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Jak zbada, czy punkt p te| tworzy z nimi wielokt wypukBy?m 2m m "X 0pa" F S,$ 0 >Je[li idc po krawdzi wielokta od pi do pi+1 (kolejno[ przeciwna do ruchu wskazwek zegara), mam p po lewej stronie prostej wyznaczonej przez t krawdz, to p mo|e by nastpnym wierzchoBkiem wielokta wypukBego.B 2%  &$2 $X s *" ,$@ 02 %X s *"  w,$@ 02 &X s *" s 9 ,$@ 02 'X s *"  g ,$@  02 (X s *"   ,$@  0 ,X 0,m" ,$  0 Tp0" 2   -X 0q"  w,$  0 Tp1" 2   .X 0Xv"  Vs ,$  0 Sp" 2   /X 0L{" V = ,$ 0 Tp2" 2   0X 0" ) 6S ,$ 0 Tp3" 2  B 1X s *Dw,$D  0B 2X s *D w,$D  0B 3X s *D 9 ,$D  0B 4X s *D g ,$D  0H X 0޽h ? @ ff3Ιd332zGG___PPT10G.P.b+[kDE' = @B DJE' = @BA?%,( < +O%,( < +D' =%(D' =%(Dg' =4@BB7BB%(D' =1:Bvisible*o3>+B#style.visibility<*X%(D' =+4 8?fCB#ppt_w*0.70BCB#ppt_wB*Y3>B ppt_w<*XD' =+4 8?\CB#ppt_hBCB#ppt_hB*Y3>B ppt_h<*XD' =-g6B fade*<3<*XD$' =%(D' =%(Dt' =A@BB7BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*X%(D' =+4 8?fCB#ppt_w*0.70BCB#ppt_wB*Y3>B ppt_w<*XD' =+4 8?\CB#ppt_hBCB#ppt_hB*Y3>B ppt_h<*XD' =-g6B fade*<3<*XD$' =%(D' =%(Dt' =A@BB7BB0B%(D' =1:Bvisible*o3>+B#style.visibility<* X%(D' =+4 8?fCB#ppt_w*0.70BCB#ppt_wB*Y3>B ppt_w<* XD' =+4 8?\CB#ppt_hBCB#ppt_hB*Y3>B ppt_h<* XD' =-g6B fade*<3<* XD{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*!X%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*!XD' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*!XD7' =%(D' =%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*$X%(D' =-o6Bdissolve*<3<*$XD3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*%X%(D' =-o6Bdissolve*<3<*%XD3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*&X%(D' =-o6Bdissolve*<3<*&XD3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*'X%(D' =-o6Bdissolve*<3<*'XD3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*(X%(D' =-o6Bdissolve*<3<*(XD@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*,X%(D' =-o6Bdissolve*<3<*,XD@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*-X%(D' =-o6Bdissolve*<3<*-XD@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*.X%(D' =-o6Bdissolve*<3<*.XD@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*/X%(D' =-o6Bdissolve*<3<*/XD@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*0X%(D' =-o6Bdissolve*<3<*0XDn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*1X%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*1XD' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*1XDn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*2X%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*2XD' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*2XDn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*3X%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*3XD' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*3XDd' =%(D ' =%(D' =4@BBBB%(D ' =+4 8?XCB ppt_xBCB ppt_xB*Y3>B ppt_x<*3XD' =+4 8?`CB ppt_yBCB1+ppt_h/2B*Y3>B ppt_y<*3XD' =1:Bhidden*o3>+B#style.visibility<*3X%(Dn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*4X%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*4XD' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*4XD' =%(D' =%(DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*"X%(D' =-6B'blinds(horizontal)*<3<*"X++0+X ++0+ X ++0+!X ++0+"X ++0+,X ++0+-X ++0+.X ++0+/X ++0+0X +O  K0 rj( q@G@ l  C ̈ p`     0ψ  7,$  0 `Problem Dla danego skoDczonego zbioru Q punkw na pBaszczyznie, znalez najmniejszy zbir wypukBy taki, |e ka|dy punkt zbioru Q znajduje si albo w jego wntrzu albo na brzegu...@ _v;@  6 @ @ @  B܈G HJ3 "`M` ,$D 0 W geometrii, dowodzi si, |e taki zbir punktw (dla |Q| >2) jest wieloktem wypukBym. ,W.@ _X XVz *      ,$D  0Z2   s *V  Z2   s * ( Z2   s *  9 Z2  s *Z2  s *f J Z2  s *  Z2  s * Z2  s **WIZ2  s *w  Z2  s *-vZB  0Do qv,$D  0B  0DoF j ,$D  0B  0Do N ,$D  0B @ 0Do g ,$D   0B @ 0Do g ,$D   0B  0Do1 v,$D   0  C W3yd?PrzykBadImpactS 6 ,$D 0N  N$_ж_жsZs pz S ,$  0 RJak znalez wierzchoBki otoczki wypukBej?**t *  06 +,$  0 $Kryterium negatywne Punkt p nie jest wierzchoBkiem otoczki wypukBej wttw le|y wewntrz trjkta o wierzchoBkach ze zbioru Q r|nych od p, lub gdy le| na odcinku Bczcym dwa r|ne od p punkty zbioru Q.H 2@ @ @ @ &DH  0޽h ?  @ ff3Ιd332z<<___PPT10<+Y D;' = @B D`;' = @BA?%,( < +O%,( < +D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*D$' =%(D' =%(Dt' =A@BB7BB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =+4 8?fCB#ppt_w*0.70BCB#ppt_wB*Y3>B ppt_w<* D' =+4 8?