není rovinný.
\proof Pro spor předpokládejme, že po~zpracování vrcholu~$v$ existuje nějaká
-zpětná hrana~$wv$, kterou algoritmus nenakreslil, čili že přístup z~$v$ k~$w$
+zpětná hrana~$wv$, kterou algoritmus nenakreslil. Tedy přístup z~$v$ k~$w$
je v~obou směrech blokován externími vrcholy. Rozborem případů ukážeme,
-že pokud jsme se důsledně řídili pravidly \#1 a \#2, graf není rovinný.
+že pokud jsme se důsledně řídili pravidly \#1 a \#2, může se to stát pouze
+v~nerovinném grafu.
+{\I Překážející blok.}
Uvažme posloupnost bloků vedoucí od vrcholu~$v$ k~$w$. V~této posloupnosti se
musí vyskytovat nějaký blok~$B$, který má na obou stranách hranice aspoň jeden
externí vrchol -- jinak bychom totiž bloky popřeklápěli tak, abychom se dostali až do~$w$.
+Označme~$x$ externí vrchol na levé straně hranice bloku~$B$, $y$~ten na pravé.
+Vrcholy $x$, $y$, $v$ a~$w$ jsou zjevně navzájem různé.
{\I Nadřazené bloky.}
Nejprve ukážeme, že kořenem bloku~$B$ musí být přímo vrchol~$v$. Kdyby totiž
{\I Podřízené bloky.}
Předtím ale rozebereme případ, kdy vrchol~$w$ neleží přímo v~bloku~$B$, nýbrž
-v~nějakém podřízeném bloku, který je připojen pod nějakou artikulaci $w'\in B$.
+v~nějakém podřízeném bloku, který je připojen pod nějakou artikulací $w'\in B$.
Tento podřízený blok přitom nemůže být externí: kdyby byl, najdeme v~grafu
-minor XXXX obsahující $K_{3,3}$ (do~$w$ jsme zkontrahovali všechny podřízené
-bloky mezi~$w'$ a~$w$). Pokud bychom tedy během obcházení hranice vstoupili do~$w'$,
-pravidlo~\#2 by zaručilo, že dojdeme do~$w$ a díky pravidlu~\#1 bychom vzápětí
-nakreslili hranu~$vw$. Tu jsme nenakreslili, takže jsme nedošli ani do~$w'$.
-Můžeme tedy bez újmy na obecnosti předpokládat, že $w=w'$.
+minor~$P$ obsahující $K_{3,3}$. Tedy není externí, takže vstoupíme-li během obcházení
+do~$w'$, pravidlo~\#2 zaručí, že dojdeme do~$w$ a díky pravidlu~\#1 vzápětí
+nakreslíme hranu~$wv$. Tu jsme nenakreslili, takže jsme nedošli ani do~$w'$.
+Můžeme proto bez újmy na obecnosti předpokládat, že hrana $wv$ vede přímo
+z~bloku~$B$.
+
+\bigskip
+\centerline{\putepdf{}{pl-minor1.pdf}\qquad\putepdf{}{pl-minor2.pdf}}
+\bigskip
+\centerline{\putepdf{}{pl-minor3.pdf}\qquad\putepdf{}{pl-minor4.pdf}}
+\bigskip
+\centerline{\putepdf{}{pl-minor5.pdf}\qquad\putepdf{}{pl-minor6.pdf}}
+\bigskip
{\I Existence plotu.}
Stačí se tedy omezit na situaci s~jediným blokem~$B$, v~němž leží vrcholy~$v$, $w$,
$x$ i~$y$. Dokážeme nyní, že uvnitř bloku existuje {\I plot} -- cesta mezi~$x$
-a~$y$, jejíž ostatní vrcholy neleží na hranici bloku.
+a~$y$, jejíž zbývající vrcholy neleží na hranici bloku.
Předpokládejme pro spor, že plot neexistuje. Před nakreslením zpětných hran
vedoucích do~$v$ ještě blok~$B$ neexistoval a jeho vrcholy patřily do několika
menších bloků. Speciálně víme, že $w$~byla artikulace oddělující $x$ od~$y$,
takže každá cesta mezi~$x$ a~$y$ musela procházet přes~$w$. Proto v~pořadí podle DFS
-musí ležet~$w$ před aspoň jedním z~vrcholů $x$ a~$y$. Tento vrchol tedy tehdy musel
+musí ležet~$w$ před aspoň jedním z~vrcholů $x$ a~$y$. Buď~$x$ nebo~$y$ tedy předtím musel
ležet v~nějakém podřízeném bloku pod~$w$. A~z~nějakého takového bloku také musela
vést zpětná hrana (tehdy ještě externí) do~$v$, která později uzavřela blok~$B$.
-K~této hraně jsme se tedy museli během obcházení dostat, a~to je možné pouze přes~$w$.
+K~této hraně jsme se museli během obcházení dostat, a~to je možné pouze přes~$w$.
+Vrchol~$w$ tedy musel být navštíven a hrana $wv$ nakreslena, což je spor.
{\I Nejvyšší plot.}
-Plot tedy existuje. Zvolíme mezi všemi ploty ten nejvyšší, tedy nejbližší k~$v$
+Plot tedy existuje. Zvolíme mezi všemi ploty ten nejvyšší, čili nejbližší k~$v$
(rozmyslete si, že to je v~rovinném nakreslení dobře definované). Označme $p_x$~vrchol,
-v~němž se plot napojuje na cestu z~$v$ do~$w$ přes~$x$, a~obdobně~$p_y$. Rozmyslíme si,
+v~němž se plot napojuje na levou cestu z~$v$ do~$w$, a~obdobně~$p_y$ na pravé cestě. Rozmyslíme si,
jak může situace vypadat.
{\bf A.} Předně žádný z~vrcholů $p_x$ a~$p_y$ nemůže být blíž k~$v$ než $x$ a~$y$. Pokud by
-bez újmy na obecnosti $p_x$~ležel mezi $v$ a~$x$, našli bychom v~graf uminor AAA obsahující $K_{3,3}$:
+bez újmy na obecnosti $p_x$~ležel mezi $v$ a~$x$, našli bychom v~grafu minor~$N_A$ obsahující $K_{3,3}$:
cestu mezi $y$ a~$p_y$ jsme zkontrahovali do~$y$, celý plot jsme zkontrahovali do hrany~$p_xy$,
ostatní vrcholy jsou utvořeny stejně jako v~předchozích minorech.
{\bf B.} Dále ukážeme, že plot může být připojen k~$v$ pouze přes $p_x$ a~$p_y$. V~opačném
-případě by se v~grafu vyskytoval minor BBB obsahujicí $K_{3,3}$: do~$x$ jsme zkontrahovali
+případě by se v~grafu vyskytoval minor~$N_B$ obsahující $K_{3,3}$: do~$x$ jsme zkontrahovali
cestu mezi $x$ a~$p_x$, podobně do~$y$ cestu mezi $y$ a~$p_y$.
-{\bf C.} Nakonec se přesvědčíme, že na \uv{dolní} cestě z~$x$ do~$y$ přes~$w$ nemůže ležet
-žádný externí vrchol. To by totiž způsobovalo minor CCC isomorfní s~$K_5$:
+{\bf C.} Nakonec se přesvědčíme, že na dolní cestě z~$x$ do~$y$ přes~$w$ nemůže ležet
+žádný externí vrchol. To by totiž způsobovalo minor~$N_C$ isomorfní s~$K_5$:
do~$w$ jsme zkontrahovali cestu mezi externím vrcholem a~$w$ a také všechny
-případné podřízené bloky, v~nichž vede externí hrana.
+případné podřízené bloky až k~externí hraně.
{\I Vnitřek bloku~$B$.}
-Podívejme se, jak graf vypadal před nakreslením vrcholu~$v$. I~tehdy musel plot
-společně s~dolní cestou ležet v~jednom bloku. Ten musel být z~jedné strany připojený
-k~$v$ přes~$p_x$, z~druhé přes~$p_y$ (a~díky vlastnosti~{\bf B} nikudy jinudy).
+Nyní uvažujme, jak graf vypadal před nakreslením vrcholu~$v$. I~tehdy musel plot
+společně s~dolní cestou ležet v~jednom bloku. Tento blok musel být z~jedné strany připojený
+nenakreslenými hranami k~$v$ přes~$p_x$, z~druhé přes~$p_y$ (a~díky vlastnosti~{\bf B} nikudy jinudy).
Říkejme tomuto bloku~$B'$ a~označme $r_1\in\{p_x,p_y\}$ jeho kořen a $r_2$ druhý
z~vrcholů $p_x,p_y$.
Jelikož jsme nakonec nakreslili zpětnou hranu uzavírající blok~$B$, museli jsme
-někdy dojít do~$p_x$. V~tomto okamžiku:
+někdy dojít do~$r_1$. V~tomto okamžiku:
\itemize\ibull
\:Díky {\bf A:} $r_2$~je předkem $x$ nebo~$y$ (případně je takovému vrcholu roven),
takže je nyní externí.
-\:Díky {\bf B:} půjdeme-li z~$r_1$ po, nejbližší \uv{zajímavý} vrchol bude~$r_2$.
