Izrek o metulju in njegove posplošitve
magistrsko delo
Abstract
Magistrsko delo obravnava izrek o metulju, ki je izrek evklidske geometrije, in številne njegove posplošitve. Poleg dokazov izreka z osnovnimi pojmi evklidske geometrije delo vključuje dokaz izreka v razširjeni evklidski ravnini. Za slednjega so v evklidski geometriji vpeljani pojmi dvorazmerje, harmonična četverka, inverzija, pol in polara ter dokazana Cevov in Menelajev izrek. Delo obravnava posplošitvi avtorjev Murray S. Klamkina in Vladimirja Volenca ter izrek o dveh metuljih. Za dokaz izreka o metulju v projektivni geometriji se delo osredotoči na pojme realne projektivne ravnine kot so harmonična četverka, projektivnost, involucija, polarnost in stožnica ter dokaže Desarguesov izrek o involuciji. Posplošitev izreka o metulju v kompleksni projektivni ravnini je dokazana s pomočjo Pascalovega izreka.
Keywords
izrek o metulju;razširjena evklidska ravnina;projektivna geometrija;Desarguesov izrek o involuciji;Pascalov izrek;
Data
| Language: | Slovenian |
|---|---|
| Year of publishing: | 2021 |
| Typology: | 2.09 - Master's Thesis |
| Organization: | UL FMF - Faculty of Mathematics and Physics |
| Publisher: | [E. Češek] |
| UDC: | 514 |
| COBISS: |
82941443
|
| Views: | 1461 |
| Downloads: | 135 |
| Average score: | 0 (0 votes) |
| Metadata: |
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Other data
| Secondary language: | English |
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| Secondary title: | Butterfly theorem and its generalizations |
| Secondary abstract: | This Master's thesis discusses the butterfly theorem of Euclidean geometry and many of its known generalizations. In addition to two proofs of the theorem with basic concepts of Euclidean geometry, the thesis also includes proof of the theorem in the extended Euclidean plane. For the latter, concepts such as cross ratio, harmonic range, inversion and pole-polar relation are introduced in Euclidean geometry and the theorems of Ceva and Menelaus are proven. The thesis addresses the generalizations of the butterfly theorem by Murray S. Klamkin's and Vladimir Volenec's and the double butterfly theorem. To prove the projective butterfly theorem, the thesis focuses on concepts of the real projective plane such as harmonic conjugacy, projectivity, involution, polarity and conics. It also proves Desargues' involution theorem. A generalization of the butterfly theorem in the complex projective plane is proved with the help of Pascal's theorem. |
| Secondary keywords: | butterfly theorem;extended Euclidean plane;projective geometry;Desarguesʼ involution theorem;Pascalʼs theorem; |
| Type (COBISS): | Master's thesis/paper |
| Study programme: | 0 |
| Thesis comment: | Univ. v Ljubljani, Fak. za matematiko in fiziko, Oddelek za matematiko, Pedagoška matematika |
| Pages: | IX, 62 str. |
| ID: | 13797463 |
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