Ray Tracing The Rest of Your Life

 

Yay Srajgdf; Sln Ynst o` Wour Bg`n ]ntnr Zlgrbny   Jopyrgflt Jopyrgf lt 9=10 ]ntnr Zlgrbny. Abb rgflts rnsnrvnc.  

Jlaptnr =; Ovnrvgnw   Gd Yay Gd Yay Srajgdf Srajgdf Gd Odn V Vnnendc nnendc adc  adc Yay  Yay Srajgdf; Srajgdf; tln Dnxt V Vnne  nne , you iugbt a ‒rnab‘ ray trajnr.   Gd tlgs vobuhn, G assuhn you wgbb in pursugdf a jarnnr rnbatnc to ray trajgdf adc wn wgbb cgvn gdto tln hatl o` jrnatgdf a vnry snrgous ray trajnr. trajnr. Vlnd you arn codn you sloubc in rnacy to start hnssgdf wgtl tln hady snrgous johhnrjgab ray trajnrs udcnrbygdf tln hovgn adc procujt cnsgfd gdcustrgns. gdcustrgns. Slnrn arn hady hady tlgd tlgdfs fs G co dot jovnr gd tlgs slort vobuhn5 G cgvn gdto odby odn odn o` hady ways to wrgtn a Ho Hodtn dtn Jarbo rnd rndcnrgdf cnrgdf profrah. G cod‗t co slacow rays (gdstnac G haen rays horn bgenby to fo towarc bgflts), igcgrnjtgodab hntlocs, Hntropobgs Hntropobgs hntlocs, or plotod happgdf. Vlat G c co o gs spn spnae ae gd tln badfuafn o` o` tln `gnbc tlat stucgns tlosn hntlocs. G tlgde o` tlgs iooe aass a cnnp cnnp nxposurn nxposurn tlat jad in your `grst o` hady hady,, adc gt wgbb nqugp you wgtl sohn o` tln jodjnpts, hatl, adc tnrhs you wgbb dnnc to stucy tln otlnrs.   www.gd1wnnendc.joh wgbb  wgbb lavn `urtlnr rnacgdfs adc rn`nrndjns. As in`orn,  in`orn, www.gd1wnnendc.joh   Ajedowbncfnhndts; tlades to Cavn Lart adc Knad Iujebny `or lnbp od tln orgfgdab Ajedowbncfnhndts; hadusjrgpt. Slades to ]aub Hnbgs aadc dc H Hattlnw attlnw Ln Lnghbgjl ghbgjl `or `gdcgdf nrrors.  

Jlaptnr 1; A Zghpbn Hodtn Jarbo ]rofrah ]rofrah  

 

Bnt‗s start wgtl odn o` tln sghpbnst Hodtn Jarbo (HJ) profrahs. HJ profrahs fgvn a Bnt‗s statgstgjab nstghatn o` ad adswnr, adc tlgs nstghatn fnts horn adc horn ajjuratn tln bodfnr you rud gt. Slgs iasgj jlarajtnrgstgj o` sghpbn profrahs procujgdf dogsy iut nvnr-inttnr adswnrs gs wlat HJ gs abb aiout, adc gt gs nspnjgabby fooc `or appbgjatgods bgen fraplgjs wlnrn frnat ajjurajy gs dot dnncnc.   As ad nxahpbn, bnt‗s nstghatn ]g nstghatn ]g.. Slnrn arn hady ways to co tlgs, wgtl tln Iu``od Dnncbn proibnh ingdf a jbassgj jasn stucy stucy.. Vn‗bb Vn‗bb co a vargatgod gdspgrnc iy tlat. Zupposn you lavn a jgrjbn gdsjrginc gdsgcn a squarn;

Dow, supposn you pgje radcoh pogdts gdsgcn tln squarn. Sln `rajtgod o` tlosn radcoh Dow, pogdts tlat ndc up gdsgcn tln jgrjbn sloubc in proportgodab to tln arna o` tln jgrjbn. Sln nxajt `rajtgod sloubc gd `ajt in tln ratgo o` tln jgrjbn arna to tln squarn arna.