+\:Díky {\bf B:} půjdeme-li z~$r_1$ po plotu, nejbližší \uv{zajímavý} vrchol bude~$r_2$.
\:Díky {\bf C:} žádný vrchol na dolní cestě není externí.
\endlist
Při vstupu~$r_1$ tedy plot vede k~externími vrcholu, zatímco dolní cesta
-k~internímu. Podle pravidla \#2 si tedy algoritmus vybere dolní cestu, kde ho
-nic nezastaví, takže dojde až do~$w$ a~nakreslí zpětnou hranu $vw$.
+k~internímu. Podle pravidla \#2 si algoritmus musí vybrat dolní cestu, kde ho
+nic nezastaví, takže dojde až do~$w$ a~nakreslí zpětnou hranu $wv$. To je opět spor.
\qed
-\bigskip
-\centerline{\putepdf{}{minor1.epdf}\qquad\putepdf{}{minor2.epdf}}
-\bigskip
-
-\bigskip
-\centerline{\putepdf{}{minor3.epdf}\qquad\putepdf{}{minor4.epdf}}
-\bigskip
-\centerline{\putepdf{}{minor5.epdf}\qquad\putepdf{}{minor6.epdf}}
-\bigskip
-
\s{Poznámka:} Podle tohoto důkazu bychom také mohli v~lineárním čase v~každém nerovinném
grafu nalézt Kuratowského podgraf, dokonce také v~$O(n)$, jelikož když je $m>3n-6$,
můžeme se omezit na~libovolných $3n-5$ hran, které určitě tvoří nerovinný podgraf.
P=11-planar
include ../Makerules
+
+$(P).pdf: $(patsubst %, pl-minor%.pdf, 1 2 3 4 5 6)
+++ /dev/null
-
- (document 0.6 ( (filename "/home/mares/tex/ga/11-planar/minor1.vrr") (name "Untitled 1")) (
- (page ( (tex_footer "\\bye
-") (tex_header "\\nopagenumbers\\vglue0pt
-") (name "minor1")) (
- (ellipse-by-center-point-rotation-eccentricity (coords -20.0 0.0) (coords -20.0 10.0) 0.0 0.0 ( (id 137789296) (conic (point-entire)) (stroke-style (stroke-style solid)) (stroke-cap (cap-style butt)) (stroke-join (join-style miter)) (stroke-width 0.100000001490116) (stroke-color (rgb 0 0 0)) (fill-color (rgb 191 191 191)) (invisible #f) (name "noname")))
- (segment (end 137789296) (coords -20.0 15.0) ( (id 138591224) (stroke-style (stroke-style solid)) (stroke-cap (cap-style butt)) (stroke-join (join-style miter)) (stroke-width 0.100000001490116) (stroke-color (rgb 0 0 0)) (fill-color (rgba 255 255 255 0)) (invisible #f) (name "noname")))
- (segment (end 138591224) (coords -20.0 20.0) ( (id 136903136) (stroke-style (stroke-style solid)) (stroke-cap (cap-style butt)) (stroke-join (join-style miter)) (stroke-width 0.100000001490116) (stroke-color (rgb 0 0 0)) (fill-color (rgba 255 255 255 0)) (invisible #f) (name "noname")))
- (top-level-group ( (name "top-level-group")) (
- (decorator-point (coords -10.0 0.0) 4 1.0 0.785399377346039 ( (stroke-style (stroke-style solid)) (stroke-cap (cap-style butt)) (stroke-join (join-style miter)) (stroke-width 0.100000001490116) (stroke-color (rgb 0 0 0)) (fill-color (rgb 255 255 255)) (invisible #f) (name "noname")))
- (decorator-point (coords -30.0 0.0) 4 1.0 0.785399377346039 ( (stroke-style (stroke-style solid)) (stroke-cap (cap-style butt)) (stroke-join (join-style miter)) (stroke-width 0.100000001490116) (stroke-color (rgb 0 0 0)) (fill-color (rgb 255 255 255)) (invisible #f) (name "noname")))
- (tex-text (coords -20.3558387756348 6.88339900970459) "$r$" ( (relative-position-x 0.5) (relative-position-y 0.5) (absolute-shift-x 0.0) (absolute-shift-y 0.0) (alignment-x (alignment-x ref-relative)) (alignment-y (alignment-y bbox-relative)) (transform (transform 1.0 0.0 0.0 1.0)) (fill-color (rgb 0 0 0)) (invisible #f) (name "noname")))
- (decorator-point (start 138591224) 0 1.0 6.0491561271192e-7 ( (stroke-style (stroke-style solid)) (stroke-cap (cap-style butt)) (stroke-join (join-style miter)) (stroke-width 0.100000001490116) (stroke-color (rgb 0 0 0)) (fill-color (rgb 0 0 0)) (invisible #f) (name "noname")))
- (tex-text (coords -33.0508766174316 -8.07580757141113) "$M$" ( (relative-position-x 0.5) (relative-position-y 0.5) (absolute-shift-x 0.0) (absolute-shift-y 0.0) (alignment-x (alignment-x ref-relative)) (alignment-y (alignment-y bbox-relative)) (transform (transform 1.0 0.0 0.0 1.0)) (fill-color (rgb 0 0 0)) (invisible #f) (name "noname")))
- (tex-text (coords -22.9824047088623 -12.3034019470215) "$w$" ( (relative-position-x 0.5) (relative-position-y 0.5) (absolute-shift-x 0.0) (absolute-shift-y 0.0) (alignment-x (alignment-x ref-relative)) (alignment-y (alignment-y bbox-relative)) (transform (transform 1.0 0.0 0.0 1.0)) (fill-color (rgb 0 0 0)) (invisible #f) (name "noname")))
- (tex-text (coords -8.20800113677979 -2.33060050010681) "$y$" ( (relative-position-x 0.5) (relative-position-y 0.5) (absolute-shift-x 0.0) (absolute-shift-y 0.0) (alignment-x (alignment-x ref-relative)) (alignment-y (alignment-y bbox-relative)) (transform (transform 1.0 0.0 0.0 1.0)) (fill-color (rgb 0 0 0)) (invisible #f) (name "noname")))
- (tex-text (coords -32.8320045471191 -1.57180035114288) "$x$" ( (relative-position-x 0.5) (relative-position-y 0.5) (absolute-shift-x 0.0) (absolute-shift-y 0.0) (alignment-x (alignment-x ref-relative)) (alignment-y (alignment-y bbox-relative)) (transform (transform 1.0 0.0 0.0 1.0)) (fill-color (rgb 0 0 0)) (invisible #f) (name "noname")))
- (tex-text (coords -22.8729648590088 14.7966032028198) "$v$" ( (relative-position-x 0.5) (relative-position-y 0.5) (absolute-shift-x 0.0) (absolute-shift-y 0.0) (alignment-x (alignment-x ref-relative)) (alignment-y (alignment-y bbox-relative)) (transform (transform 1.0 0.0 0.0 1.0)) (fill-color (rgb 0 0 0)) (invisible #f) (name "noname")))
- (tex-text (coords -22.3257656097412 22.4930038452148) "$u$" ( (relative-position-x 0.5) (relative-position-y 0.5) (absolute-shift-x 0.0) (absolute-shift-y 0.0) (alignment-x (alignment-x ref-relative)) (alignment-y (alignment-y bbox-relative)) (transform (transform 1.0 0.0 0.0 1.0)) (fill-color (rgb 0 0 0)) (invisible #f) (name "noname")))
- (cubic-bezier (coords -20.0 -10.0) 1.0 (coords -1.2038402557373 -18.5906028747559) 1.0 (coords 5.90976142883301 14.037802696228) 1.0 (end 138591224) 1.0 ( (stroke-style (stroke-style solid)) (stroke-cap (cap-style butt)) (stroke-join (join-style miter)) (stroke-width 0.100000001490116) (stroke-color (rgb 0 0 0)) (fill-color (rgba 255 255 255 0)) (invisible #f) (name "noname")))
- (cubic-bezier (coords -10.0 0.0) 1.0 (coords 0.0 5.0) 1.0 (coords 0.0 20.0) 1.0 (end 136903136) 1.0 ( (stroke-style (stroke-style solid)) (stroke-cap (cap-style butt)) (stroke-join (join-style miter)) (stroke-width 0.100000001490116) (stroke-color (rgb 0 0 0)) (fill-color (rgba 255 255 255 0)) (invisible #f) (name "noname")))
- (cubic-bezier (coords -30.0 0.0) 1.0 (coords -40.0 5.0) 1.0 (coords -40.0 20.0) 1.0 (end 136903136) 1.0 ( (stroke-style (stroke-style solid)) (stroke-cap (cap-style butt)) (stroke-join (join-style miter)) (stroke-width 0.100000001490116) (stroke-color (rgb 0 0 0)) (fill-color (rgba 255 255 255 0)) (invisible #f) (name "noname"))) 136903136 138591224
- (decorator-point (coords -20.0 20.0) 0 1.0 6.0491561271192e-7 ( (stroke-style (stroke-style solid)) (stroke-cap (cap-style butt)) (stroke-join (join-style miter)) (stroke-width 0.100000001490116) (stroke-color (rgb 0 0 0)) (fill-color (rgb 0 0 0)) (invisible #f) (name "noname")))
