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Wyobra|enie: gumowy balon, ktry najpierw napompowujemy, nastpnie wkBadamy do niego model i wreszcie spuszczamy powietrze. Wikszo[ oglnych metod, ktre pozwalaj na znajdowanie odlegBo[ci bryB, ich przeci itp. zakBada, |e bryBy s wypukBe. Algorytmy szukania - geometria obliczeniowa$p   E   $p I}OPM___PPT10i.p%++D=' = @B +"!R< *<"<0  <(     p@^0e0e?TytuB 1 `}   :Detekcja kolizjij  BA??Obraz 11  l  BA??Obraz 12"PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!ؿdrs/downrev.xmlDOK0 07e:4mwIMYX||Y;ӈ'9_[V0NRą5 Χ(2)x[bI)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!Udrs/downrev.xmlDOMk@XU*Z,UsmȎIhv6f׻yϙ:Wl`4L@V\VGPA-֞M{7L?R!EHjo#wC-m]dVJl蹤k z̗-sO:_/6_zzPK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-!Udrs/downrev.xmlPK  ,$Dp  BA??Obraz 14"PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!5,drs/downrev.xmlDOj@}mIQDL$4;]Am:ɢ78Sj ƣqauͥˤ,҇A|!bTPyR"nd[}Π+M#'Q4k >Ύ3u9l矧Y~ S;a dPK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-!5,drs/downrev.xmlPK  ,$Dj  BA??Obraz 15"þPK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!*䇽drs/downrev.xmlDOK 0 !F4ՅHm.9،m&jwxI)œjWXV0D S KˤMv+Xy!bT{_R4'n`+]lmXgR ᦔ(K+Zw\ymN:aRiS[K8@>PK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-!*䇽drs/downrev.xmlPK  ,$Dn  BA??Obraz 16"PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!$ drs/downrev.xmlDOKk1լTFQ"6n٥dD])sV\ g~f8|O@z&w 0L1;)C \PV0}C: Fo)Y9̲tXsjUElOᲘ~4snuO7:/9{PK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-!$ drs/downrev.xmlPK  ,$Ds   BA??Obraz 17"PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!tX"drs/downrev.xmlDOO0 H|HX bl+&41]Oh5^Sh ]i7[ת6V <2PETLdBۛ)6yC6JB8hZ‘8 hzL-%)^NJaC7 Ov1z:~Co\s׿Jԧze_`@PK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-!tX"drs/downrev.xmlPK  ,$Dr   BA??Obraz 18"PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!(Odrs/downrev.xmlDO0!E4UXT{Xz84c[l&ɶOYX6{7o6ΔL'3x ydeR$mXۖ$>.FU,Ks2&"}XL7E\,8\ȱCN#12j_Ǘ8ԚRA_?}ԡ ~ /PK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-!(Odrs/downrev.xmlPK  ,$D     `9Q 0e0e?<Symbol zastpczy zawarto[ci 2" PK![Content_Types].xmlJ0*miG]`Hm6 Xwt?. g#Wi3E).+7 >ON ޑ1ˋz "+R RhB.} /<^ITխ%rHK4uK~I0xM e`|X}đ I`߽N4aG2$RKIZ)4(M9`ctB{m:f@`3n|O,ܗr޾jxR0T ,0@}WBLǬ5vPK!Ǣ(drs/shapexml.xmlZn6;Y;^QS˔"YrlEo}tl'bI&)CCr8[ͅuV }LTOr5MOW;o9sV"Ko96=gvVg>Z-D6Ba[mv2V8TJ8-U}- *QbcV ~F&%BZ_-~kBf`D ( mP "IV 6, )ۏeE&t ߩ)@NQ%g]x @ZcufCL+hj<5E*IA$O9S@v6řf^ߥDu\Z)j]<ǥ04"MʩŐ:Mx~3q C]mW@/ e.)QEHSdie?