Raytracing: Intersections. Backward Tracing. Basic Ray Casting Method. Basic Ray Casting Method. Basic Ray Casting Method. Basic Ray Casting Method
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1 Rtig: Itesetios Bkwd Tig COC 4328/5327 ott A. Kig Bsi R Cstig iels i see hoot fom the ee though the iel. Fid losest -ojet itesetio. Get olo t itesetio Bsi R Cstig iels i see hoot fom the ee though the iel. Fid losest -ojet itesetio. Get olo t itesetio Bsi R Cstig iels i see hoot fom the ee though the iel. Fid losest -ojet itesetio. Get olo t itesetio Bsi R Cstig iels i see hoot fom the ee though the iel. Fid losest -ojet itesetio. Get olo t itesetio 1
2 Bsi R Cstig iels i see hoot fom the ee though the iel. Fid losest -ojet itesetio. Get olo t itesetio The Tuth! olvig itesetios e hd imle sufes ield losedfom solutio Geel se: o-lie oot fidig No simle, quik method. Eesive! Wo t lws ovege Whe eeted fo millios of s, ou WILL fid the diveget se! Good News Use imitives with losed-fom solutios. Use ojet-oieted methods. 1 itesetio method e imitive te. Ojet does its ow itesetig. ufes with losed-fom solutios. qudis: shee, lide, oe, ellisoid, oloid, et. olgos. toi, sue-qudis, low-ode slies. hee: Coe: Heoloid: Poloid: Clide: R-Ojet Itesetio Defie ojet imliitl futio f(p) = 0 Fo oit P, whe f is 0, the oit is o the sufe, o-eo defies how f w fom the sufe ou e, usull egtive elow sufe (iside ojet) M ojets e defied imliitl Give otetill ifiite esolutio Tesselltig ojets hde th usig f dietl A ifiite le is defied the futio: f(,,) = A + B + C + D A shee of dius R i 3-se: f(,,) = R 2 hee: Bsi R Model Let s tet s veto. Nmel we eeset the veto = u + vt Bsi R Model Let s tet s veto. Nmel we eeset the veto = u + vt is oit log the 2
3 Bsi R Model Bsi R Model Let s tet s veto. Nmel we eeset the veto = u + vt is oit log the u is the sttig oit Let s tet s veto. Nmel we eeset the veto = u + vt is oit log the u is the sttig oit v (uit veto) is the dietio Bsi R Model R/hee Itesetio Let s tet s veto. Nmel we eeset the veto = u + vt is oit log the u is the sttig oit v uit veto is the dietio t is diste log. stut { ve4 stt; ve4 dietio; } imle Cse: A shee of dius 1 eteed t the oigi The itesets the shee whe = u + vt stisfies the equtio fo the shee = 1 2 = 1 u 2 + 2uvt + v 2 t 2 = 1 We solve usig the qudti fomul. u uvt + v 2 t 2 = 0 ee the tig otes fo detils o voidig oud-off eos d solvig effiietl. R/hee Itesetio R/hee Itesetio Wht shee eteed t,, o dius? Is thee othe w? = 2 Plug i equtio d get =,, u + v t u + v t + u +v t 2 = 2 Ad solve usig the qudti fomul. = u + vt 2 = v 2 +v 2 +v 2 = 1 = 2 v u + v u + v u = u 2 + u 2 + u 2 2 = 2 u v = u u - 2 3
4 R/hee Itesetio R/hee Itesetio Is thee othe w? Is thee othe w? Wht if we move the shee to the oigi? Wht if we move the shee to the oigi?,, T =,, How will we get the to hit the shee t the oigi? T =,, R/hee Itesetio R/hee Itesetio Is thee othe w? Wht if we move the shee to the oigi? How will we get the to hit the shee t the oigi? Will the tsfomed lso hit the tsfomed shee? o we use the oigil equtio whih is it simle (less lultios) This oh woks fo tsfomed ojet. tt with imitive ojet. It is tsfomed (sled, otted tslted, et.) mti,. The ivese of tht mti will ut it k to its oigil stte. o the just eeds to e tsfomed 1 the the simle /ojet itesetio e used. u = 1 u v = 1 v 1,, Itesetio i Wold o Ojet e? Noml hee t oigi (ojet se) = v 2 = 1 = 2uv = u 2 1 hee eteed t,, with dius = v 2 = 1 = 2 u v = u u 2 How muh moe mth? Wht et mth tsfomig? o wh do it? How does t elte fo ojet se d wold se? 3-1* We eed the oml to lulte illumitio d efletio veto. Wht is N fo uit shee the oigi tht itesets t oit? Wht is N fte tht shee is tsfomed usig the mti? C we tsfom the oml? Rigid tsfoms fie (R,T) les use olems Emle: s M sles i.5 d 2 N ojet MN ojet Wog! N = 1, 1 2 Noml must e eediul N = 1, 1 4 lig distots oml i oosite sese of sle lied to sufe 4