@rajtgod 8 (]g*Y*Y) / ((9Y)*(9Y)) 8 ]g/3   Zgdjn tln Y tln Y jadjnbs  jadjnbs out, wn jad pgje wlatnvnr gs johputatgodabby jodvndgndt. Bnt‗s Bnt‗s fo wgtl Y wgtl  Y81 81 jndtnrnc at tln orgfgd;

 

 

Slgs fgvns hn tln adswnr Nstghatn adswnr Nstghatn o` ]g 8 :.170   G` wn jladfn tln profrah to rud `ornvnr adc kust prgdt out a ruddgdf nstghatn;

  Vn fnt vnry qugjeby dnar ]g, adc tlnd horn sbowby znro gd od gt. Slgs gs ad nxahpbn o` tln  Baw o` Cghgdgslgdf Ynturds, Ynturds, wlnrn najl sahpbn lnbps bnss tlad tln bast. Slgs gs tln worst part o` HJ. Vn jad hgtgfatn tlgs cghgdgslgdf rnturd iy stratg`ygdf stratg`ygdf tln  tln sahpbns (o`tnd jabbnc kgttnrgdf jabbnc  kgttnrgdf), ), wlnrn gdstnac o` taegdf radcoh sahpbns, wn taen a frgc adc taen odn sahpbn wgtlgd najl;  

Slgs jladfns tln sahpbn fndnratgod, iut wn dnnc to edow low hady sahpbns wn arn taegdf gd acvadjn injausn wn dnnc to edow tln frgc. frg c. Bnt‗s taen a ludcrnc hgbbgod adc try gt iotl ways;  

 

 

G fnt;    Ynfubar Nstghatn o` ]g 8 :.131> Ztratg`gnc Nstghatn o` ]g 8 :.131>7   Gdtnrnstgdfby, tln stratg`gnc hntloc gs dot odby inttnr, gt jodvnrfns wgtl a inttnr asyhptotgj Gdtnrnstgdfby, ratn! ^d`ortudatnby, ^d`ortudatnby, tlgs acvadtafn cnjrnasns wgtl tln cghndsgod o` tln proibnh (so `or nxahpbn, wgtl tln :C sp splnrn lnrn vobuhn vnrsgod tln fap woubc in bnss). Slgs gs jabbnc jabbnc tln Jursn o` Cghndsgodabgty. Cghndsgodabgty. Vn Vn arn fogdf to in vnry lgfl cghndsgodab (najl rn`bnjtgod accs two cghndsgods), so G wod‗t stratg`y gd tlgs iooe. Iut g` you arn nvnr cogdf sgdfbnrn`bnjtgod or slacowgdf or sohn strgjtby 9C proibnh, you cn`gdgtnby wadt to stratg`y stratg`y..  

Jlaptnr 9; Odn Cghndsgodab HJ Gdtnfratgo Gdtnfratgod d   Gdtnfratgod gs abb aiout johputgdf arnas adc vobuhns, so wn joubc lavn `rahnc Jlaptnr 1 gd ad gdtnfrab `orh g` wn wadtnc to haen gt haxghabby jod`usgdf. Iut sohntghns gdtnfratgod gs tln tln host da daturab turab adc jjbnad bnad way to `orhubatn tlgdf tlgdfs. s. Yndcnrgdf gs o`tnd sujl a proibnh. Bnt‗s booe at a jbassgj gdtnfrab;

 

  Gd johputnr sjgndjy dotatgod, wn hgflt wrgtn tlgs as;    G 8 arna( arna( xT9, =, 9 )   Vn joubc abso wrgtn gt as;    G 8 9*avnrafn(xT9, =, 9)   Slgs suffnsts a HJ approajl;

Slgs, as nxpnjtnc, procujns approxghatnby tln nxajt adswnr wn fnt wgtl abfnira, G abfnira, G 8 %. 9>% . Slgs fgvns rgsn to a jbnvnr oisnrvatgod tlat udcnrbgns hady hntlocs to fndnratn dodudg`orh radcoh duhinrs. Vn wadt a `udjtgod `() `udjtgod `() tlat wlnd wn jabb gt as `(cradc3% sloubc in aiovn odn. G` `() `() gs  gs gdjrnasgdf, tlnd wn woubc nxpnjt `(=.9>) 8 1.=. Slgs jad in fndnrabgznc to `gfurn out `() `or nvnry possgibn gdput;   `(](x)) 8 x

 

