--- /dev/null
+input lib
+
+figname("7-geom");
+
+figtag("male_obaly");
+beginfig(1);
+ pickup boldpen;
+ labeloffset:=1cm;
+ pair c,pos; c := (0,0); pos := c;
+ drawemptyvertex(c);
+ label.bot(btex $n=1$ etex, pos);
+ c := (2cm,0);
+ pos := pos + c;
+ pair A[];
+ A[0] := (-0.3cm, -0.2cm)+c; A[1] := (0.2cm, 0.3cm)+c;
+ draw A[0]--A[1];
+ drawemptyvertex(A[0]); drawemptyvertex(A[1]);
+ label.bot(btex $n=2$ etex, pos);
+
+ pos := pos + c;
+ A[2] := (+0.3cm, -0.4cm)+c;
+ for i := 0 upto 2: A[i] := A[i] shifted c; endfor
+ draw A[0]--A[1]--A[2]--cycle;
+ for i := 0 upto 2: drawemptyvertex(A[i]); endfor
+ label.bot(btex $n=3$ etex, pos);
+
+ pos := pos + c;
+ A[3] := (A[0]+A[1]+A[2])/3;
+ for i := 0 upto 3: A[i] := A[i] shifted c; endfor
+ draw A[0]--A[1]--A[2]--cycle;
+ for i := 0 upto 2: drawemptyvertex(A[i]); endfor
+ draw vertex(A[3]);
+
+ c := (1cm,0);
+ pos := pos + c/2;
+ A[3] := A[1]+(0.3cm,-0.2cm);
+ for i := 0 upto 3: A[i] := A[i] shifted c; endfor
+ draw A[0]--A[1]--A[3]--A[2]--cycle;
+ for i := 0 upto 3: drawemptyvertex(A[i]); endfor
+ label.bot(btex $n=4$ etex, pos);
+endfig;
+
+figtag("pridani_bodu");
+beginfig(2);
+ pair A[],B[],C,shift; shift := (4.5cm,0);
+ A[0] := (-1.7cm,1.1cm);
+ A[1] := (-1.2cm,1.2cm);
+ A[2] := (-0.4cm,1cm);
+ A[3] := (0.2cm,0.2cm);
+ A[4] := (0.4cm,-0.7cm);
+ A[5] := (-0.8cm,-1.3cm);
+ A[6] := (-1.4cm,-1.4cm);
+ B[0] := (-1.1cm, 0.7cm);
+ B[1] := (-0.6cm, 0.1cm);
+ B[2] := (-1.3cm, -0.6cm);
+ C := (1cm, 0.1cm);
+
+ % krok 1
+ pickup boldpen;
+ draw A[0] for i := 1 upto 6: --A[i] endfor;
+ for i := 1 upto 5: drawemptyvertex(A[i]); endfor
+ for i := 0 upto 2: draw vertex(B[i]); endfor
+ draw vertex(C);
+ drawarrow (C+(0.5cm,0)--C+(1.5cm,0)) shifted (0, -0.1cm) withpen normalpen;
+ for i:=0 upto 6: A[i] := A[i] shifted shift; endfor
+ for i:=0 upto 2: B[i] := B[i] shifted shift; endfor
+ C := C shifted shift;
+
+ % krok 2
+ draw for i := 0 upto 3: A[i]-- endfor C for i := 4 upto 6: --A[i] endfor;
+ draw C--A[2] dashed evenly withpen normalpen;
+ for i := 1 upto 5: drawemptyvertex(A[i]); endfor
+ for i := 0 upto 2: draw vertex(B[i]); endfor
+ drawemptyvertex(C);
+ drawarrow (C+(0.5cm,0)--C+(1.5cm,0)) shifted (0,-0.1cm) withpen normalpen;
+ for i:=0 upto 6: A[i] := A[i] shifted shift; endfor
+ for i:=0 upto 2: B[i] := B[i] shifted shift; endfor
+ C := C shifted shift;
+
+ % krok 3
+ draw for i := 0 upto 2: A[i]-- endfor C for i := 4 upto 6: --A[i] endfor;
+ for i := 1 upto 2: drawemptyvertex(A[i]); endfor