*%sסmhBy[Vߺy.e\TAZz*v:FFLN`LdH}-r*J*wڑ>[!u[ Wy!7R83OgPL!:ࢽ0LJ]Em/JaIl EP@_=ze*d KX[_Q)Nd4BՐ}'Clk5VdHiӣ4d$k+6+DEm4s0>Ǎ/̍}I\AD[pi pU8Jz/)?nJpm*ײJe_ԧ% Cv\d DϯPK-![Content_Types].xmlPK-!1_a ._rels/.relsPK-!Ǣ()drs/shapexml.xmlPK-!Pdrs/downrev.xmlPKW G z___PPT9\T yAlgorytm Jarvisa na wyznaczanie otoczki w 2D (tzw. owijanie prezentw) Istnieje uoglnienie do 3D Zadanie domowe (2D). Konkurs (3D)-"=,$p     "$pFJ?   R9Q "pole tekstowe 19"PK![Content_Types].xmlJ0*miG]`Hm6 Xwt?. g#Wi3E).+7 >ON ޑ1ˋz "+R RhB.} /<^ITխ%rHK4uK~I0xM e`|X}đ I`߽N4aG2$RKIZ)4(M9`ctB{m:f@`3n|O,ܗr޾jxR0T ,0@}WBLǬ5vPK!a<ճkr.drs/shapexml.xmlZ_o8?龃SBEKWm=TZzϕI6cGCf $nAcv>|\-V(~׏PNsu?:z1JJLdzPmɠr=8W{=#EÓ >FIKc_B-CFȈā>/C5C`W+5r/ fC)'/D}-+SgÑu3'sl2_;%B,Fѱ rIsaJ T=NF,-Ϟ&i<LHVf.29us 'U7CcJ}LZjV&(-iؒ۸Om =mȠnǠOȤK |^Uܫf BM!PbݜIƛ33*El! F=FҭL%'7E +=5ZgTNsPk w)eB=7TcS#)Uy۸2P,x7 U er֥xro%XAGEG+v(;cw0S`ih-'!,xw3ײSB!+ppQC 3p9< Gҗ]*a-;ZF.f`^PG'"6Ʒs zdQwP BM!3"[}훵|_Cd+6{h`^^8=:G0~)o0 ܦsKWf?/]h5k7xX߉aP0ޅ6}߆>3$[MW6gv]65A Ԅmz9޹y @nUP$7}Sm~c1Ii@Q;}!RHqHA [<6Xnj.ӎ |h.#ɿʝ:E a(?PK!whdrs/downrev.xmlDN0EHkQ:JC`~&z^oěa"/u|?Fm)DT&/PK-![Content_Types].xmlPK-!1_a ._rels/.relsPK-!a<ճkr.)drs/shapexml.xmlPK-!whdrs/downrev.xmlPKR PH@___PPT9" D4Jedna z wersji algorytmu Jarvisa: Wybieram punkt P0 o najmniejszej wspBrzdnej y. Je|eli jest kilka, wybieramy z nich ten o najwikszym x. Wybieram punkt P1 o najmniejszym kcie midzy poziomem, a odcinkiem Bczcym z tym punktem. Je|eli jest kilka takich punktw, wybieram najdalszy. Wybieram punkt o najmniejszym kcie midzy odcinkiem Bczcym z poprzednim punktem otoczki, a odcinkiem Bczcym z wybieranym punktem. Je|eli jest kilka punktw na jednej prostej, wybieram najdalszy. Wykonuj punkt 3. do momentu powrotu do punktu P0."  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Twierdzenie SAT (separating axis theorem)$S:S  hA  ??SAT1Obraz 3SAT1.bmpb@   A??$Bryla wkleslaObraz 4Bryla wklesla.bmp L  I}OPM___PPT10i.p%++D=' = @B +"< p$(  $ $  pR 0e0e?TytuB 1 `}   :Detekcja kolizji $  `?Q 0e0e?<Symbol zastpczy zawarto[ci 2 `  \O[ rozdzielenia (o[ na ktrej wida separacj)/// $ A ??(wierzcholek-bokObraz 5wierzcholek-bok.bmpT  0 $ T R ?Rectangle 2D<4___PPT9 `  c $ PR  pole tekstowe 9"ÏPK![Content_Types].xmlJ0*miG]`Hm6 Xwt?. g#Wi3E).+7 >ON ޑ1ˋz "+R RhB.} /<^ITխ%rHK4uK~I0xM e`|X}đ I`߽N4aG2$RKIZ)4(M9`ctB{m:f@`3n|O,ܗr޾jxR0T ,0@}WBLǬ5vPK!,.