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PONCELET CURVES ANDSURFACES
Jean Valls (UPPA)Jaca
18 /09/ 2012
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Jaca 2012. Introduction
CRCLES AND TRAN!LE
One "#$an%le %$&es 'n$'el "*+ c$#cles, +ne $nsc#$-e. C(#),
"e+"e# c$#c'sc#$-e. C(R)
Ten "e#e a#e $n$n$"el an s'c "#$an%les $nsc#$-e. $n C(R)an.
c$#c'sc#$-e. a#+'n. C(#)
T*+ #an.+ c$#cles a#e n+" "e c$#c'sc#$-e. an. "e
$nsc#$-e.c$#cles + a "#$an%le
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Jaca 2012. Introduction
PORS3
T$s 4$n. + #es'l" $s calle.porism:$" .+es n+" +cc'# $n %ene#al
-'" $ $" +cc'#s, $" +cc'#s +#$n$n$"el an cases
Te $#s" 4n+*n #es'l" + "$s "5e $s "e s+6calle. Ca55le
(+#E'le#6Ca55le 17) +#'la +# a "#$an%le $nsc#$-e. $n C(R)an.
c$#c'sc#$-e. a#+'n. C(#): 2#R;R
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Jaca 2012. Introduction
STENER PORS3
" can -e s+l&e. - an $n&e#s$+n "a" sen.s "e -l'e an.
#e.c$#cles +n "*+ c+ncen"#$c +nes
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Jaca 2012. Introduction
PONCELET PORS3
A-+'" Ca55le #es'l" P+ncele" sa$.:
> "?s a 5#+@ec"$&e #es'l"
" eans "a" $" s+'l. -e als+ "#'e +# "*+ %ene#al c+n$cs" lea.s "+
"e P+ncele" 5+#$s " $s +#e c+5l$ca"e. "anS"e$ne#?s 5+#$s
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Jaca 2012. Introduction
NAPOLEON N RUSSA P+ncele" el. 5#$s+ne# $n Sa#a"+& a"e# "e
B#asn+$?s -a""le
e %a&e "e -as$s + +.e#n al%e-#a$c %e+e"# :
.'al$",c+nse#&a"$+n + n'-e#, c+n"$n'$" 5#$nc$5le
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Jaca 2012. Introduction
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Jaca 2012. Poncelet's theorem
PONCELET?S CLOSURETEORE3
n @a$l a" Sa#a"+&, e *#+"e $s a+'s cl+s'#e "e+#e:
THM(181) Let C and D be two conics on the complexprojective
plane such that there exists one n-gon inscribedin D and
circumscribed around C. Then there are infinitelymany such
n-gons.
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Jaca 2012. Poncelet's theorem
TREE PROOFS OFPONCELET?S TEORE3
P+ncele" $sel : n6%+n "an%en" "+ nc+n$cs
an. s5ec$al$e "+ "*+ c+n$cs
Jac+-$ : $n&+l'"$+ns +n ell$5"$c c'#&e
e# : 5enc$l + 'a.#$cs $n PG
O'# a$ "+.a $s "+ 5#+5+se an+"e# +ne -ase. +n &ec"+#-'n.le
"ecn$'es
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Jaca 2012. Poncelet's theorem
SECANT VARETH OFRATONAL CURVE
Ra"$+nal n+#al c'#&e, $"s secan" &a#$e", $"s ass+c$a"e.
-'n.leA sec"$+n + "e las" +ne c+##es5+n.s "+ a 5e#5lane sec"$+n +"e
$#s" +ne : "e nn-!"#$&e#"$ces + a c+5le"e "an%en" n6%+n
A P+ncele" c'#&e : a 5enc$l + sec"$+ns
(#e+&e "*+ l$nes $n "e a.6+c a"#$I)
" %$&es a 5#++ + Da#-+'I "e+#e
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Jaca 2012. Poncelet's theorem
PROOF OF PONCELET PORS3
An $nsc#$-e. n6%+n : n5+$n"s +n C " #ea$ns
nn-!"#$-n5+$n"s T$s #ea$n$n% se" -el+n%s "+
(+ .e%#ee (n-)), s" %*CU($s a P+ncele" c'#&e
S+ an 5+$n" + C$s a &e#"eI + a c+5le"e n6%+n $nsc#$-e. $n %
+'"?s "e+#e "$s n6%+n as n&e#"$ces +n C($ n+" "en'-e# +
&e#"$ces +n ($s K 6n=nn-!"#$*$c $s $5+ss$-le +#.e%#ee
#eas+n)
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Jaca 2012. Interlude
C CURVE S APONCELET CURVE
All "e c+n$cs
All "e c'-$cs
Onl a .$&$s+# + 'a#"$cs : L'#+" 'a#"$cs
T$s .$&$s+# as .e%#ee (3+#le 1919) " $s "e $#s" D+nal.s+n
n'-e# 1M +n P
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Jaca 2012. Poncelet Surfaces
PONCELET SURFACES
Se" + 5lanes M6secan" "+ #a"$+nal n+#al c'#&e lea.s
"+P+ncele" s'#aces
Ne" + .e%#ee n.$&$s+#s +n P ; s'#ace + .e%#ee n-$
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Jaca 2012. Poncelet Surfaces
CUC PONCELET SURFACES
Te+#e :+ cubic surface % in P, is a oncelet surface.
P#++ : 5+$n"s ass+c$a"e. "+ %. T#$5le "ens+# Te a&e
a#es+l'"$+n - a 5e#se"#$c a"#$I
S++" c'-$c an.
$"s 27 l$nes
(Sal+n6Cale 189)