  Slat hnads ` kust udcons wlatnvnr wlatnvnr ]  ]  cons. Zo,   `(x) 8 ]-1(x)   Sln -1 hnads ‒gdvnrsn `udjtgod‘. ^fby dotatgod, iut stadcarc. @or our purposns wlat tlgs hnads gs, g` wn lavn pc` p() pc` p() adc  adc gts juhubatgvn cgstrgiutgod `udjtgod ]() `udjtgod ](),, tlnd g` wn co tlgs to a radcoh duhinr wn‗bb fnt wlat wn wadt;   n 8 ]-1(cradc3, tlat to sahpbn tlnta tlnta wn  wn lavn;   r9 8 GDSNFYAB_=Ttlnta 9*]g*`(t)sgd(t)   Lnrn `(t) `(t) gs  gs ad as ynt udjabjubatnc jodstadt J , so;   r9 8 GDSNFYAB_=Ttlnta 9*]g*J*sgd(t)

 

  Cogdf sohn abfnira/jabjubus tlgs ygnbcs;   r9 8 9*]g*J*(1-jos(tlnta))   Zo   jos(tlnta) 8 1 - r9/(9*]g*J)   Vn edow tlat `or Y9 `or Y9 8 1 wn 1 wn sloubc fnt tlnta_hax, so wn jad sobvn `or J;   jos(tlnta) 8 1 + r9*(jos(tlnta_hax r9*(jos(tlnta_hax)-1) )-1)    ]lg wn sahpbn bgen in`orn, so;    z 8 jos(tlnta) 8 1 + r9*(jos(tlnta_hax)-1) r9*(jos(tlnta_hax)-1)  x 8 jos(plg)*sgd(tlnta) 8 jos(9*]g*r1)*sqrt(1-zT9) jos(9*]g*r1)*sqrt(1-zT9)  8 sgd(plg)*sgd(tlnta) 8 sgd(9*]g*r1)*sqrt(1-zT9)   Dow wlat gs tlnta_hax? gs tlnta_hax?  

Vn jad snn `roh tln `gfurn tlat sgd(tlnta_hax) 8 Y / bndftl( j- p ). Zo Zo;;

 

  jos(tlnta_hax) 8 sqrt(1sqrt(1- YT9/bndfltl_squar YT9/bndfltl_squarnc(  nc( j- p ))   Vn abso dnnc to nvabuatn tln pc`  tln pc`   o` o` cgrnjtgods. @or cgrnjtgods towarc tln splnrn tlgs gs 1/sobgc_adfbn. Vlat gs tln sobgc adfbn o` tln splnrn? Gt las sohntlgdf to co wgtl tln J  aiovn. Gt, iy cn`gdgtgod, gs tln arna od tln udgt splnrn, so tln gdtnfrab gs   Zobgc_adfbn 8 GDS_=T9]g GDS_=Ttlnta_hax sgd(tlnta) 8 9*]g (1-jos(tlnta_hax))   Gt‗s fooc to jlnje tln hatl od aabb bb sujl jjabjubatgods. abjubatgods. G usuabby pbuf gd tln nxtrnhn jasns jasns (tlade you `or tlat jod jodjnpt, jnpt, Hr Hr.. Lortod– hy hy lgfl sj sjloob loob plysgjs tnajlnr). @or a znro znro racgus splnrn jos(tlnta_hax) 8 =, a =, adc dc tla tlatt wores wores.. @or a splnrn tadfndt at at   p, jos(tlnta_hax) 8 =, adc 9*]g gs tln arna o` a lnhgsplnrn, so tlat wores too.   Sln splnrn splnrn jbass  jbass dnncs tln two pc`  two  pc` -rnbatnc -rnbatnc `udjtgods;

Vgtl tln utgbgty `udjtgod;

Vn jad `grst try t ry kust sahpbgdf tln splnrn ratlnr tlad tln bgflt;

 

Slgs ygnbcs a dogsy iox, iut tln jaustgj udcnr tln splnrn gs fooc. Gt tooe `gvn tghns as bodf as sahpbgdf tln bgflt cgc `or hy jocn. Slgs gs proiaiby injausn tlosn rays tlat lgt tln fbass arn nxpndsgvn!

  Vn sloubc proiaiby kust sahpbn iotl tln splnrn adc tln bgflt. Vn jad co tlat iy jrnatgdf a hgxturn cndsgty o` tlngr two cndsgtgns. Vn Vn joubc co tlat gd tln jobor jobor `udjtgod  `udjtgod iy passgdf a bgst o` lgtaibns gd adc iugbcgdf a hgxturn pc`  hgxturn pc` , or wn joubc acc pc`  acc pc`  `udjtgods  `udjtgods to lgtaibn_bgst . G tlgde iotl tajtgjs woubc wore `gdn, iut G wgbb fo wgtl gdstruhndtgdf lgtaibn_bgst .  

 

Vn assnhibn a bgst to pass gd to jobor jobor..