-+drs/shapexml.xmlWMo1W|Ji(B!=Wf [6^ɡ¡x=c57Z2:gR禨,?N.9s^B(e7OM`&s4I\>pcYXcZ%_;LWflWc˪ -j% ͓dW.]^ksW)u(߈jņf5nٓ((j [qpw- *ÙmKiSFTa<CTxV@SȦ8+d(_uΨVR]Pah? !S$g $0BBP1}7)'ܳ[߭_F_;:]~Sq`.';uqŋiUH߀%-M@\˺*BCxuzgӈshm Alh3 g iI"5wưIT,.A*>@S;-ޔhTFU.tH%ЉAdY[L#biZ;TF"Gj_j}|+?̀q w{C[ `>P1m`M~fۧ[OF[EQEچllDٴD+-ZH茎MvK:AE pn[JE2x=PK!܈&drs/downrev.xmlDN1@&C3$ޤ Q+ '\nM[ax̛ɛ-zۊX{`4@+]`{ ;Rp sϮ6U%.RҐ89fG,&C%u3m+Y6k v4T6/`M<ߣݛ>O xU?tUo[8帺B p*#PK-![Content_Types].xmlPK-!1_a ._rels/.relsPK-!,.-+)drs/shapexml.xmlPK-!܈&drs/downrev.xmlPK Gg`B:2___PPT9  \Kolizja gdy rzut odlegBo[ci [rodkw na wszystkie osie jest mniejszy od sumy poBwek rzutw caBych bryB na t o[8r (C C AC  r  I}OPM___PPT10i.p%++D=' = @B +"?< F>(&(  ( (  pR 0e0e?TytuB 1 `}   :Detekcja kolizji` (  R 0e0e?<Symbol zastpczy zawarto[ci 2 `  jDwa gBwne typy kolizji wielo[cianw (siatki): wierzchoBek-pBaszczyzna  krawdz-krawdzF0. p Jz ( A ??(wierzcholek-bokObraz 7wierzcholek-bok.bmpCI  ( A ??(krawedz-krawedzObraz 8krawedz-krawedz.bmpp>  I}OPM___PPT10i.p%++D=' = @B +"<    , (  , ,  p`R 0e0e?TytuB 1 `}   :Detekcja kolizji" ,  R 0e0e?<Symbol zastpczy zawarto[ci 2  ,Wzory dla dwch prostopadBo[cianw: rzut [rodka prostopadBo[cianu na o[ osiem wierzchoBkw (w ukBadzie odniesienia modelu): rzuty wierzchoBkw na testowan o[: wybr maks. i min. spo[rd o[miu rzutw wierzch.R$$Y % 10 , TR ?Rectangle 2D<4___PPT9 `  c0 , TQ ?Rectangle 12D<4___PPT9 `  ct  , LA??Picture 11 TN! 0  , TR ?Rectangle 2D<4___PPT9 `  cr  , JA??Picture 1~  I}OPM___PPT10i.p%++D=' = @B +"A < H @ 0( (  0 0  pR 0e0e?TytuB 1 `}   :Detekcja kolizji 0  =Q 0e0e?<Symbol zastpczy zawarto[ci 2  Wzory dla dwch prostopadBo[cianw: ograniczenie ilo[ci sprawdzanych osi (dla dwch prostopadBo[cianw  15 osi): Ax, Ay, Az, Bx, By, Bz - wierzchoBek-pBaszczyzna (osie prostopadBe do uderzanych [cian wielo[cianu) Ax x Bx, Ax x By, Ax x Bz, itd. - krawdz-krawdz (osie prostopadBe do obu krawdzi obu wielo[cianw)&$$A0 0 T@R ?Rectangle 2D<4___PPT9 `  c0 0 TR ?Rectangle 4D<4___PPT9 `  c0 0 TR ?Rectangle 8D<4___PPT9 `  c2 0 VR ?Rectangle 10D<4___PPT9 `  c2 0 VR ?Rectangle 12D<4___PPT9 `  c I}OPM___PPT10i.p%++D=' = @B +" <   4 (  4 4  p@R 0e0e?TytuB 1 `}   :Detekcja kolizjid 4   R 0e0e?