+ for i := 4 upto 5: drawemptyvertex(A[i]); endfor
+ for i := 0 upto 2: draw vertex(B[i]); endfor
+ draw vertex(A[3]);
+ drawemptyvertex(C);
+endfig;
+
+figtag("obalky");
+beginfig(3);
+ labeloffset := 0.2cm;
+ pickup boldpen;
+ pair A[],B[];
+ A[0] := (-7cm, 0cm);
+ A[1] := (-6.2cm, 0.9cm);
+ A[2] := (-4.6cm,1.5cm);
+ A[3] := (-2.4cm,1.8cm);
+ A[4] := (-0.8cm,1.5cm);
+ A[5] := (0.4cm,0.6cm);
+ A[6] := (0.8cm,-0.10cm);
+ A[7] := (-1.6cm,-1.9cm);
+ A[8] := (-4cm,-2.1cm);
+ A[9] := (-6cm, -1.5cm);
+ A[10] := (-7cm, 0cm);
+
+ B[0] := (-2.2cm, 0.7cm);
+ B[1] := (-1.2cm, 0.1cm);
+ B[2] := (-2.6cm, -0.6cm);
+ B[3] := (-3.6cm, -0.4cm);
+ B[4] := (-3cm, 0.6cm);
+ B[5] := (-2.6cm, 1cm);
+ B[6] := (-1cm, -1.2cm);
+ B[7] := (-6.5cm, 0.2cm);
+ B[8] := (-5cm, 0.8cm);
+ B[9] := (-6cm, -0.6cm);
+ B[10] := (-5cm, -1.2cm);
+
+ draw createpath(for i := 0 upto 5: A[i]-- endfor A[6]);
+ draw (for i := 6 upto 9: A[i]-- endfor A[10]) dashed evenly;
+ for i := 0 upto 9: drawemptyvertex(A[i]); endfor
+ for i := 0 upto 10: draw vertex(B[i]); endfor
+
+ label(btex \font\myfont=csr10 \myfont horní obálka etex, ((-7cm+0.8cm)/2,2.2cm));
+ label(btex \font\myfont=csr10 \myfont dolní obálka etex, ((-7cm+0.8cm)/2,-2.5cm));
+ label.lft(btex $L$ etex, A[0]);
+ label.rt(btex $P$ etex, A[6]);
+endfig;
+
+figtag("determinant");
+beginfig(4);
+ labeloffset := 0.1cm;
+ pair A[], shift; shift := (4cm,1cm);
+
+ % det(M) > 0
+ A[0] := (-2cm, 0);
+ A[1] := (0,-1cm);
+ A[2] := (1.5cm, 0cm);
+ A[3] := A[0] + A[2] - A[1];
+
+ fill A[0]--A[1]--A[2]--A[3]--cycle withcolor 0.8white;
+ draw A[0]--A[1]--A[2]--A[3]--cycle dashed evenly;
+ drawarrow A[0]--0.96[A[0],A[1]] withpen boldpen;
+ drawarrow A[1]--0.96[A[1],A[2]] withpen boldpen;
+ for i:=0 upto 2: draw vertex(A[i]); endfor
+ label.lft(btex $h_{k-1}$ etex, A[0]);
+ label.bot(btex $h_k$ etex, A[1]);
+ label.llft(btex $\vec u$ etex, 0.5[A[0],A[1]]);
+ label.lrt(btex $\vec v$ etex, 0.5[A[1],A[2]]);
+ label.rt(btex $b$ etex, A[2]);
+ label(btex $\det(M) > 0$ etex, 0.5[A[0],A[2]]);
+
+ % det(M) = 0
+ A[0] := (-1cm, -0.5cm) + shift;
+ A[1] := (0, -1cm) + shift;
+ A[2] := (1cm, -1.5cm) + shift;
+ drawarrow A[0]--0.96[A[0],A[1]] withpen boldpen;
+ drawarrow A[1]--0.96[A[1],A[2]] withpen boldpen;
+ for i:=0 upto 2: draw vertex(A[i]); endfor
+ label.lft(btex $h_{k-1}$ etex, A[0]);
+ label.llft(btex $h_k$ etex, A[1]);
+ label.llft(btex $\vec u$ etex, 0.5[A[0],A[1]]);
+ label.llft(btex $\vec v$ etex, 0.5[A[1],A[2]]);
+ label.bot(btex $b$ etex, A[2]);
