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master's thesis
Inverzija u ravnini i primjene
Osijek: Josip Juraj Strossmayer University of Osijek, Department of Mathematics, 2018. urn:nbn:hr:126:792254
Bockovac, Marinela
Josip Juraj Strossmayer University of Osijek
Department of Mathematics
Chair of Applied Mathematics and Computer Science
Applied Mathematics and Optimization Research Group

Institutional repository: Repository of School of Applied Mathematics and Informatics

Cite this document

Bockovac, M. (2018). Inverzija u ravnini i primjene (Master's thesis). Osijek: Josip Juraj Strossmayer University of Osijek, Department of Mathematics. Retrieved from https://urn.nsk.hr/urn:nbn:hr:126:792254

Bockovac, Marinela. "Inverzija u ravnini i primjene." Master's thesis, Josip Juraj Strossmayer University of Osijek, Department of Mathematics, 2018. https://urn.nsk.hr/urn:nbn:hr:126:792254

Bockovac, Marinela. "Inverzija u ravnini i primjene." Master's thesis, Josip Juraj Strossmayer University of Osijek, Department of Mathematics, 2018. https://urn.nsk.hr/urn:nbn:hr:126:792254

Bockovac, M. (2018). 'Inverzija u ravnini i primjene', Master's thesis, Josip Juraj Strossmayer University of Osijek, Department of Mathematics, accessed 28 March 2025, https://urn.nsk.hr/urn:nbn:hr:126:792254

Bockovac M. Inverzija u ravnini i primjene [Master's thesis]. Osijek: Josip Juraj Strossmayer University of Osijek, Department of Mathematics; 2018 [cited 2025 March 28] Available at: https://urn.nsk.hr/urn:nbn:hr:126:792254

M. Bockovac, "Inverzija u ravnini i primjene", Master's thesis, Josip Juraj Strossmayer University of Osijek, Department of Mathematics, Osijek, 2018. Available at: https://urn.nsk.hr/urn:nbn:hr:126:792254

Metadata
TitleInverzija u ravnini i primjene
Title (english)Inversion in the Plane and its Applications
AuthorMarinela Bockovac
MentorZdenka Kolar-Begović (mentor)
Committee memberTomislav Marošević (predsjednik povjerenstva)
Committee memberZdenka Kolar-Begović (član povjerenstva)
Committee memberIvan Matić (član povjerenstva)
GranterJosip Juraj Strossmayer University of Osijek
Department of Mathematics
(Chair of Applied Mathematics and Computer Science)
(Applied Mathematics and Optimization Research Group)
Osijek
Defense date and country2018-07-06, Croatia
Scientific / art field,
discipline and subdiscipline		NATURAL SCIENCES
Mathematics
Geometry and Topology
Abstract
Inverzija je transformacija ravnine odredena s čvrstom točkom S koju nazivamo centar inverzije i pozitivnim realnim brojem c kojeg nazivamo konstanta inverzije. Svaka točka na kružnici polumjera rs=√c sa središtem u točki S inverzijom se preslika sama u sebe, stoga možemo reći da je inverzija jednoznačno određena kružnicom k(s,rs) koju nazivamo kružnica inverzije. Inverzija je transformacija ravnine koja skup pravaca i kružnica preslikava u taj isti skup, ali pri... More tome može pravac preslikati u kružnicu i obrnuto. Pravci koji prolaze centrom inverzije S inverzijom se preslikaju sami u sebe, dok se ostali pravci preslikaju u kružnice koje ne prolaze centrom S. Kružnice koje prolaze centrom S inverzijom se preslikavaju u pravce koji ne prolaze centrom S, a ostale kružnice se preslikavaju u kružnice koje također na prolaze centrom inverzije S. Korištenjem inverzije dokazali smo jedan od najljepših teorema geometrije trokuta, Feuerbachov teorem, koji tvrdi da Eulerova kružnica trokuta dira trokutu upisanu i sve tri pripisane kružnice. Nadalje, u radu smo primjenom inverzije dokazali najopćenitiji slučaj Apolonijevog problem. Primjenom inverzije dokazani su još i Ptolomejev i Eulerov teorem. Less
Abstract (english)
Inversion is a plane transformation determined by a fixed point S, which is called center of inversion and by a real number c called inversion constant. Each point on the circle of radius rs=√c centered at point S is mapped, by inversion, into itself, so we can say that the inversion is uniquely determined by a circle k(s,rs) which is called a circle of inversion. Inversion is a plane transformation which set of lines and circles maps to the same set or it can... More replicate the line to a circle and vice versa. Lines through the center of inversion S, by inversion replicate themselves while the other lines replicate into a circles that does not pass the center S. Circles passing through the center S are mapped, by the inversion, in a lines that does not pass the center S, and the other circles are mapped into a circles, which also does not undergo inversion center S. By using the inversion we have proved on of the most beautiful theorem in geometry of triangle, the Feuerbach’s theorem. It claims that the Euler circle of any triangle is tangent internally to the incircle and tangent externally to the three escribed circles. Further, in addition to the work, by applying inversion, we have proved the most general case of the Apollonius problem. Ptolemy’s and Euler’s theorem were also proved. Less
Keywords
Keywords (english)
Languagecroatian
URN:NBNurn:nbn:hr:126:792254
Study programmeTitle: Mathematics and Computer Science
Study programme type: university
Study level: integrated undergraduate and graduate
Academic / professional title: magistar/magistra edukacije matematike i informatike (magistar/magistra edukacije matematike i informatike)
Type of resourceText
File originBorn digital
Access conditionsOpen access
Terms of useLicence In copyright icon
Created on2018-07-10 06:32:26
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