Notes on Taxicab Geometry
| Autor: | Divjak Blaženka |
|---|---|
| Jazyk: | angličtina |
| Rok vydání: | 2000 |
| Předmět: | |
| Zdroj: | KoG Volume 5. Issue 5. |
| ISSN: | 1846-4068 1331-1611 |
| Popis: | The taxicab geometry is one of non-Euclidean geometries. Hermann Minkowski (Gesammelte Abhandlungen) introduced this geometry more than 100 years ago. In this hundred years there were the periods of marginalization and the periods of great interest and wide application of this geometry. Today there is a whole specter of application and implementation of the taxicab geometry. There are several reasons for this. First, the taxicab geometry is similar to Euclidean geometry and easy to understand. It can be observed as such a metric system where the points correspond to the intersections of the streets in the imagined city, streets run only horizontally and vertically. There are no one-way streets. From the previous description the name taxicab geometry arises. The taxicab geometry is appropriate to discuss out during the undergraduate study in the form of essays, seminar works and diploma thesis as it is described in [7] and [9]. Second, the taxicab geometry is interesting for theoretical geometry study, too. It can be analyzed by synthetic approach (introduced by David Hilbert), or by metric approach ( described by George David Birkhoff). Mentioned approaches are described and discussed in [6]. There is the third approach in geometry using abstract groups and group theory. This approach was introduced by Felix Klein and Arthur Cayley. They claimed that geometry had to be studied through acting the group of motions on the given set. Further, some propositions about ellipses in the taxicab geometry are proved. Third, the practical value of the taxicab geometry is its wide application in (urban) transportation problems, city planning and so on. This application has been described in [10]. Taxicab geometrija jedna je od neeuklidskih geometrija. Tu je geometriju još prije 100 godina razmatrao Hermann Minkowski (Gesammelte Abhandlungen), a od tada doživljavala je periode marginalizacije i gotovo zaborava, ali i periode velikog zanimanja i široke primjene. Danas se može pronaći čitav spektar upotrebe i primjene taxicab geometrije. Za to postoji više razloga. Prvo, taxicab geometrija bliska je euklidskoj geometriji i lagana za razumijevanje. Može se promatrati kao metrički sustav u kojem točke korespondiraju križanjima ulica u zamišljenom gradu gdje ulice idu samo horizontalno i vertikalno i nema jednosmjernih ulica (odatle i naziv taxicab geometrija). Kao takva pogodna je za izučavanje na dodiplomskom studiju u obliku eseja, seminarskih radova i diplomskih radova (kao što je opisano u [7] i [9]). Drugo, taxicab geometrija zanimljiva je za izučavanje i sa stanovišta teorijske geometrije. Moguće ju je opisati upotrebom sintetičkog pristupa (koji je uveo David Hilbert), ali i metričkim pristupom (za koji je zaslužan George David Birkhoff). Oba ova pristupa pojašnjena su i upotrebljena za uvođenje taxicab geometrije u [6]. Postoji i treći pristup u geometriji preko apstraktne algebre i teorije grupa koji su uveli Felix Klein i Arthur Cayley koji tvrdi da se geometrija treba proučavati preko djelovanja grupe gibanja na zadani skup. Nadalje, dokazat ćemo neke poučke o elipsi u taxicab geometriji. Treće, praktična je vrijednost taxicab geometrije njezina široka primjenjivost na stvarne (urbane) probleme transporta, planiranja gradova itd. O tim primjena govori se npr. u [10]. |
| Databáze: | OpenAIRE |
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