+ label(btex $\det(M) = 0$ etex, origin) shifted (0,0.3cm) rotated -28 shifted 0.5[A[0], A[2]];
+
+ % det(M) < 0
+ shift := (7.5cm, 1.25cm);
+ A[0] := (-1cm, -0.5cm) + shift;
+ A[1] := (1.5cm, -1cm) + shift;
+ A[2] := (2cm, -2.5cm) + shift;
+ A[3] := A[0] + A[2] - A[1];
+ fill A[0]--A[1]--A[2]--A[3]--cycle withcolor 0.8white;
+ draw A[0]--A[1]--A[2]--A[3]--cycle dashed evenly;
+ drawarrow A[0]--0.96[A[0],A[1]] withpen boldpen;
+ drawarrow A[1]--0.96[A[1],A[2]] withpen boldpen;
+ for i:=0 upto 2: draw vertex(A[i]); endfor
+ label.lft(btex $h_{k-1}$ etex, A[0]);
+ label.urt(btex $h_k$ etex, A[1]);
+ label.top(btex $\vec u$ etex, 0.5[A[0],A[1]]);
+ label.rt(btex $\vec v$ etex, 0.5[A[1],A[2]]);
+ label.rt(btex $b$ etex, A[2]);
+ label(btex $\det(M) < 0$ etex, 0.5[A[0],A[2]]);
+endfig;
+
+figtag("rybi_motivace");
+beginfig(5);
+ u := 0.3cm;
+ def draw_fish(expr pos,size,rot) =
+ draw ((-1.3u*size,0){dir 60}..{right}(u*size,-u*size/4)) rotated rot shifted pos;
+ draw ((-1.3u*size,0){dir -50}..{right}(u*size,u*size/4)) rotated rot shifted pos;
+ draw ((u*size,-u*size/4)--(u*size,u*size/4)) rotated rot shifted pos;
+ draw (-1u*size,u*size/15) rotated rot shifted pos withpen pencircle scaled (u/8);
+ for i:=1 upto 3: draw (dirs((u*size,-u*size/4+i*u*size/8), 180, u*size/6)) rotated rot shifted pos; endfor
+ enddef;
+
+ pair A[],B[];
+ A[0] := (-7cm, 0cm);
+ A[1] := (-6.2cm, 0.9cm);
+ A[2] := (-4.6cm,1.5cm);
+ A[3] := (-2.4cm,1.8cm);
+ A[4] := (-0.8cm,1.5cm);
+ A[5] := (0.4cm,0.6cm);
+ A[6] := (0.8cm,-0.10cm);
+ A[7] := (-1.6cm,-1.9cm);
+ A[8] := (-4cm,-2.1cm);
+ A[9] := (-6cm, -1.5cm);
+ A[10] := (-7cm, 0cm);
+
+ B[0] := (-2.2cm, 0.7cm);
+ B[1] := (-1.2cm, 0.1cm);
+ B[2] := (-2.6cm, -0.6cm);
+ B[3] := (-3.6cm, -0.4cm);
+ B[4] := (-3cm, 0.6cm);
+ B[5] := (-2.6cm, 1cm);
+ B[6] := (-1cm, -1.2cm);
+ B[7] := (-6.5cm, 0.2cm);
+ B[8] := (-5cm, 0.8cm);
+ B[9] := (-6cm, -0.6cm);
+ B[10] := (-5cm, -1.2cm);
+
+ for i:=0 upto 9: draw_fish(A[i], 1, 0); endfor;
+ for i:=0 upto 10: draw_fish(B[i], 1, 0); endfor;
+ draw createpath(for i:=0 upto 9: A[i]-- endfor cycle) scaled 1.13 shifted (0.4cm,0) withpen boldpen;
+endfig;
+end
--- /dev/null
+\input lecnotes.tex
+
+\prednaska{7}{Geometrické algoritmy}{(P. Klavík)}
+
+\>Uká¾eme si nìkolik základních algoritmù na øe¹ení geometrický problémù v~rovinì. Proè zrovna v~rovinì? Inu, jednorozmìrné problémy bývají triviální
+a naopak pro vy¹¹í dimenze jsou velice komplikované. Rovina je proto rozumným kompromisem mezi obtí¾ností a zajímavostí.