Adc wn fnt a cnjndt ghafn wgtl 1=== sahpbns as in`orn;

 

Ad astutn rnacnr pogdtnc out tlnrn aarn rn sohn i ibaje baje spnje spnjess gd tln ghafn aiovn. aiovn. Abb Hodtn

 

Jarbo Yay Srajnrs lavn tlgs as a hagd boop;   gxnb_jobor 8 avnrafn(hady hady sahpbns)   G` you `gdc yoursnb` fnttgdf sohn `orh o` ajdn gd tln ghafns, adc tlgs ajdn gs wlgtn or ibaje, so odn ‒iac‘ sahpbn snnhs to egbb tln wlobn pgxnb, tlat sahpbn gs proiaiby a lufn duhinr or a DaD. Slgs partgjubar ajdn gs pro proiaiby iaiby a D DaD. aD. Hgdn snnhs to john john up odjn gd nvnry 1=-1== hgbbgod rays or so.   Zo igf cnjgsgod; swnnp tlgs iuf udcnr tln ruf adc jlnje `or DaDs, or kust egbb DaDs adc lopn tlgs consd‗t consd‗t john i iaje aje to igtn us batnr batnr.. G wgbb abways opt `or tln bazy stratnfy, stratnfy, nspnjgabby wlnd G edow edow `boatgdf `boatgdf pogdt pogdt gs larc. @grst, low co wn jlnje jlnje `or a DaD? Sln odn tlgdf G abways rnhnhinr `or DaDs gs tlat ady g`  tnst   tnst wgtl a DaD gd gt gs `absn. Slgs bnacs to tln johhod trgje;

Adc wn jad gdsnrt tlat gd tln hagd boop;

  Lappgby, tln ibaje spnjes arn fodn;  

 

 

Jlaptnr 11; Sln Ynst o` Wour Bg`n   Sln purposn o` tlgs iooe was to slow tln cntagbs o` cottgdf abb tln g‗s o` tln hatl od odn way o` orfadgzgdf a plysgjabby-iasnc rndcnrnr‗s sahpbgdf approajl. Dow you jad nxpborn a bot o` cg``nrndt potndtgab patls.   G` you wadt to nxpborn Hodtn Jarbo hntlocs, booe gdto igcgrnjtgodab adc patl spajnc approajlns sujl as Hntropobgs. Wour Wour proiaigbgty spajn wod‗t in ovnr sobgc adfbn iut wgbb gdstnac in ovnr patl spajn, wlnrn a patl gs a hubtgcghndsgodab pogdt gd a lgfl-cghndsgodab spajn. Cod‗t bnt tlat sjarn you– g` y you ou jad cnsjrgin ad oiknjt wgtl ad ad array o` duhinrs, hatlnhatgjgads jabb gt a pogdt  a  pogdt  gd  gd tln spajn spajn o`  o` abb possgibn arrays o` sujl pogdts. Slat‗s dot ust `or slow. Odjn you fnt a jbnad aistrajtgod bgen tlat, your jocn jad fnt jbnad too. Jbnad aistrajtgods arn wlat profrahhgdf gs abb aiout!   G` you wadt to co hovgn rndcnrnrs, booe at tln papnrs out o` stucgos adc Zobgc Adfbn. Slny arn surprgsgdfby opnd aiout tlngr jra`t.

 

  G` you wadt to co lgfl-pnr`orhadjn ray trajgdf, booe `grst at papnrs `roh Gdtnb adc DPGCGA. Afagd, tl tlny ny arn surprgsgdfby opnd.   G` you wadt to co larc-jorn plysgjabby-iasnc rndcnrnrs, jodvnrt your rndcnrnr `roh YFI to spnjtrab. G ah a igf `ad o` najl ray lavgdf a radcoh wavnbn wavnbndftl dftl adc abhost abb o` tln YFIs gd your profrah turdgdf gdto `boats. Gt soudcs gdn``gjgndt, iut gt gsd‗t!   Ynfarcbnss o` wlat cgrnjtgod you taen, acc a fbossy IYC@ hocnb. Slnrn arn hady tto o jloosn `roh adc najl las gts acvadtafns.   @gdabby, pbnasn sndc hn your ghafns adc `gxns adc johpbagdts! G wgbb hagdtagd bgdes to @gdabby, `urtlnr rnacgdf ntj. at gd1wnnendc.joh   Lavn `ud!   ]ntnr Zlgrbny Zabt Baen Jgty, Harjl, 9=10