144 KB19 KB200820262013Dobovišek, Mirkoštevilka:1letnik:60str. 4-14ISSN:0473-7466COBISSID:16629849URN:URN:NBN:SI:doc-C75KR4HHslDruštvo matematikov, fizikov in astronomov SlovenijeObzornik za matematiko in fizikodilacijadilationgreat Poncelet theoremn-Poncelet propertyn-Ponceletova lastnostnumerical rangenumerični zakladPoncelet curvePonceletova krivuljaunitarna dilacijaunitary dilationveliki Ponceletov izrekPonceletove krivulje|In the 18th century mathematicians established the following: given two circles ?$C_1$?, inscribed to a given triangle, and ?$_2$?, circumscribed to the same triangle, then ?$C_1$? and ?$C_2$? are inscribed and circumscribed circle to an infinite number of triangles. Such a pair of curves are called Poncelet's curves. In this article the early history of Poncelet's porism is discussed. The same property occurs also in the case of an ?$n$?-sided polygon. In 1998 it was proved that the boundary of the numerical range on any ?$n times n$? matrices that admits unitary bordering is an ?$n+1$?-Poncelet's curve with respect to the unitary circle, and that such curves need not be quadrics. The example of a Poncelet's curve that is not a quadric is given in this article. It is already known that not all Poncelet's curves are boundaries of a numerical range. All Poncelet's curves with respect to a circle have not yet been classifiedV 18. stoletju so opazili, da velja naslednje: če sta krožnici včrtana in očrtana krožnica nekemu tikotniku, sta včrtana in očrtana krožnica še neskončno mnogim trikotnikom z oglišči na zunanji krožnici. Taki krivulji so poimenovali Ponceletovi krivulji. Izkazalo se je, da enako velja tudi pri ?$n$?-kotnikih. Ko so odkrili, da take krivulje niso nujno kvadrike, je raziskovanje dobilo nov zalet. Cele družine takih krivulj so dobili kot rob numerične zaloge vrednosti matrike. V članku je definicija numeričnega zaklada, prikaz, kako dobimo Ponceletovo krivuljo kot rob numeričnega zaklada, in primer, ko Ponceletova krivulja ni elipsa. Pred nekaj leti so ovrgli tudi hipotezo, da je vsaka Ponceletova krivulja rob numeričnega zaklada. Klasifikacija vseh Ponceletovih krivulj je zato še vedno odprt problemTEXTznanstveno časopisjejournalsSlovenian National E-content AggregatorNational and University Library of SloveniaDruštvo matematikov, fizikov in astronomov