\CB#ppt_hBCB#ppt_hB*Y3>B ppt_h<* D' =-g6B fade*<3<* D' =%(D' =%(Dg' =4@BB7BB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?fCB#ppt_w*0.70BCB#ppt_wB*Y3>B ppt_w<*D' =+4 8?\CB#ppt_hBCB#ppt_hB*Y3>B ppt_h<*D' =-g6B fade*<3<*Dn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<* D' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<* Dn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*Dn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*Dn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*Dn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*Dn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*Dn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*D ' =%(D9 ' =%(D' =A@BB&BB0B@HB%()?D' =1:Bvisible*o3>+B#style.visibility<*%(D' =1:B -45.0*k3>'Bstyle.rotation<*%(DS' =+4 8?lCB4BCB4BBCBB*k3>'Bstyle.rotation<*%(Dm' =+4 8?CB#ppt_y-1BCBG#ppt_y-(0.354*#ppt_w-0.172*#ppt_h)B*Y3>B ppt_y<*%(D' N=+[4 8?CBG#ppt_y-(0.354*#ppt_w-0.172*#ppt_h)BCBY#ppt_y-(0.354*#ppt_w-0.172*#ppt_h)-#ppt_h/2B*Y3>B ppt_y<*%())?)?Di' D=+4 8?CBG#ppt_y-(0.354*#ppt_w-0.172*#ppt_h)BCB#ppt_yB*Y3>B ppt_y<*%(D' =%(D' =%(DF' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-u6Bdiamond(in)*<3<*++0+ ++0+  ++0+ ++0+ +,  K0 V(  r  S  p`     0X z: |$ 0H@___PPT9" .Metoda. Dla ka|dej trjki punktw p1, p2, p3 ze zbioru Q : je[li s one wspBliniowe i p jest punktem [rodkowym tej trjki, to ze zbioru Q usuwamy punkt p, je[li nie s wspBliniowe, to oznaczmy przez d trjkt, ktrego wierzchoBkami s te punkty i usuDmy ze zbioru Q wszystkie punkty le|ce wewntrz tego trjkta. Zbir punktw, ktre pozostan w Q jest zbiorem wierzchoBkw szukanej otoczki wypukBej. = 2 2X 2 2f   [  @`J  0!V Y .Koszt: Niech |Q|=n. Jest (n nad 3) r|nych trjek punktw liczba iteracji ptli zewntrznej  O(n3). Koszt ptli wewntrznej  O(n). Razem O(n4).X 2f\ 0 @rl MP  MP,$D  0`2  0Iw`2  0TB B c $DMP`2   0 98l  Im   Im,$D  0ZB   s *DogmZB   s *Do Il 25  25 ,$D  0ZB B s *D` ZB  s *D w ZB B s *Dw` f2  62_f2  6 5 f2  6w  l gv  gv ,$D  0`2  0g  `2  0% S `2  0 I `2  0IvCH  0޽h ? @ ff3Ιd332z___PPT10.Do+7DH' = @B D' = @BA?%,( < +O%,( < +D' =%(D' =%(DF' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-u6Bdiamond(in)*<3<*Dn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<* D' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<* Dn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<* D' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<* Dn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*Dn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*Dd' =%(D ' =%(D' =4@BBBB%(D ' =+4 8?XCB ppt_xBCB ppt_xB*Y3>B ppt_x<*D' =+4 8?`CB ppt_yBCB1+ppt_h/2B*Y3>B ppt_y<*D' =1:Bhidden*o3>+B#style.visibility<*%(+8+0+ +  K0 ,,^ 0 :,(   r   S \; p`   `2 6  s *" M z ?`2 9  s *" `2 :  s *"  `2 ;  s *" pC`2 <  s *" mS`2 =  s *" c`2 >  s *" }c`2 ?  s *" `2 B  s *"  Iw `2 C  s *" Pj}`2 D  s *"  ?`2 E  s *"  `2 G  s *"  Z5 `2 L  s *" # Q `2 M  s *" V ~ `2 O  s *" 3 a `2 P  s *" '`2 S  s *" _`2 U  s *" 6df2   0" / ] f2 7  0"  # 6 f2 8  0" a" P f2 @  0"  N f2 F  0" 3 `f2 H  0" d ] f2 I  0"  \ f2 J  0"   6 f2 N  0"  # f2 Q  0"  M f2 T  0"  f2 V  0" 6 jc B W  c $D ~ w ,$D 0B X  c $D 6V ,$D 0B Y  c $DP ~ ,$D 0B Z  c $D" P,$D 0B ^  c $Dcw ,$D 0f2 e  0"   f2 f  0"  5 f2 g  0"  E l  v k  E ,$D 0h2 h  s *"`  h2 i  s *"`9 f h2 j  s *"`IvB l  c $D}~ V ,$D 0$l )<Z {  Q 6 ,$D   0n2 o  0" 9=gjn2 p  0" }-n2 q  0" ,Zn2 s  0" n2 v  0" }n2 w  0" <-in2 x  0" )MWzn2 y  0" @mB |  c $D ,$D  0B }  c $D ,$D  0B ~  c $D ,$D  0l    ,$D   0h2   s *"  ) h2   s *" W  h2   s *" g  h2   s *" g  h2   s *"  + h2   s *" vB  @ c $D  ,$D 0B   c $D6 Z~ ,$D 0B  @ c $D ,$D 04l  v  j ,$D  0h2   s *"   h2   s *" Ivh2   s *" f  h2   s *" HB   c $D Zw6 ,$D 0B   c $Dz w ,$D 0B  @ c $Dz  ,$D 0l 9    e,$D  0h2   s *" 9 f h2   s *" t  h2   s *"  B   c $Dz ,$D  0B   c $D w ,$D 0B   c $Dz w ,$D 02   s *"  ?,$D  02   s *" # Q ,$D  02   s *" 3 a ,$D  0   0V" f g JKonstrukcja otoczki metod trjktw& 2& &B   s *DFԔ w ,$D 0B   s *DFԔw ,$D 0B   s *DFԔ ,$D 0B  @ s *DFԔz ,$D 0B  @ s *DFԔ@jM P,$D  0B  @ s *DFԔ}Zj6 ,$D! 0B   s *DFԔ6 Z ,$D" 0H   0޽h ? @ ff3Ιd332z___PPT10}.IPa+ GDQ' = @B D ' = @BA?%,( < +O%,( < +D' =%(D' =%(Dg' =4@BB7BB%(D' =1:Bvisible*o3>+B#style.visibility<*W %(D' =+4 8?fCB#ppt_w*0.70BCB#ppt_wB*Y3>B ppt_w<*W D' =+4 8?