- (decorator-point (coords -20.0 15.0) 0 1.0 6.0491561271192e-7 ( (stroke-style (stroke-style solid)) (stroke-cap (cap-style butt)) (stroke-join (join-style miter)) (stroke-width 0.100000001490116) (stroke-color (rgb 0 0 0)) (fill-color (rgb 0 0 0)) (invisible #f) (name "noname")))
- (decorator-point (coords -20.0 -10.0) 0 1.0 6.0491561271192e-7 ( (stroke-style (stroke-style solid)) (stroke-cap (cap-style butt)) (stroke-join (join-style miter)) (stroke-width 0.100000001490116) (stroke-color (rgb 0 0 0)) (fill-color (rgb 0 0 0)) (invisible #f) (name "noname"))) 137789296))))))
\ No newline at end of file
+++ /dev/null
-
- (document 0.6 ( (filename "/home/mares/tex/ga/11-planar/minor2.vrr") (name "Untitled 1")) (
- (page ( (tex_footer "\\bye
-") (tex_header "\\nopagenumbers\\vglue0pt
-") (name "minor2")) (
- (cubic-bezier (coords -30.0 0.0) 1.0 (coords -40.0 5.0) 1.0 (coords -40.0 15.0) 1.0 (coords -20.0 15.0) 1.0 ( (id 139157832) (stroke-style (stroke-style solid)) (stroke-cap (cap-style butt)) (stroke-join (join-style miter)) (stroke-width 0.100000001490116) (stroke-color (rgb 0 0 0)) (fill-color (rgba 255 255 255 0)) (invisible #f) (name "noname")))
- (ellipse-by-center-point-rotation-eccentricity (coords -20.0 0.0) (coords -20.0 10.0) 0.0 0.0 ( (id 139031080) (conic (point-entire)) (stroke-style (stroke-style solid)) (stroke-cap (cap-style butt)) (stroke-join (join-style miter)) (stroke-width 0.100000001490116) (stroke-color (rgb 0 0 0)) (fill-color (rgb 191 191 191)) (invisible #f) (name "noname")))
- (top-level-group ( (name "top-level-group")) (
- (decorator-point (coords -10.0 0.0) 4 1.0 0.785399377346039 ( (stroke-style (stroke-style solid)) (stroke-cap (cap-style butt)) (stroke-join (join-style miter)) (stroke-width 0.100000001490116) (stroke-color (rgb 0 0 0)) (fill-color (rgb 255 255 255)) (invisible #f) (name "noname")))
- (decorator-point (coords -30.0 0.0) 4 1.0 0.785399377346039 ( (stroke-style (stroke-style solid)) (stroke-cap (cap-style butt)) (stroke-join (join-style miter)) (stroke-width 0.100000001490116) (stroke-color (rgb 0 0 0)) (fill-color (rgb 255 255 255)) (invisible #f) (name "noname")))
- (segment (start 139031080) (controlpoint-4 139157832) ( (stroke-style (stroke-style solid)) (stroke-cap (cap-style butt)) (stroke-join (join-style miter)) (stroke-width 0.100000001490116) (stroke-color (rgb 0 0 0)) (fill-color (rgb 0 0 0)) (invisible #f) (name "noname")))
- (tex-text (coords -20.3558387756348 6.88339900970459) "$v$" ( (relative-position-x 0.5) (relative-position-y 0.5) (absolute-shift-x 0.0) (absolute-shift-y 0.0) (alignment-x (alignment-x ref-relative)) (alignment-y (alignment-y bbox-relative)) (transform (transform 1.0 0.0 0.0 1.0)) (fill-color (rgb 0 0 0)) (invisible #f) (name "noname")))
- (decorator-point (coords -20.0000019073486 10.0) 0 1.0 6.0491561271192e-7 ( (stroke-style (stroke-style solid)) (stroke-cap (cap-style butt)) (stroke-join (join-style miter)) (stroke-width 0.100000001490116) (stroke-color (rgb 0 0 0)) (fill-color (rgb 0 0 0)) (invisible #f) (name "noname")))
- (tex-text (coords -31.2179622650147 -8.45826148986816) "$N$" ( (relative-position-x 0.5) (relative-position-y 0.5) (absolute-shift-x 0.0) (absolute-shift-y 0.0) (alignment-x (alignment-x ref-relative)) (alignment-y (alignment-y bbox-relative)) (transform (transform 1.0 0.0 0.0 1.0)) (fill-color (rgb 0 0 0)) (invisible #f) (name "noname")))
- (tex-text (coords -22.9824047088623 -12.3034019470215) "$w$" ( (relative-position-x 0.5) (relative-position-y 0.5) (absolute-shift-x 0.0) (absolute-shift-y 0.0) (alignment-x (alignment-x ref-relative)) (alignment-y (alignment-y bbox-relative)) (transform (transform 1.0 0.0 0.0 1.0)) (fill-color (rgb 0 0 0)) (invisible #f) (name "noname")))
- (tex-text (coords -8.20800113677979 -2.33060050010681) "$y$" ( (relative-position-x 0.5) (relative-position-y 0.5) (absolute-shift-x 0.0) (absolute-shift-y 0.0) (alignment-x (alignment-x ref-relative)) (alignment-y (alignment-y bbox-relative)) (transform (transform 1.0 0.0 0.0 1.0)) (fill-color (rgb 0 0 0)) (invisible #f) (name "noname")))
- (tex-text (coords -32.8320045471191 -1.57180035114288) "$x$" ( (relative-position-x 0.5) (relative-position-y 0.5) (absolute-shift-x 0.0) (absolute-shift-y 0.0) (alignment-x (alignment-x ref-relative)) (alignment-y (alignment-y bbox-relative)) (transform (transform 1.0 0.0 0.0 1.0)) (fill-color (rgb 0 0 0)) (invisible #f) (name "noname")))
- (tex-text (coords -21.8017578125 17.310417175293) "$u$" ( (relative-position-x 0.5) (relative-position-y 0.5) (absolute-shift-x 0.0) (absolute-shift-y 0.0) (alignment-x (alignment-x ref-relative)) (alignment-y (alignment-y bbox-relative)) (transform (transform 1.0 0.0 0.0 1.0)) (fill-color (rgb 0 0 0)) (invisible #f) (name "noname")))
- (cubic-bezier (coords -20.0 -10.0) 1.0 (coords -1.2038402557373 -18.5906028747559) 1.0 (coords 5.90976142883301 14.037802696228) 1.0 (end 139031080) 1.0 ( (stroke-style (stroke-style solid)) (stroke-cap (cap-style butt)) (stroke-join (join-style miter)) (stroke-width 0.100000001490116) (stroke-color (rgb 0 0 0)) (fill-color (rgba 255 255 255 0)) (invisible #f) (name "noname")))
- (cubic-bezier (coords -10.0 0.0) 1.0 (coords 0.0 5.0) 1.0 (coords 0.0 15.0) 1.0 (controlpoint-4 139157832) 1.0 ( (stroke-style (stroke-style solid)) (stroke-cap (cap-style butt)) (stroke-join (join-style miter)) (stroke-width 0.100000001490116) (stroke-color (rgb 0 0 0)) (fill-color (rgba 255 255 255 0)) (invisible #f) (name "noname"))) 139157832
- (decorator-point (coords -20.0 15.0) 0 1.0 6.0491561271192e-7 ( (stroke-style (stroke-style solid)) (stroke-cap (cap-style butt)) (stroke-join (join-style miter)) (stroke-width 0.100000001490116) (stroke-color (rgb 0 0 0)) (fill-color (rgb 0 0 0)) (invisible #f) (name "noname")))
- (decorator-point (coords -20.0 -10.0) 0 1.0 6.0491561271192e-7 ( (stroke-style (stroke-style solid)) (stroke-cap (cap-style butt)) (stroke-join (join-style miter)) (stroke-width 0.100000001490116) (stroke-color (rgb 0 0 0)) (fill-color (rgb 0 0 0)) (invisible #f) (name "noname"))) 139031080))))))
\ No newline at end of file
+++ /dev/null
-
- (document 0.6 ( (filename "/home/mares/tex/ga/11-planar/minor3.vrr") (name "Untitled 1")) (
- (page ( (tex_footer "\\bye
-") (tex_header "\\nopagenumbers\\vglue0pt
-") (name "minor3")) (
- (cubic-bezier (coords -20.0 -15.0) 1.0 (coords -1.2038402557373 -18.5906028747559) 1.0 (coords 5.90976142883301 14.037802696228) 1.0 (coords -19.957347869873 9.84761619567871) 1.0 ( (id 137787336) (stroke-style (stroke-style solid)) (stroke-cap (cap-style butt)) (stroke-join (join-style miter)) (stroke-width 0.100000001490116) (stroke-color (rgb 0 0 0)) (fill-color (rgba 255 255 255 0)) (invisible #f) (name "noname")))
