Ohrwi, phr an mo -r l Tim of olliion: r n + & ar ign ian r n+ l Rpon: rfl aroun n, an mo - r rrfl or n i ign ian

28 BONUS Dynami Sparaing Axi Thorm l SAT: on axi a a im for orlap l Sam wih DSAT, bu: U a rlai ym whr B i fix i.., ompu A rlai moion o B. N o aju A projion on h axi o ha h inral mo on h axi a wll l N o am ax a wih SAT l Sam riria for orlap/ijoin: If no orlap on axi > ijoin If orlap on all ax > obj orlap

29 BONUS Dynami Swp-an-Prun l hp://graphi.ia.uai.u/~oming/papr/oming_aa_riphy5.pf

30 Exri l Cra a funion by wriing o on papr ha for inrion bwn: wo phr a ray an a phr iw fruum an a phr

31 San Lin Fill S ai g o AB an AC For y A.y, A.y-,...,C.y If yb.y xhang AB wih BC Compu xar an xn. Inrpola olor, ph, xoor for poin xar,y an xn,y For x xar, xar+,...,xn Compu olor, ph for x,y uing inrpolaion. xn Thi i on morn way o rariz a riangl

32 Uing Inrpolaion C C C 3 pifi by glcolor or by rx haing C 4 rmin by inrpolaing bwn C an C 3 C 5 rmin by inrpolaing bwn C an C 3 inrpola bwn C 4 an C 5 along pan C an lin C 4 C C 5 pan C 3

33 Rarizing a Triangl Conx Polygon only Nononx polygon aum o ha bn lla Shar rul.g. olor ha bn ompu for h ri. Dph oluion rol wih z-buffr. Marh aro an lin inrpolaing rx har oupu paramr, a inpu o h fragmn har. Inrmnal work mall

34 Floo Fill Fill an b on rurily if w know a poin loa ini WHITE San onr g ino buffr in g/ini olor BLACK floo_fillin x, in y { ifra_pixlx,y WHITE { wri_pixlx,y,black; floo_fillx-, y; floo_fillx+, y; floo_fillx, y+; floo_fillx, y-; } }

35 Wha you n o know Analyi : B abl o ompu ray phr or ohr formula ray riangl Gomrial Ray/box wih lab- Ray/polygon 3D->D AABB/AABB Ohr: Poin/plan Sphr/plan Box/plan, AABB/plan SAT Know wha a ynami i Unran floofill