\CB#ppt_hBCB#ppt_hB*Y3>B ppt_h<*W D' =-g6B fade*<3<*W D' =%(D' =%(Dg' =4@BB7BB%(D' =1:Bvisible*o3>+B#style.visibility<*^ %(D' =+4 8?fCB#ppt_w*0.70BCB#ppt_wB*Y3>B ppt_w<*^ D' =+4 8?\CB#ppt_hBCB#ppt_hB*Y3>B ppt_h<*^ D' =-g6B fade*<3<*^ D' =%(D' =%(Dg' =4@BB7BB%(D' =1:Bvisible*o3>+B#style.visibility<*X %(D' =+4 8?fCB#ppt_w*0.70BCB#ppt_wB*Y3>B ppt_w<*X D' =+4 8?\CB#ppt_hBCB#ppt_hB*Y3>B ppt_h<*X D' =-g6B fade*<3<*X D' =%(D' =%(Dg' =4@BB7BB%(D' =1:Bvisible*o3>+B#style.visibility<*k %(D' =+4 8?fCB#ppt_w*0.70BCB#ppt_wB*Y3>B ppt_w<*k D' =+4 8?\CB#ppt_hBCB#ppt_hB*Y3>B ppt_h<*k D' =-g6B fade*<3<*k D' =%(D' =%(Dg' =4@BB7BB%(D' =1:Bvisible*o3>+B#style.visibility<*Z %(D' =+4 8?fCB#ppt_w*0.70BCB#ppt_wB*Y3>B ppt_w<*Z D' =+4 8?\CB#ppt_hBCB#ppt_hB*Y3>B ppt_h<*Z D' =-g6B fade*<3<*Z D' =%(D' =%(Dg' =4@BB7BB%(D' =1:Bvisible*o3>+B#style.visibility<*Y %(D' =+4 8?fCB#ppt_w*0.70BCB#ppt_wB*Y3>B ppt_w<*Y D' =+4 8?\CB#ppt_hBCB#ppt_hB*Y3>B ppt_h<*Y D' =-g6B fade*<3<*Y D' =%(D' =%(Dg' =4@BB7BB%(D' =1:Bvisible*o3>+B#style.visibility<*l %(D' =+4 8?fCB#ppt_w*0.70BCB#ppt_wB*Y3>B ppt_w<*l D' =+4 8?\CB#ppt_hBCB#ppt_hB*Y3>B ppt_h<*l D' =-g6B fade*<3<*l Dn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*{ %(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*{ D' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*{ D' =%(D' =%(Dg' =4@BB7BB%(D' =1:Bvisible*o3>+B#style.visibility<*| %(D' =+4 8?fCB#ppt_w*0.70BCB#ppt_wB*Y3>B ppt_w<*| D' =+4 8?\CB#ppt_hBCB#ppt_hB*Y3>B ppt_h<*| D' =-g6B fade*<3<*| D' =%(D' =%(Dg' =4@BB7BB%(D' =1:Bvisible*o3>+B#style.visibility<*} %(D' =+4 8?fCB#ppt_w*0.70BCB#ppt_wB*Y3>B ppt_w<*} D' =+4 8?\CB#ppt_hBCB#ppt_hB*Y3>B ppt_h<*} D' =-g6B fade*<3<*} D' =%(D' =%(Dg' =4@BB7BB%(D' =1:Bvisible*o3>+B#style.visibility<*~ %(D' =+4 8?fCB#ppt_w*0.70BCB#ppt_wB*Y3>B ppt_w<*~ D' =+4 8?\CB#ppt_hBCB#ppt_hB*Y3>B ppt_h<*~ D' =-g6B fade*<3<*~ Dn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<* D' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<* D' =%(D' =%(Dg' =4@BB7BB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =+4 8?fCB#ppt_w*0.70BCB#ppt_wB*Y3>B ppt_w<* D' =+4 8?\CB#ppt_hBCB#ppt_hB*Y3>B ppt_h<* D' =-g6B fade*<3<* D' =%(D' =%(Dg' =4@BB7BB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =+4 8?fCB#ppt_w*0.70BCB#ppt_wB*Y3>B ppt_w<* D' =+4 8?\CB#ppt_hBCB#ppt_hB*Y3>B ppt_h<* D' =-g6B fade*<3<* D' =%(D' =%(Dg' =4@BB7BB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =+4 8?fCB#ppt_w*0.70BCB#ppt_wB*Y3>B ppt_w<* D' =+4 8?\CB#ppt_hBCB#ppt_hB*Y3>B ppt_h<* D' =-g6B fade*<3<* Dn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<* D' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<* D' =%(D' =%(Dg' =4@BB7BB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =+4 8?fCB#ppt_w*0.70BCB#ppt_wB*Y3>B ppt_w<* D' =+4 8?\CB#ppt_hBCB#ppt_hB*Y3>B ppt_h<* D' =-g6B fade*<3<* D' =%(D' =%(Dg' =4@BB7BB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =+4 8?fCB#ppt_w*0.70BCB#ppt_wB*Y3>B ppt_w<* D' =+4 8?\CB#ppt_hBCB#ppt_hB*Y3>B ppt_h<* D' =-g6B fade*<3<* D' =%(D' =%(Dg' =4@BB7BB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =+4 8?fCB#ppt_w*0.70BCB#ppt_wB*Y3>B ppt_w<* D' =+4 8?\CB#ppt_hBCB#ppt_hB*Y3>B ppt_h<* D' =-g6B fade*<3<* Dn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<* D' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<* Dn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<* D' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<* D' =%(D' =%(Dg' =4@BB7BB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =+4 8?fCB#ppt_w*0.70BCB#ppt_wB*Y3>B ppt_w<* D' =+4 8?\CB#ppt_hBCB#ppt_hB*Y3>B ppt_h<* D' =-g6B fade*<3<* D' =%(D' =%(Dg' =4@BB7BB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =+4 8?fCB#ppt_w*0.70BCB#ppt_wB*Y3>B ppt_w<* D' =+4 8?\CB#ppt_hBCB#ppt_hB*Y3>B ppt_h<* D' =-g6B fade*<3<* Dn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<* D' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<* Dn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<* D' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<* Dn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<* D' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<* D' =%(D' =%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-o6Bdissolve*<3<* D' =%(D' =%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-o6Bdissolve*<3<* D' =%(D' =%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-o6Bdissolve*<3<* D' =%(D' =%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-o6Bdissolve*<3<* D' =%(D' =%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-o6Bdissolve*<3<* D' =%(D' =%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-o6Bdissolve*<3<* D' =%(D' =%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-o6Bdissolve*<3<* +<=  