- (cubic-bezier (coords -30.0 0.0) 1.0 (coords -40.0 5.0) 1.0 (coords -40.0 15.0) 1.0 (coords -20.0 15.0) 1.0 ( (id 137835008) (stroke-style (stroke-style solid)) (stroke-cap (cap-style butt)) (stroke-join (join-style miter)) (stroke-width 0.100000001490116) (stroke-color (rgb 0 0 0)) (fill-color (rgba 255 255 255 0)) (invisible #f) (name "noname")))
- (segment (controlpoint-1 137787336) (coords -20.0 -15.0) ( (id 137835488) (stroke-style (stroke-style solid)) (stroke-cap (cap-style butt)) (stroke-join (join-style miter)) (stroke-width 0.100000001490116) (stroke-color (rgb 0 0 0)) (fill-color (rgba 255 255 255 0)) (invisible #f) (name "noname")))
- (ellipse-by-center-point-rotation-eccentricity (coords -25.0 -5.0) (controlpoint-1 137835008) 2.35620069503784 0.970000028610229 ( (id 137836968) (conic (point-entire)) (stroke-style (stroke-style solid)) (stroke-cap (cap-style butt)) (stroke-join (join-style miter)) (stroke-width 0.100000001490116) (stroke-color (rgb 0 0 0)) (fill-color (rgb 191 191 191)) (invisible #f) (name "noname")))
- (cubic-bezier (coords -10.0 0.0) 1.0 (coords 0.0 5.0) 1.0 (coords 0.0 15.0) 1.0 (controlpoint-4 137835008) 1.0 ( (id 137837576) (stroke-style (stroke-style solid)) (stroke-cap (cap-style butt)) (stroke-join (join-style miter)) (stroke-width 0.100000001490116) (stroke-color (rgb 0 0 0)) (fill-color (rgba 255 255 255 0)) (invisible #f) (name "noname")))
- (segment (coords -20.0 10.0) (start 137836968) ( (id 137838632) (stroke-style (stroke-style solid)) (stroke-cap (cap-style butt)) (stroke-join (join-style miter)) (stroke-width 0.100000001490116) (stroke-color (rgb 0 0 0)) (fill-color (rgba 0 0 0 0)) (invisible #f) (name "noname")))
- (ellipse-by-center-point-rotation-eccentricity (coords -15.0 -5.0) (controlpoint-1 137837576) 0.785400629043579 0.970000028610229 ( (id 137837848) (conic (point-entire)) (stroke-style (stroke-style solid)) (stroke-cap (cap-style butt)) (stroke-join (join-style miter)) (stroke-width 0.100000001490116) (stroke-color (rgb 0 0 0)) (fill-color (rgb 191 191 191)) (invisible #f) (name "noname")))
- (segment (start 137838632) (end 137837848) ( (id 137842912) (stroke-style (stroke-style solid)) (stroke-cap (cap-style butt)) (stroke-join (join-style miter)) (stroke-width 0.100000001490116) (stroke-color (rgb 0 0 0)) (fill-color (rgba 0 0 0 0)) (invisible #f) (name "noname")))
- (top-level-group ( (name "top-level-group")) (
- (decorator-point (coords -20.0 -10.0) 0 1.0 6.0491561271192e-7 ( (stroke-style (stroke-style solid)) (stroke-cap (cap-style butt)) (stroke-join (join-style miter)) (stroke-width 0.100000001490116) (stroke-color (rgb 0 0 0)) (fill-color (rgb 0 0 0)) (invisible #f) (name "noname")))
- (decorator-point (coords -20.0 -15.0) 4 1.0 0.785399377346039 ( (stroke-style (stroke-style solid)) (stroke-cap (cap-style butt)) (stroke-join (join-style miter)) (stroke-width 0.100000001490116) (stroke-color (rgb 0 0 0)) (fill-color (rgb 255 255 255)) (invisible #f) (name "noname")))
- (decorator-point (coords -10.0 0.0) 4 1.0 0.785399377346039 ( (stroke-style (stroke-style solid)) (stroke-cap (cap-style butt)) (stroke-join (join-style miter)) (stroke-width 0.100000001490116) (stroke-color (rgb 0 0 0)) (fill-color (rgb 255 255 255)) (invisible #f) (name "noname")))
- (decorator-point (coords -30.0 0.0) 4 1.0 0.785399377346039 ( (stroke-style (stroke-style solid)) (stroke-cap (cap-style butt)) (stroke-join (join-style miter)) (stroke-width 0.100000001490116) (stroke-color (rgb 0 0 0)) (fill-color (rgb 255 255 255)) (invisible #f) (name "noname")))
- (segment (start 137842912) (controlpoint-4 137837576) ( (stroke-style (stroke-style solid)) (stroke-cap (cap-style butt)) (stroke-join (join-style miter)) (stroke-width 0.100000001490116) (stroke-color (rgb 0 0 0)) (fill-color (rgba 0 0 0 0)) (invisible #f) (name "noname")))
- (segment (coords -20.0 -10.0) (end 137835488) ( (stroke-style (stroke-style solid)) (stroke-cap (cap-style butt)) (stroke-join (join-style miter)) (stroke-width 0.100000001490116) (stroke-color (rgb 0 0 0)) (fill-color (rgba 255 255 255 0)) (invisible #f) (name "noname")))
- (cubic-bezier (end 137835488) 1.0 (coords -46.3190803527832 -15.3501586914063) 1.0 (coords -53.9030075073242 18.7536754608154) 1.0 (controlpoint-4 137837576) 1.0 ( (stroke-style (stroke-style solid)) (stroke-cap (cap-style butt)) (stroke-join (join-style miter)) (stroke-width 0.100000001490116) (stroke-color (rgb 0 0 0)) (fill-color (rgba 255 255 255 0)) (invisible #f) (name "noname"))) 137835488 137842912 137838632 137837848 137836968
- (tex-text (coords -20.3558387756348 6.88339900970459) "$v$" ( (relative-position-x 0.5) (relative-position-y 0.5) (absolute-shift-x 0.0) (absolute-shift-y 0.0) (alignment-x (alignment-x ref-relative)) (alignment-y (alignment-y bbox-relative)) (transform (transform 1.0 0.0 0.0 1.0)) (fill-color (rgb 0 0 0)) (invisible #f) (name "noname")))
- (decorator-point (coords -20.0000019073486 10.0) 0 1.0 6.0491561271192e-7 ( (stroke-style (stroke-style solid)) (stroke-cap (cap-style butt)) (stroke-join (join-style miter)) (stroke-width 0.100000001490116) (stroke-color (rgb 0 0 0)) (fill-color (rgb 0 0 0)) (invisible #f) (name "noname")))
- (tex-text (coords -31.2179622650147 -8.45826148986816) "$N_1$" ( (relative-position-x 0.5) (relative-position-y 0.5) (absolute-shift-x 0.0) (absolute-shift-y 0.0) (alignment-x (alignment-x ref-relative)) (alignment-y (alignment-y bbox-relative)) (transform (transform 1.0 0.0 0.0 1.0)) (fill-color (rgb 0 0 0)) (invisible #f) (name "noname")))
- (tex-text (coords -22.9824047088623 -12.3034019470215) "$w$" ( (relative-position-x 0.5) (relative-position-y 0.5) (absolute-shift-x 0.0) (absolute-shift-y 0.0) (alignment-x (alignment-x ref-relative)) (alignment-y (alignment-y bbox-relative)) (transform (transform 1.0 0.0 0.0 1.0)) (fill-color (rgb 0 0 0)) (invisible #f) (name "noname")))
- (tex-text (coords -8.20800113677979 -2.33060050010681) "$y$" ( (relative-position-x 0.5) (relative-position-y 0.5) (absolute-shift-x 0.0) (absolute-shift-y 0.0) (alignment-x (alignment-x ref-relative)) (alignment-y (alignment-y bbox-relative)) (transform (transform 1.0 0.0 0.0 1.0)) (fill-color (rgb 0 0 0)) (invisible #f) (name "noname")))
- (tex-text (coords -32.8320045471191 -1.57180035114288) "$x$" ( (relative-position-x 0.5) (relative-position-y 0.5) (absolute-shift-x 0.0) (absolute-shift-y 0.0) (alignment-x (alignment-x ref-relative)) (alignment-y (alignment-y bbox-relative)) (transform (transform 1.0 0.0 0.0 1.0)) (fill-color (rgb 0 0 0)) (invisible #f) (name "noname")))
- (tex-text (coords -21.8017578125 17.310417175293) "$u$" ( (relative-position-x 0.5) (relative-position-y 0.5) (absolute-shift-x 0.0) (absolute-shift-y 0.0) (alignment-x (alignment-x ref-relative)) (alignment-y (alignment-y bbox-relative)) (transform (transform 1.0 0.0 0.0 1.0)) (fill-color (rgb 0 0 0)) (invisible #f) (name "noname"))) 137787336 137837576 137835008