5 Review Noml The mti, tsfoms ojet se ito wold se Theefoe the ivese goes fom wold 1 se to ojet se If q is the oesodig oit to i wold se, the q = wold = ojet q w = o Usig the ivese = 1 1 q o = 1 wold se w ojet = 1 wold ojet se Fo le tht sses though the oigi, d oit,, o the le, N = 0 I mti fom this eomes, N T = 0 If q is the wod se oit to q = = 1 q N T 1 q = 0 desies le i wold se whose oml is N T 1 Let N w (wold se oml) e the oml of the tsfomed le, so N w = N T 1 N w = 1T N 1 goes fom ojet se to wold se = t N T 1 q = 0 o, the tsose of the ivese tkes ou ojet (AB) T se = B T Aoml T ito wold se! R/Tigle Itesetio Tigle defied veties,,,. 3 oits defies le with oml = Fo oit,, i the le =? Whee the itesets the le, = u + vt stisfies the ove equtio so u + vt = 0 vt = u u t = v to view otted view R/Tigle Itesetio If llel to the le, v=0 C t solve fo t d o itesetio to view v v otted view R/Tigle Itesetio If llel to the le, v=0 C t solve fo t d o itesetio Othewise we hve itesetio v withi the le. Does t me tigle is iteseted. v to view otted view R/Tigle Itesetio If llel to the le, v=0 C t solve fo t d o itesetio Othewise we hve itesetio withi the le. Does t me tigle is iteseted. If the thee dot oduts ll hve the sme sig, the oit is iside the tigle. Wh? 5
6 Qudis R: (t) = o + t d, (t) = o + t d (t) = o + t d Tous hee: Coe: Clide: Heoloid: Poloid: Vitios? Use the tsfomtio tik Podut of two imliit iles ( R) = 0 ( + R) = 0 (( R) )(( + R) ) = ( 2 2R + R ) ( 2 + 2R + R ) = R R R 2 + R 4 = ( R 2 ) R R 2 ufe of ottio: ele 2 with f(,,) = ( R 2 ) 2 + 4R 2 ( 2 2 ) Quti!!! (ee Ghis Gems V fo solve) U to fou tous itesetios R R-Ojet Itesetio Retus itesetio i hit eod i Net field eles hit eod to hold list of itesetios List ol o-egtive itesetio metes R lws oigites side If fist t = 0 the oigited iside Pit lssifies segmets Odd segmets i Eve segmets i i i Bsi R Cstig iels i see hoot fom the ee though the iel. Fid losest -ojet itesetio. Get olo t itesetio Illumitio Model C123 INTRODUCTION TO COMPUTER GRAPHIC umm uttig it ll togethe imle, o-eusive te P = eept fo eh smle of imge: Comute d fo eh ojet: Iteset P+td with ojet // Of ll the ojets tht iteset, whih oe is visile? elet ojet with smllest o-egtive t-vlue (visile ojet) Illumitio Model Fo ojet whee does the light ome fom? Diet fom light soue Though the ojet. Refleted fom othe ojet Iidet illumitio (miet) Fo this ojet, fid ojet se itesetio oit Comute oml t tht oit Tsfom oml to wold se Use wold se oml fo lightig omuttios Adies v Dm Otoe 29, of 50 6
7 Iidet Illumitio Tes of Rs Light Whee does this ome fom? How light tsmitted (efted) though othe ojet. How light ouig off of o-efletive sufe. Fo ow we wo t wo this iidet illumitio, it is the sujet of othe methods (glol illumitio, diosit, hoto mig). We ll just ll it miet light. To te the light kwd we eed to efom the illumitio lultios. To do this we eed few et tes. Pim s - C light dietl to iel. eod s get light to oit hdow s - Big light fom the light soue. Refletio s - Big light efleted fom othe sufe. Tsmissio s - C light though ojet. Pim R Oque ojet 2 hdow R R 2 R 1 P Ee 1 3 T 1 T 2 R 3 Tset ojet Reusive Rs Reusive R Tig Demo (2d) RCst(see) fo ll iels (,) i see: te(fomeethough(,)) hde(oit, ) Colo = kgoud; fo eh light if!hdow(oit,, light) Colo += PhogIllumitio(oit,, light) if seulmteil te(eflet(oit, )) if eftive te(eftio(oit,)) etu Colo te() if (itesetio= losestitesetio()) etu hde(itesetio, ) else etu kgoudcolo losestitesetio() fo ll ojets fid itesetio fo losest itesetio etu the itesetio oit, sufe oml, sufe, sufe ttiutes, et. htt:// eils/hegh/te/t_jv/ te.html 7
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