K0 &,d(  r  S  p`     0tMJ,$ 0 <1. W zbiorze Q wybieramy punkt p o najmniejszej wspBrzdnej y. Je|eli jest kilka takich punktw, to wybieramy ten z nich, ktry ma najmniejsz wspBrzdn x. 2   0tY( ,$ 0 t2. Sortujemy pozostaBe punkty ze wzgldu na kt jaki tworzy wektor p0p z dodatnim kierunkiem osi x. Je[li kilka punktw tworzy ten sam kt, usuwamy wszystkie z wyjtkiem najbardziej oddalonego od p0. Niech uzyskanym cigiem bdzie p1, p2,...pm. V 2A  1   0З d&` ,$ 0 z3. Do pocztkowo pustego stosu s wkBadamy punkty p0, p1, p2.`> 20    > l  Z  +I ,$D 0ZB  s *D  TB   c $DVc  TB   c $DV  TB   c $D& V `2   0 & `2   0  `2  06 Ic @   BjCDEFF5%?[bjy[[L=@  c"$` 9  P   BCjDE(FA5% j*]K-- @   c"$` V6 : H   BCDE4FAA5% j .c0=M=K @    " c    0T" s 9Z  Tp0" 2  `2  0V    0" s Z  Gx 2   0x" I +  Gp 2   0" s 7W,$ 0 ,4. for i:=3 to n do while pi le|y na prawo od prostej (top(pop(s)), top(s)) do s := pop(s) od; s:= push(pi); od;J 2! _    h4GHdA Papeteria" q) ,$D 0 sprawdzamy, czy po doBczeniu nowego punktu, punkty na stosie tworz zbir wypukBySS?=[ SZl   , ,$D 0h2  s *" t  0" W Tp0" 2  ZB  s *Dvh2  s *" I v   0(" I0 Tp1" 2  h2 ! s *"  8  " 0"    Tp2" 2  ZB $ s *D 9 I`B % 0Dg  h2 & s *" V  ZB 'B s *D g ZB ( s *DF V  ) 0 "    bpi" 2 h2 * s *"  F h2 + s *" g  H  0޽h ? @ ff3Ιd332z___PPT10.Jw]+Dt' = @B D/' = @BA?%,( < +O%,( < +D' =%(D' =%(DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B'blinds(horizontal)*<3<*D' =%(D' =%(DP' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B#blinds(vertical)*<3<*D' =%(D' =%(Dg' =4@BB7BB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?fCB#ppt_w*0.70BCB#ppt_wB*Y3>B ppt_w<*D' =+4 8?\CB#ppt_hBCB#ppt_hB*Y3>B ppt_h<*D' =-g6B fade*<3<*D$' =%(D' =%(Dt' =A@BB7BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?fCB#ppt_w*0.70BCB#ppt_wB*Y3>B ppt_w<*D' =+4 8?\CB#ppt_hBCB#ppt_hB*Y3>B ppt_h<*D' =-g6B fade*<3<*D$' =%(D' =%(Dt' =A@BB7BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?fCB#ppt_w*0.70BCB#ppt_wB*Y3>B ppt_w<*D' =+4 8?\CB#ppt_hBCB#ppt_hB*Y3>B ppt_h<*D' =-g6B fade*<3<*D' =%(D' =%(D@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-o6Bdissolve*<3<*D' =%(D' =%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*,%(D' =-o6Bdissolve*<3<*,++0+ ++0+ ++0+ ++0+ ++0+ +D  K0 "l(  r  S  p`     0x" ,$ 0 1. Ka|dy punkt, ktry zostaB usunity ze stosu nie nale|y do otoczki wypukBej.O 2L Ol G"   "G ,$D 0  <" t ^pj  <" tp ^pk  <" tpS  G...    <" tS 6  Pp0    C W3yd?stosImpact" G6 K ZB  B s *D   0" g" Itop 2   0 " m,$ 0 Przypu[my, |e rozwa|amy punkt pi zbioru Q i |e le|y on na prawo od prostej pkpj. Zgodnie z algorytmem, ze stosu usuwamy punkt pj. 2  -   Z,/l    ,$D  0h2  s *"  9 f h2  s *" S  h2  s *"   h2  s *"     0"   Tp0" 2    0,"  S :  bpk" 2   00!"  c # J  bpj" 2   0%"   bpi" 2 B @ 0D E ,$D 0B  s *DԔ 9 ,$D 0B  s *DԔS ,$D 0B @ s *DԔ f ,$D  0  0," pc,$ 0 *Rzeczywi[cie, jak pokazuje rysunek punkt pj nie mo|e nale|e do otoczki wypukBej zbioru Q.,[ 2* 0&)08l }g !,$D  0ZB  s *Do}gZB   s *Do}:`l " 0L3" F ,$  0 Punkty przegldamy w porzdku rosncych wspBrzdnych biegunowych.C 2C CH  0޽h ? @ ff3Ιd332z|,t,___PPT10T,.J0v+DH+' = @B D+' = @BA?%,( < +O%,( < +D' =%(D' =%(DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B'blinds(horizontal)*<3<*D' =%(D' =%(DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B'blinds(horizontal)*<3<*D' =%(D' =%(DG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-6B'blinds(horizontal)*<3<* Dn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*D' =%(D' =%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-o6Bdissolve*<3<*D' =%(D' =%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-o6Bdissolve*<3<*D' =%(D' =%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-o6Bdissolve*<3<*D' =%(D' =%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-o6Bdissolve*<3<*D' =%(D' =%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-o6Bdissolve*<3<*D' =%(D' =%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-o6Bdissolve*<3<*D' =%(D' =%(Dg' =4@BB7BB%(D' =1:Bvisible*o3>+B#style.visibility<*!%(D' =+4 8?fCB#ppt_w*0.70BCB#ppt_wB*Y3>B ppt_w<*!D' =+4 8?