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Ponceletov izrek : diplomsko delo
Kos, Silva, matematičarka
Pokazali bomo enega najpomembnejših in najlepših izrekov v klasični projektivni geometriji, to je Ponceletov izrek. V ta namen si bomo najprej pogledali nekaj osnovnih definicij in izrekov ...projektivne geometrije. Poseben poudarek bomo namenili stožnicam, šopom stožnic in involucijam, ki nam bodo v pomoč pri dokazu Ponceletovega izreka. Klasični Ponceletov izrek pravi, da če za par stožnic ▫$C$▫ in ▫$\Gamma$▫ obstaja tak ▫$n$▫-kotnik, ki je včrtan stožnici ▫$C$▫ in očrtan stožnici ▫$\Gamma$▫, potem obstaja neskončno mnogo takih ▫$n$▫-kotnikov. Ideja dokaza je indukcija na število stranic ▫$n$▫-kotnika, vendar je dobra le v bolj splošnem primeru za poligone, ki so včrtani stožnici ▫$C$▫ in katerih stranice so tangente na ▫$n$▫ stožnic ▫$\Gamma_1, \Gamma_2, \ldots, \Gamma_n$▫ v šopu stožnic, ki vsebuje tudi stožnici ▫$C$▫ in ▫$\Gamma$▫. Trditev Ponceletovega izreka je poseben primer zgornjega za ▫$\Gamma_1 = \Gamma_2 = \ldots = \Gamma_n$▫. Tudi mi bomo pokazali splošnejši primer. V zadnjem delu naloge pa bomo poiskali eksplicitni pogoj za stožnici ▫$C$▫ in ▫$\Gamma$▫, pod katerim tak ▫$n$▫-kotnik obstaja.
Type of material - undergraduate thesis
Publication and manufacture - Ljubljana : [S. Kos], 1999
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