- (decorator-point (coords -20.0 15.0) 0 1.0 6.0491561271192e-7 ( (stroke-style (stroke-style solid)) (stroke-cap (cap-style butt)) (stroke-join (join-style miter)) (stroke-width 0.100000001490116) (stroke-color (rgb 0 0 0)) (fill-color (rgb 0 0 0)) (invisible #f) (name "noname")))))))))
\ No newline at end of file
+++ /dev/null
-
- (document 0.6 ( (filename "/home/mares/tex/ga/11-planar/minor4.vrr") (name "Untitled 1")) (
- (page ( (tex_footer "\\bye
-") (tex_header "\\nopagenumbers\\vglue0pt
-") (name "minor4")) (
- (cubic-bezier (coords -20.0 -10.0) 1.0 (coords -1.2038402557373 -18.5906028747559) 1.0 (coords 5.90976142883301 14.037802696228) 1.0 (coords -20.0000019073486 10.0) 1.0 ( (id 138484704) (stroke-style (stroke-style solid)) (stroke-cap (cap-style butt)) (stroke-join (join-style miter)) (stroke-width 0.100000001490116) (stroke-color (rgb 0 0 0)) (fill-color (rgba 255 255 255 0)) (invisible #f) (name "noname")))
- (cubic-bezier (coords -30.0 0.0) 1.0 (coords -40.0 5.0) 1.0 (coords -40.0 15.0) 1.0 (coords -20.0 15.0) 1.0 ( (id 138496008) (stroke-style (stroke-style solid)) (stroke-cap (cap-style butt)) (stroke-join (join-style miter)) (stroke-width 0.100000001490116) (stroke-color (rgb 0 0 0)) (fill-color (rgba 255 255 255 0)) (invisible #f) (name "noname")))
- (parametric-point 138484704 1.0 ( (id 138485080) (name "noname")))
- (cubic-bezier (coords -10.0 0.0) 1.0 (coords 0.0 5.0) 1.0 (coords 0.0 15.0) 1.0 (controlpoint-4 138496008) 1.0 ( (id 139196672) (stroke-style (stroke-style solid)) (stroke-cap (cap-style butt)) (stroke-join (join-style miter)) (stroke-width 0.100000001490116) (stroke-color (rgb 0 0 0)) (fill-color (rgba 255 255 255 0)) (invisible #f) (name "noname")))
- (ellipse-by-3-points-rotation-eccentricity (controlpoint-1 138484704) (controlpoint-1 138496008) (controlpoint-1 139196672) 0.0 0.800000011920929 ( (id 139198464) (conic (point-entire)) (stroke-style (stroke-style solid)) (stroke-cap (cap-style butt)) (stroke-join (join-style miter)) (stroke-width 0.100000001490116) (stroke-color (rgb 0 0 0)) (fill-color (rgb 191 191 191)) (invisible #f) (name "noname")))
- (parametric-point 139198464 0.499403983354568 ( (id 139004072) (name "noname")))
- (parametric-point 139198464 0.571895003318787 ( (id 138498176) (name "noname")))
- (parametric-point 139198464 0.426486879587173 ( (id 139195136) (name "noname")))
- (segment (center 138485080) (center 139004072) ( (id 139358224) (stroke-style (stroke-style solid)) (stroke-cap (cap-style butt)) (stroke-join (join-style miter)) (stroke-width 0.100000001490116) (stroke-color (rgb 0 0 0)) (fill-color (rgba 255 255 255 0)) (invisible #f) (name "noname")))
- (parametric-point 139358224 0.00882991496473551 ( (id 139202856) (name "noname")))
- (quadratic-bezier (center 139202856) 1.0 (coords -25.0 10.0) 1.0 (center 138498176) 1.0 ( (id 138498392) (stroke-style (stroke-style solid)) (stroke-cap (cap-style butt)) (stroke-join (join-style miter)) (stroke-width 0.100000001490116) (stroke-color (rgb 0 0 0)) (fill-color (rgba 0 0 0 0)) (invisible #f) (name "noname")))
- (parametric-point 138498392 0.00549423275515437 ( (id 139044504) (name "noname")))
- (top-level-group ( (name "top-level-group")) (
- (decorator-point (coords -20.0 -10.0) 0 1.0 6.0491561271192e-7 ( (stroke-style (stroke-style solid)) (stroke-cap (cap-style butt)) (stroke-join (join-style miter)) (stroke-width 0.100000001490116) (stroke-color (rgb 0 0 0)) (fill-color (rgb 0 0 0)) (invisible #f) (name "noname")))
- (decorator-point (coords -10.0 0.0) 4 1.0 0.785399377346039 ( (stroke-style (stroke-style solid)) (stroke-cap (cap-style butt)) (stroke-join (join-style miter)) (stroke-width 0.100000001490116) (stroke-color (rgb 0 0 0)) (fill-color (rgb 255 255 255)) (invisible #f) (name "noname")))
- (decorator-point (coords -30.0 0.0) 4 1.0 0.785399377346039 ( (stroke-style (stroke-style solid)) (stroke-cap (cap-style butt)) (stroke-join (join-style miter)) (stroke-width 0.100000001490116) (stroke-color (rgb 0 0 0)) (fill-color (rgb 255 255 255)) (invisible #f) (name "noname"))) 139195136
- (quadratic-bezier (center 139044504) 1.0 (coords -15.0 10.0) 1.0 (center 139195136) 1.0 ( (stroke-style (stroke-style solid)) (stroke-cap (cap-style butt)) (stroke-join (join-style miter)) (stroke-width 0.100000001490116) (stroke-color (rgb 0 0 0)) (fill-color (rgba 0 0 0 0)) (invisible #f) (name "noname"))) 139044504 138498176 138498392 139202856
- (tex-text (coords -18.8895301818848 1.29010832309723) "$z$" ( (relative-position-x 0.5) (relative-position-y 0.5) (absolute-shift-x 0.0) (absolute-shift-y 0.0) (alignment-x (alignment-x ref-relative)) (alignment-y (alignment-y bbox-relative)) (transform (transform 1.0 0.0 0.0 1.0)) (fill-color (rgb 0 0 0)) (invisible #f) (name "noname")))
- (decorator-point (end 139358224) 0 1.0 0.785399377346039 ( (stroke-style (stroke-style solid)) (stroke-cap (cap-style butt)) (stroke-join (join-style miter)) (stroke-width 0.100000001490116) (stroke-color (rgb 0 0 0)) (fill-color (rgb 0 0 0)) (invisible #f) (name "noname"))) 139004072 139358224
- (parametric-point 139198464 0.387289702892303 ( (name "noname")))
- (parametric-point 139198464 0.61937540769577 ( (name "noname"))) 138485080 139198464
- (segment (coords -20.0 10.0) (controlpoint-4 138496008) ( (stroke-style (stroke-style solid)) (stroke-cap (cap-style butt)) (stroke-join (join-style miter)) (stroke-width 0.100000001490116) (stroke-color (rgb 0 0 0)) (fill-color (rgb 0 0 0)) (invisible #f) (name "noname")))
- (tex-text (coords -22.3842029571533 11.57834815979) "$v$" ( (relative-position-x 0.5) (relative-position-y 0.5) (absolute-shift-x 0.0) (absolute-shift-y 0.0) (alignment-x (alignment-x ref-relative)) (alignment-y (alignment-y bbox-relative)) (transform (transform 1.0 0.0 0.0 1.0)) (fill-color (rgb 0 0 0)) (invisible #f) (name "noname")))
- (decorator-point (coords -20.0000019073486 10.0) 0 1.0 6.0491561271192e-7 ( (stroke-style (stroke-style solid)) (stroke-cap (cap-style butt)) (stroke-join (join-style miter)) (stroke-width 0.100000001490116) (stroke-color (rgb 0 0 0)) (fill-color (rgb 0 0 0)) (invisible #f) (name "noname")))
- (tex-text (coords -33.4506683349609 -8.51737308502197) "$N_2$" ( (relative-position-x 0.5) (relative-position-y 0.5) (absolute-shift-x 0.0) (absolute-shift-y 0.0) (alignment-x (alignment-x ref-relative)) (alignment-y (alignment-y bbox-relative)) (transform (transform 1.0 0.0 0.0 1.0)) (fill-color (rgb 0 0 0)) (invisible #f) (name "noname")))
- (tex-text (coords -22.9824047088623 -12.3034019470215) "$w$" ( (relative-position-x 0.5) (relative-position-y 0.5) (absolute-shift-x 0.0) (absolute-shift-y 0.0) (alignment-x (alignment-x ref-relative)) (alignment-y (alignment-y bbox-relative)) (transform (transform 1.0 0.0 0.0 1.0)) (fill-color (rgb 0 0 0)) (invisible #f) (name "noname")))