\CB#ppt_hBCB#ppt_hB*Y3>B ppt_h<*!D' =-g6B fade*<3<*!D' =%(D' =%(DP' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*"%(D' =-6B#blinds(vertical)*<3<*"D' =%(D' =%(DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B'blinds(horizontal)*<3<*++0+ ++0+ ++0+ ++0+" +P  K0 ~'v'DT@@&(  @r @ S K p`     @ 0TM" M,$ 0 h2. Gdy do cigu wierzchoBkw wielokta wypukBego doBczamy nowy punkt ze wskazanego na rysunkach obszaru, to otrzymamy znw wielokt wypukBy. 2 l  ) ,@& :J ,$D  0h2  @ s *" IFv @ 6T"  ) bpi" 2 l  6 0 C@@: : ,$D 0 @ 6Y" ` 6 p  bpk" 2 ZB @ s *D F a ) `B @ 0D F CV ZB @B s *D Cn2 @ 0"  ) V n2  @ 0" a  E n2  @ 0" CV p n2  @ 0"    @ 6_"  I 0 Tp0" 2   @ 6@d"  s Z  bpj" 2  @ 6Di" q)   Tp1" 2  l 7  -@`=9 ,$D 0`B  @B 0D:`B @ 0Dg6 `B @B 0D:c Z `B @B 0DCg) `B  @B 0D `B !@B 0D:S w `B "@B 0D V `B #@B 0DG `B $@B 0Dp `B %@B 0D7 `B &@B 0DCW) `B '@B 0DC ) `B (@B 0D* `B )@B 0DS 9 `B *@B 0Dg  `B +@B 0DtF B .@@ s *Do`h ,$D 0B /@ s *Do h ,$D 0l  m9  D@ ,$D 0n2 @ 0" )Vn2 @ 0" SP}n2 @ 0" CpBn2 @ 0" ) V h2 0@ s *" 6c c TB 1@B c $D6 V TB 2@ c $DVV TB 3@ c $DSP)TB 4@B c $Dp}S =@ 6u"  6   Tp0" 2   >@ 6z"  t bpj" 2  ?@ 6@" ) 9g  Tp1" 2   @@ 0D" &m6T bpk" 2 l I0 B@S p: ,$D   0n2 ;@ 0" v A@ 0" I0 bpi" 2 Hl  C6  P@P  ,$D  0ZB E@B s *D# CC ZB F@ s *DpC6c ZB G@B s *D 6 ZB H@B s *D`  ZB I@B s *DCS Ss ZB J@B s *DS  ZB K@B s *D p ZB M@B s *D &s ZB N@B s *D  ZB O@B s *D6  B Q@@ s *Dop ,$D  0B R@ s *Do p ,$D  0d S@ 0"  d,$  0 *Zgodnie z algorytmem, je[li doBczamy pi do stosu, to pi le|y po lewej stronie prostej pkpj. Ponadto kt jaki tworzy pip0 z dodatnim kierunkiem osi OX jest wikszy ni| kt jaki tworzy pjp0 z osi OX. Mamy wic gwarancj, |e doBczony punkt pi nale|y do zakreskowanego obszaru. 2'    @5"t&#H @ 0޽h ? @ ff3Ιd332z&((___PPT10'.+"Db'' = @B D'' = @BA?%,( < +O%,( < +D' =%(D' =%(DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*@%(D' =-6B'blinds(horizontal)*<3<*@D' =%(D' =%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*C@%(D' =-o6Bdissolve*<3<*C@D' =%(D' =%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*-@%(D' =-o6Bdissolve*<3<*-@Dn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*,@%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*,@D' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*,@D' =%(D' =%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*.@%(D' =-o6Bdissolve*<3<*.@D' =%(D' =%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*/@%(D' =-o6Bdissolve*<3<*/@D' =%(D' =%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*D@%(D' =-o6Bdissolve*<3<*D@D' =%(D' =%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*P@%(D' =-o6Bdissolve*<3<*P@Dn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*B@%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*B@D' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*B@D' =%(D' =%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*Q@%(D' =-o6Bdissolve*<3<*Q@D' =%(D' =%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*R@%(D' =-o6Bdissolve*<3<*R@D' =%(D' =%(DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*S@%(D' =-6B'blinds(horizontal)*<3<*S@+p+0+@ ++0+S@ +9  K0 wo  (  r  S 2 p`     0l8" z4,$ 0 Z powy|szych rozwa|aD wynika, |e niezmiennikiem ptli w 4tym kroku algorytmu Grahama jest wBasno[:d 2d dn  V<APapeteria" ,$D 0 `Zbir punktw na stosie tworzy wielokt wypukBy.1 21 1  C W3yd?Koszt algorytmuImpact" 2M ,$D  0  0A" p* ,$ 0 Operacje dominujce to porwnywanie wspBrzdnych lub badanie poBo|enia punktu wzgldem prostej.b 2b b3   0\E"   ,$ 0 %O(n) + O(nlg n) + O(1) + O(n-3)& 2&40   hIGDHA Papeteria"  GJ ,$D 0 Vszukanie minimum *   hNG(&HA Papeteria" V / f ,$D 0 P sortowanie   3   hOGT"HwA Papeteria" ) p ,$D  0 Yinicjalizacja stosu %   hTGHA Papeteria"  S ,$D  0 Kkrok4    0XX"  tY ,$  0 h = O(nlg n) 2 &H  0޽h ?O@     @ ff3Ιd332z**___PPT10*.J3r+Jݦ%D}(' = @B D8(' = @BA?%,( < +O%,( < +D' =%(D' =%(DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B'blinds(horizontal)*<3<*D' =%(D' =%(D@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-o6Bdissolve*<3<*Dn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*D' =%(D' =%(DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B'blinds(horizontal)*<3<*D' =%(D' =%(D@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-o6Bdissolve*<3<* D$' =%(D' =%(Dt' =A@BB7BB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =+4 8?fCB#ppt_w*0.70BCB#ppt_wB*Y3>B ppt_w<* D' =+4 8?