<Symbol zastpczy zawarto[ci 2   nDTwierdzenie SAT stosowane jest w przypadku obszarw ograniczajcych OBB Testy AABB mog by rozumiane jako szczeglny przypadek twierdzenia SAT (tylko trzy osie)0 4 TR ?Rectangle 2D<4___PPT9 `  c0 4 TR ?Rectangle 4D<4___PPT9 `  c0 4 T`R ?Rectangle 8D<4___PPT9 `  c2 4 V R ?Rectangle 10D<4___PPT9 `  c2 4 V R ?Rectangle 12D<4___PPT9 `  c I}OPM___PPT10i.p%++D=' = @B +"<GC 8(  8 8  pR 0e0e?TytuB 1 `}   :Detekcja kolizjiL 8  `R 0e0e?<Symbol zastpczy zawarto[ci 2N  V,Naiwna (czytelna) implementacja SAT dla OBB:-0 8 TR ?Rectangle 2D<4___PPT9 `  c0 8 T@R ?Rectangle 4D<4___PPT9 `  cy 8 NR ?Rectangle 1FI___PPT9 Gvoid ObliczRzutProstopadloscianuNaProsta(Prostopadloscian* p,Wektor n,double& min, double& max,bool nastepnePolozenie) const { Macierz macierzObrotu; if(!nastepnePolozenie) macierzObrotu=p->MacierzObrotu(); else macierzObrotu=p->NastepnaMacierzObrotu(); Wektor osX=macierzObrotu.KolumnaX(); Wektor osY=macierzObrotu.KolumnaY(); Wektor osZ=macierzObrotu.KolumnaZ(); Wektor polowaRozmiaru=p->Rozmiar()/2; double polowaDlugosciRzutu = polowaRozmiaru.X*fabs(n*osX)+polowaRozmiaru.Y*fabs(n*osY)+polowaRozmiaru.Z*fabs(n*osZ); double rzutSrodkaMasy; if(!nastepnePolozenie) rzutSrodkaMasy=n*p->PolozenieSrodkaMasy(); else rzutSrodkaMasy=n*p->NastepnePolozenieSrodkaMasy(); min=rzutSrodkaMasy-polowaDlugosciRzutu; max=rzutSrodkaMasy+polowaDlugosciRzutu; } nH c c c  c c c ;c c 1 c  $c$ '(c( ,c, '0c0 4c4 '8c8 I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!wdrs/downrev.xmlDN0DHHܨ 6ԭZ$פ6$ڦMX8z=;,FRV~05$J֫ۛ%^LI*,B/P@Pp4ĮNcѵ\:Dy$9רL|pc P:6an< 4'_+QRSe)YDhM~PK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-!wdrs/downrev.xmlPK ju$x0___PPT108___PPT9 9bool CzyOdcinkiNaProstejPokrywajaSie(double minA,double maxA,double minB,double maxB) const { return !((maxA < minB) || (minA > maxB)); }  c T[  ,  I}OPM___PPT10t.p%++,D' = @B D' = @BA?%,( < +O%,( < +D' = @B%(D' = @B%(D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*8%(D' =-g6B fade*<3<*8+8+0+8 +"p<GC 4 , < (  < <  p`R 0e0e?TytuB 1 `}   :Detekcja kolizjiL <   R 0e0e?<Symbol zastpczy zawarto[ci 2N  V,Naiwna (czytelna) implementacja SAT dla OBB:-0 < TR ?Rectangle 2D<4___PPT9 `  c0 < TR ?Rectangle 4D<4___PPT9 `  c < N@R ?Rectangle 1"PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!Ldrs/downrev.xmlDN0DHu ꤈B !lhڰķiݿbь,VɌlO>hgImXh%Β#X-OOXKwk7Q F}Syz2fn"c^qf󢨸AmC= ͷЦ5 Sʢi[!H)Ka"\WW\cdlM/PK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-!Ldrs/downrev.xmlPKF# p$<4___PPT9 3wbool