- (tex-text (coords -6.94939517974854 -2.17135572433472) "$y$" ( (relative-position-x 0.5) (relative-position-y 0.5) (absolute-shift-x 0.0) (absolute-shift-y 0.0) (alignment-x (alignment-x ref-relative)) (alignment-y (alignment-y bbox-relative)) (transform (transform 1.0 0.0 0.0 1.0)) (fill-color (rgb 0 0 0)) (invisible #f) (name "noname")))
- (tex-text (coords -32.8320045471191 -1.57180035114288) "$x$" ( (relative-position-x 0.5) (relative-position-y 0.5) (absolute-shift-x 0.0) (absolute-shift-y 0.0) (alignment-x (alignment-x ref-relative)) (alignment-y (alignment-y bbox-relative)) (transform (transform 1.0 0.0 0.0 1.0)) (fill-color (rgb 0 0 0)) (invisible #f) (name "noname")))
- (tex-text (coords -21.8017578125 17.310417175293) "$u$" ( (relative-position-x 0.5) (relative-position-y 0.5) (absolute-shift-x 0.0) (absolute-shift-y 0.0) (alignment-x (alignment-x ref-relative)) (alignment-y (alignment-y bbox-relative)) (transform (transform 1.0 0.0 0.0 1.0)) (fill-color (rgb 0 0 0)) (invisible #f) (name "noname"))) 138484704 139196672 138496008
- (decorator-point (coords -20.0 15.0) 0 1.0 6.0491561271192e-7 ( (stroke-style (stroke-style solid)) (stroke-cap (cap-style butt)) (stroke-join (join-style miter)) (stroke-width 0.100000001490116) (stroke-color (rgb 0 0 0)) (fill-color (rgb 0 0 0)) (invisible #f) (name "noname")))))))))
\ No newline at end of file
+++ /dev/null
-
- (document 0.6 ( (filename "/home/mares/tex/ga/11-planar/minor5.vrr") (name "Untitled 1")) (
- (page ( (tex_footer "\\bye
-") (tex_header "\\nopagenumbers\\vglue0pt
-") (name "minor5")) (
- (cubic-bezier (coords -20.0 -10.0) 1.0 (coords -1.2038402557373 -18.5906028747559) 1.0 (coords 5.90976142883301 14.037802696228) 1.0 (coords -20.0000019073486 10.0) 1.0 ( (id 139648776) (stroke-style (stroke-style solid)) (stroke-cap (cap-style butt)) (stroke-join (join-style miter)) (stroke-width 0.100000001490116) (stroke-color (rgb 0 0 0)) (fill-color (rgba 255 255 255 0)) (invisible #f) (name "noname")))
- (cubic-bezier (coords -30.0 0.0) 1.0 (coords -40.0 5.0) 1.0 (coords -40.0 15.0) 1.0 (coords -20.0 15.0) 1.0 ( (id 139649408) (stroke-style (stroke-style solid)) (stroke-cap (cap-style butt)) (stroke-join (join-style miter)) (stroke-width 0.100000001490116) (stroke-color (rgb 0 0 0)) (fill-color (rgba 255 255 255 0)) (invisible #f) (name "noname")))
- (parametric-point 139648776 1.0 ( (id 139614728) (name "noname")))
- (cubic-bezier (coords -10.0 0.0) 1.0 (coords 0.0 5.0) 1.0 (coords 0.0 15.0) 1.0 (controlpoint-4 139649408) 1.0 ( (id 139650176) (stroke-style (stroke-style solid)) (stroke-cap (cap-style butt)) (stroke-join (join-style miter)) (stroke-width 0.100000001490116) (stroke-color (rgb 0 0 0)) (fill-color (rgba 255 255 255 0)) (invisible #f) (name "noname")))
- (ellipse-by-3-points-rotation-eccentricity (controlpoint-1 139648776) (controlpoint-1 139649408) (controlpoint-1 139650176) 0.0 0.800000011920929 ( (id 139650600) (conic (point-entire)) (stroke-style (stroke-style solid)) (stroke-cap (cap-style butt)) (stroke-join (join-style miter)) (stroke-width 0.100000001490116) (stroke-color (rgb 0 0 0)) (fill-color (rgb 191 191 191)) (invisible #f) (name "noname")))
- (parametric-point 139650600 0.61937540769577 ( (id 139652136) (name "noname")))
- (top-level-group ( (name "top-level-group")) (
- (decorator-point (coords -10.0 0.0) 4 1.0 0.785399377346039 ( (stroke-style (stroke-style solid)) (stroke-cap (cap-style butt)) (stroke-join (join-style miter)) (stroke-width 0.100000001490116) (stroke-color (rgb 0 0 0)) (fill-color (rgb 255 255 255)) (invisible #f) (name "noname")))
- (decorator-point (coords -30.0 0.0) 4 1.0 0.785399377346039 ( (stroke-style (stroke-style solid)) (stroke-cap (cap-style butt)) (stroke-join (join-style miter)) (stroke-width 0.100000001490116) (stroke-color (rgb 0 0 0)) (fill-color (rgb 255 255 255)) (invisible #f) (name "noname")))
- (decorator-point (coords -20.0 -10.0) 0 1.0 6.0491561271192e-7 ( (stroke-style (stroke-style solid)) (stroke-cap (cap-style butt)) (stroke-join (join-style miter)) (stroke-width 0.100000001490116) (stroke-color (rgb 0 0 0)) (fill-color (rgb 0 0 0)) (invisible #f) (name "noname")))
- (quadratic-bezier (center 139614728) 1.0 (coords -10.3469953536987 6.19384908676147) 1.0 (controlpoint-1 139650176) 1.0 ( (stroke-style (stroke-style solid)) (stroke-cap (cap-style butt)) (stroke-join (join-style miter)) (stroke-width 0.100000001490116) (stroke-color (rgb 0 0 0)) (fill-color (rgba 0 0 0 0)) (invisible #f) (name "noname")))
- (quadratic-bezier (controlpoint-4 139648776) 1.0 (coords -28.0145111083984 8.78994655609131) 1.0 (center 139652136) 1.0 ( (stroke-style (stroke-style solid)) (stroke-cap (cap-style butt)) (stroke-join (join-style miter)) (stroke-width 0.100000001490116) (stroke-color (rgb 0 0 0)) (fill-color (rgba 0 0 0 0)) (invisible #f) (name "noname")))
- (tex-text (coords -25.5876560211182 0.617045819759369) "$z$" ( (relative-position-x 0.5) (relative-position-y 0.5) (absolute-shift-x 0.0) (absolute-shift-y 0.0) (alignment-x (alignment-x ref-relative)) (alignment-y (alignment-y bbox-relative)) (transform (transform 1.0 0.0 0.0 1.0)) (fill-color (rgb 0 0 0)) (invisible #f) (name "noname")))
- (decorator-point (center 139652136) 0 1.0 0.785399377346039 ( (stroke-style (stroke-style solid)) (stroke-cap (cap-style butt)) (stroke-join (join-style miter)) (stroke-width 0.100000001490116) (stroke-color (rgb 0 0 0)) (fill-color (rgb 0 0 0)) (invisible #f) (name "noname")))
- (parametric-point 139650600 0.499403983354568 ( (name "noname")))
- (parametric-point 139650600 0.387289702892303 ( (name "noname"))) 139652136 139614728 139650600
- (segment (coords -20.0 10.0) (controlpoint-4 139649408) ( (stroke-style (stroke-style solid)) (stroke-cap (cap-style butt)) (stroke-join (join-style miter)) (stroke-width 0.100000001490116) (stroke-color (rgb 0 0 0)) (fill-color (rgb 0 0 0)) (invisible #f) (name "noname")))
- (tex-text (coords -22.6754245758057 11.3860445022583) "$v$" ( (relative-position-x 0.5) (relative-position-y 0.5) (absolute-shift-x 0.0) (absolute-shift-y 0.0) (alignment-x (alignment-x ref-relative)) (alignment-y (alignment-y bbox-relative)) (transform (transform 1.0 0.0 0.0 1.0)) (fill-color (rgb 0 0 0)) (invisible #f) (name "noname")))
- (decorator-point (coords -20.0000019073486 10.0) 0 1.0 6.0491561271192e-7 ( (stroke-style (stroke-style solid)) (stroke-cap (cap-style butt)) (stroke-join (join-style miter)) (stroke-width 0.100000001490116) (stroke-color (rgb 0 0 0)) (fill-color (rgb 0 0 0)) (invisible #f) (name "noname")))
- (tex-text (coords -33.4506683349609 -8.51737308502197) "$N_3$" ( (relative-position-x 0.5) (relative-position-y 0.5) (absolute-shift-x 0.0) (absolute-shift-y 0.0) (alignment-x (alignment-x ref-relative)) (alignment-y (alignment-y bbox-relative)) (transform (transform 1.0 0.0 0.0 1.0)) (fill-color (rgb 0 0 0)) (invisible #f) (name "noname")))