\CB#ppt_hBCB#ppt_hB*Y3>B ppt_h<* D' =-g6B fade*<3<* D$' =%(D' =%(Dt' =A@BB7BB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =+4 8?fCB#ppt_w*0.70BCB#ppt_wB*Y3>B ppt_w<* D' =+4 8?\CB#ppt_hBCB#ppt_hB*Y3>B ppt_h<* D' =-g6B fade*<3<* D$' =%(D' =%(Dt' =A@BB7BB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =+4 8?fCB#ppt_w*0.70BCB#ppt_wB*Y3>B ppt_w<* D' =+4 8?\CB#ppt_hBCB#ppt_hB*Y3>B ppt_h<* D' =-g6B fade*<3<* D$' =%(D' =%(Dt' =A@BB7BB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =+4 8?fCB#ppt_w*0.70BCB#ppt_wB*Y3>B ppt_w<* D' =+4 8?\CB#ppt_hBCB#ppt_hB*Y3>B ppt_h<* D' =-g6B fade*<3<* DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(++0+ ++0+ ++0+ ++0+  ++0+  ++0+  ++0+  ++0+  ++0+ +/  K0 770dx()7(  (r ( S q p`   l    <(  ,$D 0}@    (  h2 ( s *" S   ( 0`s"    GA 2 N    (  h2 ( s *" S    ( 0x"    GB 2 N     ( `Gh2  ( s *" S    ( 0|"    GC 2 N     ( g@J'h2 ( s *" S   ( 0x"    GD 2 N    ( { h2 ( s *" S   ( 0"    GE 2 N    ( @ # h2 ( s *" S   ( 0"    GF 2 N    (   h2 ( s *" S   ( 0"    GH 2 N    (   h2 ( s *" S   ( 0H"    GI 2 N    (  qh2 ( s *" S   ( 0|"    GJ 2 N    (  %  h2  ( s *" S   !( 0"    GK 2 N    "(   h2 #( s *" S   $( 0"    GG 2  ;( 0" Mz G [Dany zbir punktw Q: 2  =( C W3yd?stos Impact"   ,$D  0  >( 0Ħ"  SY ,$  0 jPo posortowaniu: BKIJFGHECDA 2  B ?( c $D s ,$D 0B @( c $Dp s ,$D 0B A( c $D Cs ,$D 0B B( c $DC s ,$D 0B C( c $DP s ,$D 0B D( c $D s ,$D 0B E( c $DM s ,$D  0B F( c $D s ,$D  0B G( c $DJ s ,$D  0B H( c $DS  s ,$D  0Ql  I L( I,$D 0b@ V  6( ITB %( c $DV TB &( c $DZB '( s *Df  I( <"  9  GI  J( <" 9   GK  K( <Զ"  I GB l  gI P( gI,$D 0pN  g 7(  gITB 8( c $D TB 9( c $DgZB :( s *Dgg@  9 A I O(9 ;I M( <`"  9 A  GK  N( <п"  A I GB Ql J -J T( J-J,$D 0b@  g 1(J -ITB .( c $D TB /( c $DgZB 0( s *Dgg Q( <" w 9  GJ  R( < " w9   GK  S( <" w J GB Bl ( J r(( J,$D 0pN  g 2(  ITB 3( c $D TB 4( c $DgZB 5( s *Dgg b( B" = 9  GJ  c( B" =9   GK  d( B" = J GB  m( <@" >(   GF _l  I s( I,$D 0pN  g U(  ITB V( c $D TB W( c $DgZB X( s *Dgg n( <<"  8  GJ  o( <"  8  GK  p( <"  I GB 0l  ) ~ J w() ~ J,$D 0pN  g Y(  ~ ITB Z( c $D TB [( c $DgZB \( s *Dgg q( <"  ) Q  GG  t( <"  Q 9  GJ  u( <"  9 Q  GK  v( <p"  Q J GB  x( C W3yd?itd Impact"   D ,$D  0H ( 0޽h ? @ ff3Ιd332zJJ___PPT10I.Jw+DDI' = @B DII' = @BA?%,( < +O%,( < +D' =%(D' =%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*<(%(D' =-o6Bdissolve*<3<*<(D' =%(D' =%(Dg' =4@BB7BB%(D' =1:Bvisible*o3>+B#style.visibility<*?(%(D' =+4 8?fCB#ppt_w*0.70BCB#ppt_wB*Y3>B ppt_w<*?(D' =+4 8?\CB#ppt_hBCB#ppt_hB*Y3>B ppt_h<*?(D' =-g6B fade*<3<*?(D' =%(D' =%(Dg' =4@BB7BB%(D' =1:Bvisible*o3>+B#style.visibility<*@(%(D' =+4 8?fCB#ppt_w*0.70BCB#ppt_wB*Y3>B ppt_w<*@(D' =+4 8?\CB#ppt_hBCB#ppt_hB*Y3>B ppt_h<*@(D' =-g6B fade*<3<*@(D' =%(D' =%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*A(%(D' =-o6Bdissolve*<3<*A(D' =%(D' =%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*B(%(D' =-o6Bdissolve*<3<*B(D' =%(D' =%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*C(%(D' =-o6Bdissolve*<3<*C(D' =%(D' =%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*D(%(D' =-o6Bdissolve*<3<*D(D' =%(D' =%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*E(%(D' =-o6Bdissolve*<3<*E(D' =%(D' =%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*F(%(D' =-o6Bdissolve*<3<*F(D' =%(D' =%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*G(%(D' =-o6Bdissolve*<3<*G(D' =%(D' =%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*H(%(D' =-o6Bdissolve*<3<*H(D ' =%(D9 ' =%(D' =A@BB&BB0B@HB%()?D' =1:Bvisible*o3>+B#style.visibility<*>(%(D' =1:B -45.0*k3>'Bstyle.rotation<*>(%(DS' =+4 8?lCB4BCB4BBCBB*k3>'Bstyle.rotation<*>(%(Dm' =+4 8?CB#ppt_y-1BCBG#ppt_y-(0.354*#ppt_w-0.172*#ppt_h)B*Y3>B ppt_y<*>(%(D' =+[4 8?CBG#ppt_y-(0.354*#ppt_w-0.172*#ppt_h)BCBY#ppt_y-(0.354*#ppt_w-0.172*#ppt_h)-#ppt_h/2B*Y3>B ppt_y<*>(%())?)