CzyRzutyProstopadloscianowNaOsNakladajaSie(Wektor n,Prostopadloscian* pA,Prostopadloscian* pB,bool nastepnePolozenie) const { double minA,maxA,minB,maxB; ObliczRzutProstopadloscianuNaProsta(pA,n,minA,maxA,nastepnePolozenie); ObliczRzutProstopadloscianuNaProsta(pB,n,minB,maxB,nastepnePolozenie); return CzyOdcinkiNaProstejPokrywajaSie(minA,maxA,minB,maxB); } x xc b    I}OPM___PPT10.p%++nS_D0' = @B D' = @BA?%,( < +O%,( < +D"' = @B%(%(D' = @B%(D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*<%(D' =-g6B fade*<3<*<+8+0+< +"<GC   @ (  @ @  pR 0e0e?TytuB 1 `}   :Detekcja kolizjiL @  R 0e0e?<Symbol zastpczy zawarto[ci 2N  V,Naiwna (czytelna) implementacja SAT dla OBB:-0 @ TR ?Rectangle 2D<4___PPT9 `  c0 @ TR ?Rectangle 4D<4___PPT9 `  c @ NR ?Rectangle 1"PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK! Adrs/downrev.xmlLN0DHuؠIB !PPķIiݿbXftFgzd{rF@>πiM+}85$HVӓV̆2,AP@Tq4ȤngƐkrxHp="J7É:jԱ^ǡ/op$MacierzObrotu(); macierzObrotuB=pB->MacierzObrotu(); } else { macierzObrotuA=pA->NastepnaMacierzObrotu(); macierzObrotuB=pB->NastepnaMacierzObrotu(); } switch(indeks) { ... } }   c        I}OPM___PPT10.p%++nS_D0' = @B D' = @BA?%,( < +O%,( < +D"' = @B%(%(D' = @B%(D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*@%(D' =-g6B fade*<3<*@+8+0+@ +"<GC rjDR(  D D  p`R 0e0e?TytuB 1 `}   :Detekcja kolizjiL D   R 0e0e?<Symbol zastpczy zawarto[ci 2N  V,Naiwna (czytelna) implementacja SAT dla OBB:-0 D T`R ?Rectangle 2D<4___PPT9 `  c0 D TR ?Rectangle 4D<4___PPT9 `  cP  D N@R ?Rectangle 1"PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!DCdrs/downrev.xmlDN0DHuTIBݪCT]4%Mvj{,|伲F@>̀iT|4kH<,s,=в1D]cɹo:v$uC)gYVpʤGZwշP=x:+B#style.visibility<*D%(D' =-g6B fade*<3<*D+8+0+D +"D<GC   H (  H H  p@ 0e0e?TytuB 1 `}   :Detekcja kolizjiL H  6 0e0e?<Symbol zastpczy zawarto[ci 2N  V,Naiwna (czytelna) implementacja SAT dla OBB:-0 H T; ?Rectangle 2D<4___PPT9 `  c0 H T ?Rectangle 4D<4___PPT9 `  c H N\ ?Rectangle 1"PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!Wdrs/downrev.xmlDN0DHu ꤈B!X5m&16chFgtGv 5YLc2h$֐yXϖXJ{4:Tc b|sD&uuC1YVpʤ'z/-[Tsk!./=@1?c+QRmq3ρ/} ' %d |PK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-!Wdrs/downrev.xmlPKF p$<4___PPT9 Kbool TestNakrywaniaDwochProstopadloscianow(Prostopadloscian* pA,Prostopadloscian* pB,bool nastepnePolozenie) const { for(int i=1;i<=15;++i) if(!CzyRzutyProstopadloscianowNaOsNakladajaSie( OsRzutowania(i,pA,pB, nastepnePolozenie).Unormowany(),pA,pB), nastepnePolozenie) return false; return true; } L Lc b,     I}OPM___PPT10.p%++nS_D0' = @B D' = @BA?