- (tex-text (coords -22.9824047088623 -12.3034019470215) "$w$" ( (relative-position-x 0.5) (relative-position-y 0.5) (absolute-shift-x 0.0) (absolute-shift-y 0.0) (alignment-x (alignment-x ref-relative)) (alignment-y (alignment-y bbox-relative)) (transform (transform 1.0 0.0 0.0 1.0)) (fill-color (rgb 0 0 0)) (invisible #f) (name "noname")))
- (tex-text (coords -6.94939517974854 -1.30598974227905) "$y$" ( (relative-position-x 0.5) (relative-position-y 0.5) (absolute-shift-x 0.0) (absolute-shift-y 0.0) (alignment-x (alignment-x ref-relative)) (alignment-y (alignment-y bbox-relative)) (transform (transform 1.0 0.0 0.0 1.0)) (fill-color (rgb 0 0 0)) (invisible #f) (name "noname")))
- (tex-text (coords -32.8320045471191 -1.57180035114288) "$x$" ( (relative-position-x 0.5) (relative-position-y 0.5) (absolute-shift-x 0.0) (absolute-shift-y 0.0) (alignment-x (alignment-x ref-relative)) (alignment-y (alignment-y bbox-relative)) (transform (transform 1.0 0.0 0.0 1.0)) (fill-color (rgb 0 0 0)) (invisible #f) (name "noname")))
- (tex-text (coords -21.8017578125 17.310417175293) "$u$" ( (relative-position-x 0.5) (relative-position-y 0.5) (absolute-shift-x 0.0) (absolute-shift-y 0.0) (alignment-x (alignment-x ref-relative)) (alignment-y (alignment-y bbox-relative)) (transform (transform 1.0 0.0 0.0 1.0)) (fill-color (rgb 0 0 0)) (invisible #f) (name "noname"))) 139648776 139650176 139649408
- (decorator-point (coords -20.0 15.0) 0 1.0 6.0491561271192e-7 ( (stroke-style (stroke-style solid)) (stroke-cap (cap-style butt)) (stroke-join (join-style miter)) (stroke-width 0.100000001490116) (stroke-color (rgb 0 0 0)) (fill-color (rgb 0 0 0)) (invisible #f) (name "noname")))))))))
\ No newline at end of file
+++ /dev/null
-
- (document 0.6 ( (filename "/home/mares/tex/ga/11-planar/minor6.vrr") (name "Untitled 1")) (
- (page ( (tex_footer "\\bye
-") (tex_header "\\nopagenumbers\\vglue0pt
-") (name "minor6")) (
- (cubic-bezier (coords -30.0 0.0) 1.0 (coords -40.0 5.0) 1.0 (coords -40.0 15.0) 1.0 (coords -20.0 15.0) 1.0 ( (id 139741312) (stroke-style (stroke-style solid)) (stroke-cap (cap-style butt)) (stroke-join (join-style miter)) (stroke-width 0.100000001490116) (stroke-color (rgb 0 0 0)) (fill-color (rgba 255 255 255 0)) (invisible #f) (name "noname")))
- (cubic-bezier (coords -20.0 -10.0) 1.0 (coords -1.2038402557373 -18.5906028747559) 1.0 (coords 5.90976142883301 14.037802696228) 1.0 (coords -20.0000019073486 10.0) 1.0 ( (id 139619464) (stroke-style (stroke-style solid)) (stroke-cap (cap-style butt)) (stroke-join (join-style miter)) (stroke-width 0.100000001490116) (stroke-color (rgb 0 0 0)) (fill-color (rgba 255 255 255 0)) (invisible #f) (name "noname")))
- (cubic-bezier (coords -10.0 0.0) 1.0 (coords 0.0 5.0) 1.0 (coords 0.0 15.0) 1.0 (controlpoint-4 139741312) 1.0 ( (id 139742080) (stroke-style (stroke-style solid)) (stroke-cap (cap-style butt)) (stroke-join (join-style miter)) (stroke-width 0.100000001490116) (stroke-color (rgb 0 0 0)) (fill-color (rgba 255 255 255 0)) (invisible #f) (name "noname")))
- (segment (coords -20.0 10.0) (controlpoint-4 139741312) ( (id 139742504) (stroke-style (stroke-style solid)) (stroke-cap (cap-style butt)) (stroke-join (join-style miter)) (stroke-width 0.100000001490116) (stroke-color (rgb 0 0 0)) (fill-color (rgb 0 0 0)) (invisible #f) (name "noname")))
- (parametric-point 139619464 1.0 ( (id 139619720) (name "noname")))
- (ellipse-by-3-points-rotation-eccentricity (controlpoint-1 139619464) (controlpoint-1 139741312) (controlpoint-1 139742080) 0.0 0.800000011920929 ( (id 139743272) (conic (point-entire)) (stroke-style (stroke-style solid)) (stroke-cap (cap-style butt)) (stroke-join (join-style miter)) (stroke-width 0.100000001490116) (stroke-color (rgb 0 0 0)) (fill-color (rgb 191 191 191)) (invisible #f) (name "noname")))
- (top-level-group ( (name "top-level-group")) (
- (decorator-point (coords -30.0 0.0) 4 1.0 0.785399377346039 ( (stroke-style (stroke-style solid)) (stroke-cap (cap-style butt)) (stroke-join (join-style miter)) (stroke-width 0.100000001490116) (stroke-color (rgb 0 0 0)) (fill-color (rgb 255 255 255)) (invisible #f) (name "noname"))) 139741312
- (decorator-point (coords -10.0 0.0) 4 1.0 0.785399377346039 ( (stroke-style (stroke-style solid)) (stroke-cap (cap-style butt)) (stroke-join (join-style miter)) (stroke-width 0.100000001490116) (stroke-color (rgb 0 0 0)) (fill-color (rgb 255 255 255)) (invisible #f) (name "noname")))
- (decorator-point (coords -20.0 -10.0) 4 1.0 0.785400629043579 ( (stroke-style (stroke-style solid)) (stroke-cap (cap-style butt)) (stroke-join (join-style miter)) (stroke-width 0.100000001490116) (stroke-color (rgb 0 0 0)) (fill-color (rgb 255 255 255)) (invisible #f) (name "noname")))
- (cubic-bezier (controlpoint-1 139619464) 1.0 (coords -54.4186973571777 -18.9017639160156) 1.0 (coords -50.9240226745605 21.8665885925293) 1.0 (end 139742504) 1.0 ( (stroke-style (stroke-style solid)) (stroke-cap (cap-style butt)) (stroke-join (join-style miter)) (stroke-width 0.100000001490116) (stroke-color (rgb 0 0 0)) (fill-color (rgba 255 255 255 0)) (invisible #f) (name "noname")))
- (quadratic-bezier (center 139619720) 1.0 (coords -10.0 5.0) 1.0 (controlpoint-1 139742080) 1.0 ( (stroke-style (stroke-style solid)) (stroke-cap (cap-style butt)) (stroke-join (join-style miter)) (stroke-width 0.100000001490116) (stroke-color (rgb 0 0 0)) (fill-color (rgba 0 0 0 0)) (invisible #f) (name "noname")))
- (quadratic-bezier (controlpoint-4 139619464) 1.0 (coords -30.0 5.0) 1.0 (controlpoint-1 139741312) 1.0 ( (stroke-style (stroke-style solid)) (stroke-cap (cap-style butt)) (stroke-join (join-style miter)) (stroke-width 0.100000001490116) (stroke-color (rgb 0 0 0)) (fill-color (rgba 0 0 0 0)) (invisible #f) (name "noname")))
- (parametric-point 139743272 0.499403983354568 ( (name "noname")))
- (parametric-point 139743272 0.387289702892303 ( (name "noname")))
- (parametric-point 139743272 0.61937540769577 ( (name "noname"))) 139619720 139743272 139742504
- (tex-text (coords -22.7725009918213 10.2322235107422) "$v$" ( (relative-position-x 0.5) (relative-position-y 0.5) (absolute-shift-x 0.0) (absolute-shift-y 0.0) (alignment-x (alignment-x ref-relative)) (alignment-y (alignment-y bbox-relative)) (transform (transform 1.0 0.0 0.0 1.0)) (fill-color (rgb 0 0 0)) (invisible #f) (name "noname")))
- (decorator-point (coords -20.0000019073486 10.0) 0 1.0 6.0491561271192e-7 ( (stroke-style (stroke-style solid)) (stroke-cap (cap-style butt)) (stroke-join (join-style miter)) (stroke-width 0.100000001490116) (stroke-color (rgb 0 0 0)) (fill-color (rgb 0 0 0)) (invisible #f) (name "noname")))
- (tex-text (coords -37.1394920349121 -3.80593585968018) "$N_4$" ( (relative-position-x 0.5) (relative-position-y 0.5) (absolute-shift-x 0.0) (absolute-shift-y 0.0) (alignment-x (alignment-x ref-relative)) (alignment-y (alignment-y bbox-relative)) (transform (transform 1.0 0.0 0.0 1.0)) (fill-color (rgb 0 0 0)) (invisible #f) (name "noname")))