+
+Celou kapitolou nás bude provázet pohádka ze ¾ivota letních medvìdù. Pokusíme se vyøe¹it jejich \uv{ka¾dodenní} problémy~\dots
+
+\h{Hledání konvexního obalu}
+
+{\I Daleko na severu ¾ili lední medvìdi. Ve vodách tamního moøe byla hojnost ryb a jak je známo, ryby jsou oblíbenou pochoutkou ledních medvìdù.
+Proto¾e medvìdi z~na¹í pohádky rozhodnì nejsou ledajací a ani chytrost jim neschází, rozhodli se v¹echny ryby pochytat. Znají pøesná místa výskytu
+ryb a rádi by vyrobili obrovskou sí», do které by je v¹echny chytili. Pomozte medvìdùm zjistit, jaký nejmen¹í obvod taková sí» mù¾e mít.}
+
+\figure{7-geom5_rybi_motivace.eps}{Problém ledních mìdvìdù: Jaký je nejmen¹í obvod sítì?}{2.5in}
+
+Neboli v~øeèi matematické, chceme pro zadanou mno¾inu bodù v~rovinì nalézt její konvexní obal. Co je to konvexní obal? Mno¾ina bodù je {\I konvexní},
+pokud pro ka¾dé dva body obsahuje i celou úseèku mezi nimi. {\I Konvexní obal} je nejmen¹í konvexní podmno¾ina roviny, která obsahuje v¹echny zadané
+body.\foot{Pamatujete si na lineární obaly ve vektorových prostorech? Lineární obal mno¾iny vektorù je nejmen¹í vektorový podprostor, který tyto
+vektory obsahuje. Není náhoda, ¾e tato definice pøipomíná definici konvexního obalu. Na druhou stranu ka¾dý vektor z~lineárního obalu lze vyjádøit
+jako lineární kombinaci daných vektorù. Podobnì platí i pro konvexní obaly, ¾e ka¾dý bod z~obalu je konvexní kombinací daných bodù. Ta se li¹í od
+lineární v~tom, ¾e v¹echny koeficienty jsou v~intervalu $[0,1]$ a navíc souèet v¹ech koeficientù je $1$. Tento algebraický pohled mù¾e mnohé vìci
+zjednodu¹it. Zkuste si dokázat, ¾e obì definice konvexního obalu jsou ekvivalentní.} Z~algoritmického hlediska nás v¹ak bude zajímat jenom jeho
+hranice, kterou budeme dále oznaèovat jako konvexní obal.
+
+Na¹ím úkolem je nalézt konvexní obal koneèné mno¾iny bodù. To je v¾dy konvexní mnohoúhelník, navíc s~vrcholy v~zadaných bodech. Øe¹ením problému tedy
+bude posloupnost bodù, které tvoøí konvexní obal. Pro malé mno¾iny je konvexní obal nakreslen na obrázku, pro více bodù je v¹ak situace mnohem
+slo¾itìj¹í.
+
+\figure{7-geom1_male_obaly.eps}{Konvexní obaly malých mno¾in.}{3in}
+
+Pro jednoduchost budeme pøedpokládat, ¾e v¹echny body mají rùzné $x$-ové souøadnice. Tedy utøídìní bodù zleva doprava je urèené jednoznaènì.\foot{To si
+mù¾eme dovolit pøedpokládat, nebo» se v¹emi body staèí nepatrnì pootoèit. Tím konvexní obal urèitì nezmìníme. Av¹ak jednodu¹¹í øe¹ení je naprogramovat
+tøídìní lexikograficky (druhotnì podle souøadnice $y$) a vyøadit identické body.} Tím máme zaji¹tìné, ¾e existují dva body, nejlevìj¹í a
+nejpravìj¹í, pro které platí následující invariant:
+
+\s{Invariant:} Nejlevìj¹í a nejpravìj¹í body jsou v¾dy v~konvexním obalu.