%,( < +O%,( < +D"' = @B%(%(D' = @B%(D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*H%(D' =-g6B fade*<3<*H+8+0+H +"# < * "  L (  L L  pR 0e0e?TytuB 1 `}   :Detekcja kolizji L  R 0e0e?<Symbol zastpczy zawarto[ci 2 `  4Optymalizacje przeprowadzenie testw w ukBadzie lokalnym A rozpisanie 15 konkretnych przypadkw, co pozwala na uniknicie wielu niepotrzebnych obliczeD&0 L T R ?Rectangle 2D<4___PPT9 `  c0 L TR ?Rectangle 4D<4___PPT9 `  c0 L TR ?Rectangle 2D<4___PPT9 `  cr L JA??Picture 1 0 L T`R ?Rectangle 9D<4___PPT9 `  c2  L VR ?Rectangle 11D<4___PPT9 `  c2  L V`R ?Rectangle 12D<4___PPT9 `  ct  L LA??Picture 11@ eC I}OPM___PPT10i.p%++D=' = @B +"*< %% Pg%(  P P  pR 0e0e?TytuB 1 `}   :Detekcja kolizji P  @R 0e0e?<Symbol zastpczy zawarto[ci 2 `  6Optymalizacje Zaczynamy od pytania o to, czy dBugo[ poBowy rzutw obu prostopadBo[cianw jest mniejsza od odlegBo[ci [rodkw tych prostopadBo[cianw? 60 P T R ?Rectangle 2D<4___PPT9 `  c0 P TR ?Rectangle 4D<4___PPT9 `  c0 P TR ?Rectangle 2D<4___PPT9 `  c0 P T`R ?Rectangle 9D<4___PPT9 `  c2 P VR ?Rectangle 11D<4___PPT9 `  ct  P LA??Picture 10$ _  x ^ .:z2XQΙƴ w58?}Wr` ڠ( H;˛kXl~:BBt  ]7BH=s&ԼXLje?tO5TFJ6ԜՒey>uC%PeAM7(2fIhL·O<_t,Ts >l hqI,_|H;`k|bojOS"ڱtð~@ز0Zr8g4r>ɐa"I&h9YFH&_?!+߉H֋8ϼR_~O:7rNq2'vPK!|Ddrs/downrev.xmlDOMk@^n*Ťއ4 fgCvnmkR؇pSECC KoQ`h8^>[ާddrs/connectorxml.xmlj0нw7='vE䑐c﭅Y^nRJEj /R֦Q/o |l@ׯ6c![&f@Ƶ(t:juLcAagEB805>wV7Z5J1v;X_qQVb` 4˝I O>74'Pdb5-b%bt43/ E;7rVq9O[PK!Gdrs/downrev.xmlDOMk@ e^nk%uQZb&lnM׻yϙ-ZS+ծ`8@V)8?_ GXZ&7rw;3m@g"QA}KҜ ck>:&Rh" rhSzI~+{S}&?o;^z?p?PK-!%[Content_Types].xmlPK-!3X 1_rels/.relsPK-!sq?>.drs/connectorxml.xmlPK-!G3drs/downrev.xmlPK";-I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!|drs/downrev.xmlDA0oVQEd]<ȂUXhmy)M,Yo{SWZVO"ę% x ydeR `M}.] DJdMlMm \7FB,9,XӾ_vYٞ29ŤhV <~Z  7OPK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-!|drs/downrev.xmlPK 5r,$Dz P JA??Picture 1,٭I8r~Sz P JA??Picture 5,3CU_z P JA??Picture 7@_{΄h/@PP I}OPMWO___PPT10/.p%++oD' = @B D' = @BA?%,( < +O%,( < +D' = @B%(D' = @B%(D+' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*P%(D' =-g6B fade*<3<*P+"y< x0T`(  T T  pR 0e0e?TytuB 1 `}   :Detekcja kolizji T  ` 0e0e?<Symbol zastpczy zawarto[ci 2 `  Optymalizacje Je|eli te wielko[ci wyrazimy we wspBrzdnych ukBadu lokalnego zwizanego z prostopadB. 