From: Martin Mares Date: Sun, 15 Nov 2009 11:22:15 +0000 (+0100) Subject: Prvni verze konvexnich obalu. X-Git-Url: http://mj.ucw.cz/gitweb/?a=commitdiff_plain;h=f0d1a6d462eed5cc2b829c36cf8b33348c5c2719;p=ads2.git Prvni verze konvexnich obalu. --- diff --git a/7-geom/7-geom.mp b/7-geom/7-geom.mp new file mode 100644 index 0000000..4337fd0 --- /dev/null +++ b/7-geom/7-geom.mp @@ -0,0 +1,225 @@ +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 diff --git a/7-geom/7-geom.tex b/7-geom/7-geom.tex new file mode 100644 index 0000000..f294889 --- /dev/null +++ b/7-geom/7-geom.tex @@ -0,0 +1,113 @@ +\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 diff --git a/7-geom/7-geom1_male_obaly.eps b/7-geom/7-geom1_male_obaly.eps new file mode 100644 index 0000000..f156331 --- /dev/null +++ b/7-geom/7-geom1_male_obaly.eps @@ -0,0 +1,324 @@ +%!PS +%%BoundingBox: -13 -35 216 12 +%%HiResBoundingBox: -12.12236 -34.76685 215.33812 11.24376 +%%Creator: MetaPost 0.993 +%%CreationDate: 2009.11.14:2207 +%%Pages: 1 +%*Font: cmmi10 9.96265 9.96265 6e:8 +%*Font: cmr10 9.96265 9.96265 31:f008 +%%BeginProlog +%%EndProlog +%%Page: 1 1 + 1 1 1 setrgbcolor +newpath 1.99252 0 moveto +1.99252 0.52847 1.78256 1.03523 1.4089 1.4089 curveto +1.03523 1.78256 0.52847 1.99252 0 1.99252 curveto +-0.52847 1.99252 -1.03523 1.78256 -1.4089 1.4089 curveto +-1.78256 1.03523 -1.99252 0.52847 -1.99252 0 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a/7-geom/lib.mp b/7-geom/lib.mp new file mode 100644 index 0000000..2536305 --- /dev/null +++ b/7-geom/lib.mp @@ -0,0 +1,82 @@ +% 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;