+
+Algoritmus na nalezení konvexního obalu funguje na následujícím jednoduchém principu, kterému se nìkdy øíká {\I zametání roviny}. Procházíme body
+zleva doprava a postupnì roz¹iøujeme doposud nalezený konvexní obal o~dal¹í body. Na zaèátku bude konvexní obal jediného bodu samotný bod. Na konci
+$k$-tého kroku algoritmu známe konvexní obal prvních $k$ bodù. Kdy¾ algoritmus skonèí, známe hledaný konvexní obal. Podle invariantu musíme v~$k$-tém
+kroku pøidat do obalu $k$-tý nejlevìj¹í bod. Zbývá si jen rozmyslet, jak pøesnì tento bod pøidat.
+
+Poslední bod napojíme na nejbli¾¹í bod konvexního obalu, který je nad ním a pod ním. Èasto se v¹ak stane, ¾e obal pøestane být konvexní. To se dá v¹ak
+snadno napravit, staèí odebírat pøedcházející body tak dlouho, dokud nezískáme konvexní obal. Pøíklad pøidání bodu je na obrázku ní¾e. Mnohdy je
+potøeba odebrat i více ne¾ jeden bod.
+
+\figure{7-geom2_pridani_bodu.eps}{Pøidání bodu do konvexního obalu.}{4.5in}
+
+Pro pøípadnou implementaci a rozbor slo¾itosti si nyní popí¹eme algoritmus detailnìji. Aby se lépe popisoval, rozdìlíme si konvexní obal na dvì èásti
+spojující nejlevìj¹í a nejpravìj¹í bod obalu. Budeme jim øíkat {\I horní obálka} a {\I dolní obálka}.
+
+\figure{7-geom3_obalky.eps}{Horní a dolní obálka konvexního obalu.}{3.4in}
+
+Obì obálky jsou lomené èáry, navíc horní obálka poøád zatáèí doprava a dolní naopak doleva. Pro udr¾ování bodù v~obálkách staèí dva zásobníky.
+V~$k$-tém kroku algoritmu pøidáme zvlá¹» $k$-tý bod do horní i dolní obálky. Pøidáním $k$-tého bodu se v¹ak mù¾e poru¹it smìr, ve kterém obálka
+zatáèí. Proto budeme nejprve body z~obálky odebírat a $k$-tý bod pøidáme a¾ ve chvíli, kdy jeho pøidání smìr zatáèení neporu¹í.
+
+\s{Algoritmus:}
+
+\algo
+
+\:Setøídíme body podle $x$-ové souøadnice, oznaème body $b_1, \ldots, b_n$.
+\:Vlo¾íme do horní a dolní obálky bod $b_1$: $H = D = (b_1)$.
+\:Pro ka¾dý dal¹í bod $b = b_2,\ldots,b_n$:
+\::Pøepoèítáme horní obálku:
+\:::Dokud $\vert H\vert \ge 2$, $H = (\ldots, h_{k-1}, h_k)$ a úhel $h_{k-1} h_k b$ je orientovaný doleva:
+\::::Odebereme poslední bod $h_k$ z~obálky $H$.
+\:::Pøidáme bod $b$ do obálky $H$.
+\::Symetricky pøepoèteme dolní obálku (s orientací doprava).
+\: Výsledný obal je tvoøen body v~obálkách $H$ a $D$.
+
+\endalgo
+
+Rozebereme si èasovou slo¾itost algoritmu. Setøídit body podle $x$-ové souøadnice doká¾eme v~èase $\O(n \log n)$. Pøidání dal¹ího bodu do obálek
+trvá lineárnì vzhledem k~poètu odebraných bodù. Zde vyu¾ijeme obvyklý postup: Ka¾dý bod je odebrán nejvý¹e jednou, a tedy v¹echna odebrání trvají
+dohromady $\O(n)$. Konvexní obal doká¾eme sestrojit v~èase $\O(n \log n)$, a pokud bychom mìli seznam bodù ji¾ utøídený, doká¾eme to dokonce $\O(n)$.