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W przypadku wielo[cianw mo|na u|y metody V-clip (Briana Mitrich). Rozpoczyna si od podziaBu przestrzeni wokB wielo[cianu na obszary. Ka|dy z obszarw w tym podziale skBada si z punktw, ktre s bli|ej jednego wierzchoBka, krawdzi lub [ciany ni| innych. Dwuwymiarowy analog tego problemu nazywany jest teselacj Woronoja i prowadzi do sBynnego diagramu Woronoja. W detekcji kolizji fakt, |e do jednego obszaru nale| punkty bli|sze jednemu elementowi wielo[cianu ni| innym, ma istotne znaczenie i jest podstaw wydajno[ci tego algorytmu, pozwalajc na zastpienie operacji na elementach wielo[cianu operacjami na zwizanych z nimi obszarach. n p p&+pW0 d T1 ?Rectangle 2D<4___PPT9 `  c0 d T3 ?Rectangle 4D<4___PPT9 `  c0 d T ?Rectangle 2D<4___PPT9 `  c0 d T1 ?Rectangle 9D<4___PPT9 `  c I}OPM___PPT10i.p%++D=' = @B +"<  h (  h h  p3 0e0e?TytuB 1 `}   :Detekcja kolizji h   0e0e?<Symbol zastpczy zawarto[ci 2 `  Jak znalez punkt styku i normaln zderzenia (lini akcji), czyli ukBad odniesienia zderzenia? Najbardziej poprawn odpowiedz daje iteracyjna metoda GJK (Gilberta, Johnsona i Keerthi). Dla bryB wypukBych. dla ktrych umiemy zdefiniowa tzw. funkcj odwzorowania wspierajcego (ang. support mapping function), zwracajc punkt siatki modelu, ktry jest najbardziej wysunity we wskazanym w jej argumencie kierunku. Algorytm rozpoczyna si, przynajmniej formalnie, od wyznaczenia tzw. r|nicy Minkowskiego siatek dwch modeli, ktrych kolizja jest brana pod uwag. R|nica Minkowskiego to zbir wszystkich wektorowych r|nic poBo|eD punktw z obu siatek. Modele nakBadaj si na siebie, je|eli w ich r|nicy Minkowskiego znajduje si punkt bliski pocztkowi ukBadu wspBrzdnych.  p p&6p[}p 0 h T 4 ?Rectangle 2D<4___PPT9 `  c0 h T4 ?Rectangle 4D<4___PPT9 `  c0 h T4 ?Rectangle 2D<4___PPT9 `  c0 h T@5 ?Rectangle 9D<4___PPT9 `  c I}OPM___PPT10i.p%++D=' = @B +"U< \Tl<(  l l  p 7 0e0e?TytuB 1 `}   :Detekcja kolizji l  5 0e0e?