?Di' =+4 8?CBG#ppt_y-(0.354*#ppt_w-0.172*#ppt_h)BCB#ppt_yB*Y3>B ppt_y<*>(%(`Dn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*=(%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*=(D' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*=(D' =%(D' =%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*L(%(D' =-o6Bdissolve*<3<*L(D' =%(D' =%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*P(%(D' =-o6Bdissolve*<3<*P(D' =%(D' =%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*T(%(D' =-o6Bdissolve*<3<*T(D' =%(D' =%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*r(%(D' =-o6Bdissolve*<3<*r(D' =%(D' =%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*s(%(D' =-o6Bdissolve*<3<*s(D' =%(D' =%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*w(%(D' =-o6Bdissolve*<3<*w(Dn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*x(%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*x(D' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*x(+8+0+>( +>(  K0 0 (   (  r  S d p`     0" J ,$ 0 p*Obroty kartk papieru 2 Z  C ZwG UNd)?AlgorytmArial Black$ k !2" @Y ,$D  0  0" T,$ 0 1. Znalez punkty o najmniejszej i najwikszej wspBrzdnej y: p0 i q0. BI 2@   I  0X" `  ,$ 0 2. Skonstruowa dwa cigi: lewy cig p0,... , pm=q0 i prawy cig q0,...,qk = p0, takie |e razem tworz otoczk wypukBy 2 f   f   ( y"   0/"  VU ,$ 0 p i+1 jest tym punktem zbioru Q, ktry ma najmniejszy kt biegunowy ze wzgldu na pi liczc od dodatniej pBosi OX w lewo.B{ 2 N '&R'   C W3yd?MetodaImpact" ,$D  0    07"  V,$ 0 ~q i+1 jest tym punktem zbioru Q, ktry ma najmniejszy kt biegunowy ze wzgldu na qi liczc od ujemnej pBosi OX w prawo.Bz 2 N &&R&H  0޽h ? @ ff3Ιd332z___PPT10v.JP8+6vD2' = @B D' = @BA?%,( < +O%,( < +Dn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<* D' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<* D' =%(D' =%(DX' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B+checkerboard(across)*<3<*Dn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*D' =%(D' =%(DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B'blinds(horizontal)*<3<*D' =%(D' =%(DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B'blinds(horizontal)*<3<*D' =%(D' =%(DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-6B'blinds(horizontal)*<3<* D' =%(D' =%(DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-6B'blinds(horizontal)*<3<* ++0+ ++0+ ++0+ ++0+  ++0+  +  K0     (   r   S TR p`   P   0S" /w),$ 0 hNiech k bdzie liczb wierzchoBkw szukanej otoczki.5 25 5   0X" @,$  0 Ka|dy krok ptli kosztuje O(n), bo szukamy minimum kta biegunowego.F 2F&)7   V]APapeteria" /,5,$D 0 o Razem O(n*k) 2& :   0Da" 2W  ,$ 0 Uwaga W algorytmie Jarvisa podobnie jak w algorytmie Grhama nie musimy liczy ktw. Wystarczy wiedzie, |e jeden kt jest wikszy od drugiego. 2@U   VfAPapeteria"  w%,$D 0 BPoprawka (Eddy&Floyd) 1.Wybra cztery ekstremalne punkty. 2.Usun wszystkie punkty le|ce wewntrz tego czworokta. 3. Dalej tak jak w alg. Jarvisa lub Grahama. 2@  yH   0޽h ? @ ff3Ιd332z1)___PPT10 .Jp+PD' = @B D' = @BA?%,( < +O%,( < +D$' =%(D' =%(Dt' =A@BB7BB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =+4 8?fCB#ppt_w*0.70BCB#ppt_wB*Y3>B ppt_w<* D' =+4 8?\CB#ppt_hBCB#ppt_hB*Y3>B ppt_h<* D' =-g6B fade*<3<* D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<* D' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<* D' =%(D' =%(DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-6B'blinds(horizontal)*<3<* D' =%(D' =%(DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-6B'blinds(horizontal)*<3<* D' =%(D' =%(D@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-o6Bdissolve*<3<* ++0+  ++0+  ++0+  ++0+  ++0+  +  K0 ,+$+ @|$*(  $r $ S | p`   8 /S  2$]* T    $#  wg h2 $ s *" S   $ 0~"    GA 2 T    $# z ] h2  $ s *" S    $ 0"    GB 2 T     $# = h2  $ s *" S    $ 0\"    GC 2 T    $# h2 $ s *" S   $ 0"    GD 2 T    $#  ? &h2 $ s *" S   $ 00"    GE 2 T    $#  C * h2 $ s *" S   $ 0d"    GF 2 T    $#  /m h2 $ s *" S   $ 0"    GH 2 T    $#  q h2 $ s *" S   $ 0̝"    GI 2 T    $# h2 $ s *" S   $ 0"    GJ 2 T     $# p S h2 !$ s *" S   "$ 04"    GK 2 T    #$# @ }# dh2 $$ s *" S   %$ 0h"    GG 2 RB 0$ s *D GY  3$ 0T" cY,$ 0 Q ABCDEFGHIJK 2   `$ 0L" cY,$ 0 a BACDEFGHIJK& 2f  B a$ s *DL,$D  0B b$ s *D,$D  0 c$ 0 " PJ,$ 0 _ BKCDEFGHIJA$ 2f  B d$ s *Dww},$D   0 e$ 0" ,$  0 _ BKJDEFGHICA$ 2f   f$ 08"  ,$ 0 _ BKJHEFGDIJA$ 2f  g$ 0" 6 0 ,$ 0 _ BKJHDFGEIJA$ 2f  h$ 0P"  cY ,$ 0 _ BKJHDAGEIJF$ 2f B i$ s *D`,$D   0B k$ 0Do  ,$D 0B l$ 0Do s ,$D 0B m$ 0DoV  ,$D 0B n$ c $D% V ,$D 0~ o$ 0" 9  Zamiast kta wzgldem osi wystarczy bada kt wzgl. przedBu|enia ostatnio znalezionego boku otoczki.f 2f fT p$  BC DEFAA5%[ >&&@  " c  ,$D 0B q$ c $D/` 6 ,$D  0T r$  BCsDEFAA5%svTj5-55@  " ; S ,$D  0B s$ c $D- P,$D 0T t$  BCDEFAA5%Eb"j.@  "  ` m,$D 0B u$@ c $DG ,$D  0T v$  BCDEFAA5%DK-q@  " ,$D 0T y$  BCDEFAA5% H[c@  " 2t ,$D 0B z$ c $D6  V ,$D 0T {$  B>CDEFAA5%[@yy'>@  "   ,$D 0LB |$@ c $D}:) H $ 0޽h ? @ ff3Ιd332zOXGX___PPT10'X.Jp|+N0DsV' = @B D.V' = @BA?