+
+\s{Algebraický dodatek:} Existuje jednoduchý postup, jak zjistit orientaci úhlu? Uká¾eme si jeden zalo¾ený na lineární algebøe. Budou se hodit
+vlastnosti determinantu. Absolutní hodnota determinantu je objem pøíslu¹ného rovnobì¾nostìnu tvoøeného øádkovými vektory matice. Dùle¾itìj¹í v¹ak je,
+¾e znaménko determinantu urèuje \uv{orientaci} vektorù, zda je levotoèivá èi pravotoèivá. Proto¾e ná¹ problém je rovinný, budeme uva¾ovat determinanty
+matic $2 \times 2$.
+
+Uva¾me souøadnicový systém v~rovinì, kde $x$-ová souøadnice roste smìrem doprava a~$y$-ová smìrem nahoru. Chceme zjistit orientaci úhlu $h_{k-1} h_k
+b$. Polo¾me $\vec u = (x_1, y_1)$ jako rozdíl souøadnic $h_k$ a~$h_{k-1}$ a podobnì $\vec v = (x_2, y_2)$ je rozdíl souøadnic $b$ a~$h_k$. Matice $M$
+je definována následovnì:
+$$M = \pmatrix{\vec u \cr \vec v} = \pmatrix {x_1&y_1\cr x_2&y_2}.$$
+Úhel $h_{k-1} h_k b$ je orientován doleva, právì kdy¾ $\det M = x_1y_2 - x_2y_1$ je nezáporný,\foot{Neboli vektory $\vec u$ a $\vec v$ odpovídají
+rozta¾ení a zkosení vektorù báze $\vec x = (1,0)$ a $\vec y = (0,1)$, pro nì¾ je determinant nezáporný.} a spoèítat hodnotu determinantu je jednoduché.
+Mo¾né situace jsou nakresleny na obrázku. Poznamenejme, ¾e k~podobnému vzorci se lze také dostat pøes vektorový souèin vektorù $\vec u$ a $\vec v$.
+
+\figure{7-geom4_determinant.eps}{Jak vypadají determinanty rùzných znamének v~rovinì.}{4.6in}
+
+\s{©lo by to vyøe¹it rychleji?} Také vám vrtá hlavou, zda existují rychlej¹í algoritmy? Nejrychlej¹í známý algoritmus, jeho¾ autorem je T.~Chan,
+funguje v~èase $\O(n \log h)$, kde $h$ je poèet bodù le¾ících v~konvexním obalu, a pøitom je pøekvapivì jednoduchý. Zde si naznaèíme, jak tento
+algoritmus funguje.
+
+Algoritmus pøichází s~následující my¹lenkou. Pøedpokládejme, ¾e bychom znali velikost konvexního obalu $h$. Rozdìlíme body libovolnì do $\lceil {n
+\over h} \rceil$ mno¾in $Q_1, \ldots, Q_k$ tak, ¾e $\vert Q_i \vert \le h$. Pro ka¾dou z~tìchto mno¾in nalezneme konvexní obal pomocí vý¹e popsaného
+algoritmu. To doká¾eme pro jednu v~èase $\O(h \log h)$ a pro v¹echny v~èase $\O(n \log h)$. Nakonec vezmeme tyto konvexní obaly a pomocí jiného
+algoritmu je v~èase $\O(n \log h)$ spojíme do konvexního obalu celé mno¾iny.
+
+Popsanému algoritmu schází jedna dùle¾itá vìc: Ve skuteènosti vìt¹inou neznáme velikost $h$. Budeme proto algoritmus iterovat s~rostoucí hodnotou $h$,
+dokud konvexní obal nesestrojíme. Pokud pøi slepování konvexních obalù zjistíme, ¾e konvexní obal je vìt¹í ne¾ $h$, výpoèet ukonèíme. Zbývá je¹tì
+zvolit, jak rychle má $h$ rùst. Pokud by rostlo moc pomalu, budeme poèítat zbyteènì mnoho fází, naopak pøi rychlém rùstu by nás poslední fáze mohla
+stát pøíli¹ mnoho.