<Symbol zastpczy zawarto[ci 2 `  (nJak znalez punkt styku i normaln zderzenia (lini akcji), czyli ukBad odniesienia zderzenia? Po wyznaczeniu r|nicy Minkowskiego tworzymy bryB opisan na maksymalnie czterech dowolnie wybranych jej punktach. Nastpnie korzystajc ze wspomnianej przed chwil funkcji, szukamy punktu P nale|cego do nowego obszaru (niekoniecznie jego wierzchoBka), ktry jest najbli|ej pocztku ukBadu wspBrzdnych. Je|eli ten punkt znajduje si w pocztku ukBadu wspBrzdnych - znalezli[my punkt styku. W przeciwnym razie redukujemy obszar do odcinka lub trjkta, na ktrym znajduje si znaleziony przed chwil punkt P. Jak wida w obliczeniach nie ma potrzeby wyznaczania caBej r|nicy Minkowskiego. Potrzebujemy tylko czterech jej punktw i tylko te punkty obliczamy korzystajc z funkcji odwzorowania wspierajcego (wydajno[!). 8 p p&@p80 l T6 ?Rectangle 2D<4___PPT9 `  c0 l T7 ?Rectangle 4D<4___PPT9 `  c0 l T@8 ?Rectangle 2D<4___PPT9 `  c0 l T9 ?Rectangle 9D<4___PPT9 `  c I}OPM___PPT10i.p%++D=' = @B +" <   p (  p p  p`9 0e0e?TytuB 1 `}   :Detekcja kolizjit p  `6 0e0e?<Symbol zastpczy zawarto[ci 2 `  ~Jak znalez punkt styku i normaln zderzenia (lini akcji), czyli ukBad odniesienia zderzenia? W nastpnym kroku szukamy wierzchoBka modelu, ktry jest najbardziej odlegBy od punktu P. Je|eli jego odlegBo[ od pocztku ukBadu wspBrzdnych jest taka sama, jak punktu P, to znaczy, |e nowy wierzchoBek odpowiada r|nicy dwch najbli|szych siebie punktw z obu siatek. To jest punkt styku. Je|eli tak nie jest, wBczamy nowy wierzchoBek do podobszaru i powstaje nowy obszar (trjkt lub czworo[cian), w ktrym szukamy punktu najbli|szego pocztkowi ukBadu wspBrzdnych. 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Wydobywanie punktu styku i normalnej z twierdzenia SAT: Normaln zderzenia mo|na ustali [ledzc zmiany, jakie nastpuj wzdBu| ka|dej z pitnastu osi separacji. Nale|y ustali, w ktrym kierunku nakrywanie pojawiBo si jako ostatnie. Ta o[ bdzie wskazywaBa normaln.to p p&9ppo0 x T @ ?Rectangle 2D<4___PPT9 `  c0 x T = ?Rectangle 4D<4___PPT9 `  c0 x T= ?Rectangle 2D<4___PPT9 `  c0 x T@ ?Rectangle 9D<4___PPT9 `  c x A-??&SAT - normalnaObraz 7SAT - normalna.bmp e I}OPM___PPT10i.p%++D=' = @B +"< |(  | |  p@ 0e0e?TytuB 1 `}   :Detekcja kolizji |  ? 0e0e?<Symbol zastpczy zawarto[ci 2 `   Jak znalez punkt styku i normaln zderzenia (lini akcji), czyli ukBad odniesienia zderzenia? W praktyce sytuacja jest bardziej skomplikowana ze wzgldu na kwantyzacj czasu w symulacjach. Zazwyczaj tu| przed zderzeniem to nie jedna, ale dwie osie pokazuj separacj. Typow sytuacj jest, |e jedna z osi odpowiada zderzeniu typu wierzchoBek-[ciana, a druga  krawdz-krawdz. 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