%,( < +O%,( < +D' =%(D' =%(D@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*3$%(D' =-o6Bdissolve*<3<*3$Dn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*a$%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*a$D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*a$D' =%(D' =%(D@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*`$%(D' =-o6Bdissolve*<3<*`$Dn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*b$%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*b$D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*b$D' =%(D' =%(Dg' =4@BB7BB%(D' =1:Bvisible*o3>+B#style.visibility<*p$%(D' =+4 8?fCB#ppt_w*0.70BCB#ppt_wB*Y3>B ppt_w<*p$D' =+4 8?\CB#ppt_hBCB#ppt_hB*Y3>B ppt_h<*p$D' =-g6B fade*<3<*p$D' =%(D' =%(Dg' =4@BB7BB%(D' =1:Bvisible*o3>+B#style.visibility<*n$%(D' =+4 8?fCB#ppt_w*0.70BCB#ppt_wB*Y3>B ppt_w<*n$D' =+4 8?\CB#ppt_hBCB#ppt_hB*Y3>B ppt_h<*n$D' =-g6B fade*<3<*n$D' =%(D' =%(D@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*c$%(D' =-o6Bdissolve*<3<*c$Dn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*d$%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*d$D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*d$D' =%(D' =%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*q$%(D' =-o6Bdissolve*<3<*q$D' =%(D' =%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*r$%(D' =-o6Bdissolve*<3<*r$D' =%(D' =%(D@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*e$%(D' =-o6Bdissolve*<3<*e$Dn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*i$%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*i$D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*i$D' =%(D' =%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*t$%(D' =-o6Bdissolve*<3<*t$D' =%(D' =%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*s$%(D' =-o6Bdissolve*<3<*s$D' =%(D' =%(D@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*f$%(D' =-o6Bdissolve*<3<*f$D' =%(D' =%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*k$%(D' =-o6Bdissolve*<3<*k$Dn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*u$%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*u$D' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*u$D' =%(D' =%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*v$%(D' =-o6Bdissolve*<3<*v$D' =%(D' =%(D@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*g$%(D' =-o6Bdissolve*<3<*g$D' =%(D' =%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*l$%(D' =-o6Bdissolve*<3<*l$D' =%(D' =%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*y$%(D' =-o6Bdissolve*<3<*y$D' =%(D' =%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*z$%(D' =-o6Bdissolve*<3<*z$D' =%(D' =%(D@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*h$%(D' =-o6Bdissolve*<3<*h$D' =%(D' =%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*{$%(D' =-o6Bdissolve*<3<*{$D' =%(D' =%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*m$%(D' =-o6Bdissolve*<3<*m$++0+3$ ++0+`$ ++0+c$ ++0+e$ ++0+f$ ++0+g$ ++0+h$ +  K0 @,(  ,r , S  p`   H , 0޽h ? @ ff3Ιd332z___PPT10i.J G+D=' = @B + 0 vn(  X  C    n  S x$ @   |Je[li |Q|=n, to mo|na utworzy (n nad 3) trjkty. Zatem ptla gBwna wykona O(n3) iteracji. W wewntrznej ptli sprawdzamy wszystkie punkty zbioru Q. Poniewa| sprawdzenie, czy punkt le|y wewntrz trjkta mo|na wykona z kosztem staBym, zatem koszt ptli wewntrznej- O(n). Ostatecznie koszt caBego algorytmy O(n4). (> 2>@/H  0޽h ? ̙3380___PPT10.HF 0  p8(  8X 8 C      8 S 1 @   JAlgorytm ten wykorzystuje stos jako struktur pomocnicz. Ka|dy punkt zbioru Q jest raz wkBadany na st os i o ile stwierdzimy, |e nie mo|e by wierzchoBkiem otoczki, jest usuwany ze stosu.Na zakoDczenie stos zawiera wszystkie wierzchoBki otoczki w porzdku przeciwnym do ruchu wskazwek zegara@dPhH 8 0޽h ? ̙3380___PPT10.[P M 0 < (  <X < C     < S d< @   "Poniewa| przegldamy punkty w porzdku rosncych wspBrzdnych biegunowych wzgldem p0, to dla punktu pj istnieje trjkt we wntrzu ktrego znalazBoby si pj. Zatem pj nie mo|e by wierzchoBkiem otoczki wypukBej.Zg4.H < 0޽h ? ̙3380___PPT10.P| 0 L(  LX L C     L S XE @   4Niech p0 bdzie punktem o najmniejszej wspBrzdnej y. Zaczynajc od p0, przykBadamy kartk papieru rwnolegle do X i obracamy j w lewo, tak dBugo a| napotkamy jaki[ punkt zbioru Q. Ten znaleziony punkt bdzie wierzchoBkiem otoczki. Powtarzamy postpowanie od znalezionego punktu. ( 2H L 0޽h ? ̙3380___PPT10./0IO]r|0' 4L. Q7)CF͑r5r~A P0˖gU0:? 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