+
+V~$k$-té iteraci polo¾íme $h = 2^{2^k}$. Dostáváme celkovou slo¾itost algoritmu:
+$$\sum_{m=0}^{\O(\log \log h)} \O(n \log 2^{2^m}) = \sum_{m=0}^{\O(\log \log h)} \O(n \cdot 2^m) = \O(n \log h),$$
+kde poslední rovnost dostaneme jako souèet prvních $\O(\log \log h)$ èlenù geometrické øady $\sum 2^m$.
+
+\bye
--- /dev/null
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+showpage
+%%EOF
--- /dev/null
+P=7-geom
+
+include ../Makerules
--- /dev/null
+% implementation of figure naming and figure transparency
+string name,tag; name := ""; tag := "";
+def updatefigname =
+ if tag="": filenametemplate (name & "%c.eps");
+ else: filenametemplate (name & "%c_" & tag & ".eps");
+ fi;
+enddef;
+
+def figname(expr n) = name := n; updatefigname; enddef;
+def figtag(expr t) = tag := t; updatefigname; enddef;
+
+picture transparent_picture;
+color transparent_color; transparent_color := 0.9white;
+def drawtransparent(expr num) =
+ transparent_picture := currentpicture;
+endfig;
+
+if tag="": filenametemplate (name & "%c_transparent.eps");
+else: filenametemplate (name & "%c_" & tag & "_transparent.eps");
+fi;
+beginfig(num);
+ draw transparent_picture withcolor transparent_color;
+endfig;
+
+updatefigname;
+enddef;
+
+def from(expr p,d,len) =
+ p+dir(d)*len
+enddef;
+
+def dirs(expr p,d,len) =
+ p--from(p,d,len)
+enddef;
+
+def drawvertices(expr s,n) =
+ for i:=s upto n:
+ draw vertex(PQ[i]);
+ endfor
+enddef;
+
+def drawfvertices(expr s,n,flags) =
+ for i:=s upto n:
+ draw vertex(PQ[i]) flags;
+ endfor
+enddef;
+
+def vertex(expr p) = p withpen pencircle scaled 4pt enddef;
+def drawemptyvertex(expr p) = unfill fullcircle scaled 4pt shifted p; draw fullcircle scaled 4pt shifted p; enddef;
+def drawendpointvertex(expr p) = draw vertex(p) withcolor red; draw fullcircle scaled 6pt shifted p; enddef;
+def createpath(expr p) = shakepath(p, 0.015cm,0.1cm) enddef;
+vardef shakepath(expr p,d,l) =
+ save r,b;
+ path r; r := point(arctime 0 of p) of p;
+ b := -1;
+ for i:=l step l until arclength(p):
+ r := r--(point(arctime i of p) of p)+dir(angle(direction(arctime i of p) of p rotated 90))*d*b;
+ b := -b;
+ endfor
+ r--point(arctime arclength(p) of p) of p
+enddef;
+
+pen normalpen; normalpen := pencircle scaled 0.6pt;
+pen boldpen; boldpen := pencircle scaled 1.5pt;
+pen bolderpen; bolderpen := pencircle scaled 2pt;
+def dotline = withdots scaled 0.82 withpen boldpen enddef;
+
+vardef unclosedbubblec(expr p,c) =
+ bubblec((p..reverse p..cycle),c)
+enddef;
+
+vardef bubblec(expr p,c) =
+ save r;
+ path r; r := (point(arctime 0 of p) of p)+dir(angle(direction(arctime 0 of p) of p rotated 90))*c;
+ for i:=0.01cm step 0.025cm until arclength(p):
+ r := r..(point(arctime i of p) of p)+dir(angle(direction(arctime i of p) of p rotated 90))*c;
+ endfor
+ r..(point(arctime arclength(p) of p) of p)+dir(angle(direction(arctime arclength(p) of p) of p rotated 90))*c..cycle
+enddef;
+
+vardef bubble(expr p) = bubblec(p,0.12cm) enddef;
+vardef unclosedbubble(expr p) = unclosedbubblec(p,0.12cm) enddef;
projects / ads2.git / commitdiff