- (tex-text (coords -20.248571395874 -12.9403553009033) "$w$" ( (relative-position-x 0.5) (relative-position-y 0.5) (absolute-shift-x 0.0) (absolute-shift-y 0.0) (alignment-x (alignment-x ref-relative)) (alignment-y (alignment-y bbox-relative)) (transform (transform 1.0 0.0 0.0 1.0)) (fill-color (rgb 0 0 0)) (invisible #f) (name "noname")))
- (tex-text (coords -6.94939517974854 -1.59444499015808) "$y$" ( (relative-position-x 0.5) (relative-position-y 0.5) (absolute-shift-x 0.0) (absolute-shift-y 0.0) (alignment-x (alignment-x ref-relative)) (alignment-y (alignment-y bbox-relative)) (transform (transform 1.0 0.0 0.0 1.0)) (fill-color (rgb 0 0 0)) (invisible #f) (name "noname")))
- (tex-text (coords -32.8320045471191 -1.57180035114288) "$x$" ( (relative-position-x 0.5) (relative-position-y 0.5) (absolute-shift-x 0.0) (absolute-shift-y 0.0) (alignment-x (alignment-x ref-relative)) (alignment-y (alignment-y bbox-relative)) (transform (transform 1.0 0.0 0.0 1.0)) (fill-color (rgb 0 0 0)) (invisible #f) (name "noname")))
- (tex-text (coords -21.8017578125 17.310417175293) "$u$" ( (relative-position-x 0.5) (relative-position-y 0.5) (absolute-shift-x 0.0) (absolute-shift-y 0.0) (alignment-x (alignment-x ref-relative)) (alignment-y (alignment-y bbox-relative)) (transform (transform 1.0 0.0 0.0 1.0)) (fill-color (rgb 0 0 0)) (invisible #f) (name "noname"))) 139619464 139742080
- (decorator-point (coords -20.0 15.0) 0 1.0 6.0491561271192e-7 ( (stroke-style (stroke-style solid)) (stroke-cap (cap-style butt)) (stroke-join (join-style miter)) (stroke-width 0.100000001490116) (stroke-color (rgb 0 0 0)) (fill-color (rgb 0 0 0)) (invisible #f) (name "noname")))))))))
\ No newline at end of file
--- /dev/null
+unitsize(1cm);
+currentpen = black + roundcap + 0.2mm;
+
+real vsize = 0.1;
+real xsize = 0.08;
+
+void vert(pair v)
+{
+ filldraw(circle(v, vsize), white, black);
+}
+
+void xert(pair v)
+{
+ filldraw(box(v-(xsize,xsize), v+(xsize,xsize)), white, black);
+}
+
+pair r = (0, 0);
+pair v = (0, 0.5);
+pair u = (0, 1);
+pair w = (0, -2);
+pair x = (-1, -1);
+pair y = (1, -1);
+
+draw(r -- v);
+draw(v -- u);
+filldraw(circle((0,-1), 1), gray(0.75));
+
+draw(x {dir(180)} .. {dir(0)} u);
+draw(y {dir(0)} .. {dir(180)} u);
+draw(w {dir(-45)} .. {dir(180)} v);
+
+vert(r); label("$r$", r, 2S);
+vert(v); label("$v$", v, 1.5W);
+vert(u); label("$u$", u, 2N);
+vert(w); label("$w$", w, 2SW);
+xert(x); label("$x$", x, 1.5SW);
+xert(y); label("$y$", y, 1.5SE);
+
+label("$M$", (-1.3, -1.9));
--- /dev/null
+unitsize(1cm);
+currentpen = black + roundcap + 0.2mm;
+
+real vsize = 0.1;
+real xsize = 0.08;
+
+void vert(pair v)
+{
+ filldraw(circle(v, vsize), white, black);
+}
+
+void xert(pair v)
+{
+ filldraw(box(v-(xsize,xsize), v+(xsize,xsize)), white, black);
+}
+
+pair v = (0, 0);
+pair u = (0, 0.5);
+pair w = (0, -2);
+pair x = (-1, -1);
+pair y = (1, -1);
+
+draw(v -- u);
+filldraw(circle((0,-1), 1), gray(0.75));
+
+draw(x {dir(180)} .. {dir(0)} u);
+draw(y {dir(0)} .. {dir(180)} u);
+draw(w {dir(-40)} .. {dir(210)} v);
+
+vert(v); label("$v$", v, 2S);
+vert(u); label("$u$", u, 2N);
+vert(w); label("$w$", w, 2SW);
+xert(x); label("$x$", x, 1.5SW);
+xert(y); label("$y$", y, 1.5SE);
+
+label("$N$", (-1.3, -1.9));
--- /dev/null
+unitsize(1cm);
+currentpen = black + roundcap + 0.2mm;
+
+real vsize = 0.1;
+real xsize = 0.08;
+
+void vert(pair v)
+{
+ filldraw(circle(v, vsize), white);
+}
+
+void xert(pair v)
+{
+ filldraw(box(v-(xsize,xsize), v+(xsize,xsize)), white);
+}
+
+pair v = (0, 0);
+pair u = (0, 0.5);
+pair ww = (0, -2);
+pair w = (0, -2.5);
+pair x = (-1, -1);
+pair y = (1, -1);
+
+draw(v -- u);
+draw(ww -- w);
+filldraw(circle((0,-1), 1), gray(0.75));
+
+draw(x {dir(180)} .. {dir(0)} u);
+draw(y {dir(0)} .. {dir(180)} u);
+draw(w {dir(-20)} .. {dir(210)} v);
+draw(w {dir(200)} .. {dir(-20)} u);
+
+vert(v); label("$v$", v, 2S);
+vert(u); label("$u$", u, 2N);
+vert(ww); label("$w'$", ww, 2N);
+xert(w); label("$w$", w, 2S);
+xert(x); label("$x$", x, 1.5SW);
+xert(y); label("$y$", y, 1.5SE);
+
+label("$P$", (-1.3, -1.9));
--- /dev/null
+unitsize(1cm);
+currentpen = black + roundcap + 0.2mm;
+
+real vsize = 0.1;
+real xsize = 0.08;
+
+void vert(pair v)
+{
+ filldraw(circle(v, vsize), white, black);
+}
+
+void xert(pair v)
+{
+ filldraw(box(v-(xsize,xsize), v+(xsize,xsize)), white, black);
+}
+
+pair v = (0, 0);
+pair u = (0, 0.5);
+pair w = (0, -2);
+pair x = (-1, -1);
+pair y = (1, -1);
+pair px = (0,-1) + dir(150);
+
+draw(v -- u);
+filldraw(circle((0,-1), 1), gray(0.75));
+draw(px -- y);
+
+draw(x {dir(180)} .. {dir(0)} u);
+draw(y {dir(0)} .. {dir(180)} u);
+draw(w {dir(-40)} .. {dir(210)} v);
+
+vert(v); label("$v$", v, 2S);
+vert(u); label("$u$", u, 2N);
+vert(w); label("$w$", w, 2SW);
+xert(x); label("$x$", x, 1.5SW);
+xert(y); label("$y$", y, 1.5SE);
+vert(px); label("$p_x$", px, 1.5NNW);
+
+label("$N_A$", (-1.3, -1.9));
--- /dev/null
+unitsize(1cm);
+currentpen = black + roundcap + 0.2mm;
+
+real vsize = 0.1;
+real xsize = 0.08;
+
+void vert(pair v)
+{
+ filldraw(circle(v, vsize), white, black);
+}
+
+void xert(pair v)
+{
+ filldraw(box(v-(xsize,xsize), v+(xsize,xsize)), white, black);
+}
+
+pair v = (0, 0);
+pair u = (0, 0.5);
+pair w = (0, -2);
+pair x = (-1, -1);
+pair y = (1, -1);
+pair z = (0,-1);
+
+draw(v -- u);
+filldraw(circle((0,-1), 1), gray(0.75));
+draw(x -- z);
+draw(z -- y);
+draw(v -- z);
+
+draw(x {dir(180)} .. {dir(0)} u);
+draw(y {dir(0)} .. {dir(180)} u);
+draw(w {dir(-40)} .. {dir(210)} v);
+
+vert(v); label("$v$", v, 2*dir(-120));
+vert(u); label("$u$", u, 2N);
+vert(w); label("$w$", w, 2SW);
+xert(x); label("$x$", x, 1.5SW);
+xert(y); label("$y$", y, 1.5SE);
+vert(z); label("$z$", z, 2S);
+
+label("$N_B$", (-1.3, -1.9));
--- /dev/null
+unitsize(1cm);
+currentpen = black + roundcap + 0.2mm;
+
+real vsize = 0.1;
+real xsize = 0.08;
+
+void vert(pair v)
+{
+ filldraw(circle(v, vsize), white, black);
+}
+
+void xert(pair v)
+{
+ filldraw(box(v-(xsize,xsize), v+(xsize,xsize)), white, black);
+}
+
+pair v = (0, 0);
+pair u = (0, 0.5);
+pair w = (0, -2);
+pair x = (-1, -1);
+pair y = (1, -1);
+
+draw(v -- u);
+filldraw(circle((0,-1), 1), gray(0.75));
+draw(x -- y);
+
+draw(x {dir(180)} .. {dir(0)} u);
+draw(y {dir(0)} .. {dir(180)} u);
+draw(w {dir(-40)} .. {dir(210)} v);
+draw(w {dir(220)} .. {dir(-20)} u);
+
+vert(v); label("$v$", v, 2S);
+vert(u); label("$u$", u, 2N);
+xert(w); label("$w$", w, 2S);
+xert(x); label("$x$", x, 1.5SW);
+xert(y); label("$y$", y, 1.5SE);
+
+label("$N_C$", (-1.3, -1.9));
%-uncompressed.pdf: %.pdf
qpdf --object-streams=disable --stream-data=uncompress $< $@
+%.pdf: %.asy
+ asy -f pdf -tex pdftex $<
+
mostlyclean:
rm -f *.dvi *.log *~ core *.o *.aux *.bbl *.blg *.toc
projects / ga.git / commitdiff