Spherical geometry

U ovom diplomskom radu smo kroz četiri poglavlja obradili osnovne rezultate sferne geometrije izgradene na \(\mathbb{R}^3\). U drugom poglavlju smo definirali osnovne pojmove: sfera, velika kružznica, antipodalne točke i udaljenost izmedu dviju točaka. U trećem poglavlju smo proučavali sfernu trigonometriju. Definirali smo sferni trokut i dokazali analogone kosinusova, sinusova i Pitagorina poučka. Potom smo pokazali nekoliko rezultata za pravokutni sferni trokut i dualni teorem o kosinusu koji nema analogona u euklidskoj geometriji. U petom poglavlju smo obradili grupu izometrija sfere. Definirali smo izometriju na sferi, ortogonalni operator na \(\mathbb{R}^3\) i pokazali da je grupa izometrija na sferi izomorfna ortogonalnoj grupi na \(\mathbb{R}^3\). Zatim smo definirali osnu simetriju i pokazali da se svaka izometrija sfere može prikazati kao kompozicija najviše tri osne simetrije. Na samom kraju smo dokazali da na sferi, uz standardne teoreme o sukladnosti trokuta, vrijedi i \( KKK\) teorem o sukladnosti.In this thesis divided in four chapters we studied fundamental results of spherical geometry considered as a subset of \(\mathbb{R}^3\). In the second chapter we defined elementary terms: sphere, great circles, antipodal points and the distance between two points. In the third chapter we studied spherical trigonometry. We defined the spherical triangle and proved analogues of the cosine theorem, the sine theorem and the Pythagoras' theorem. Then, we proved several results about right-angled triangles and proved the dual cosine theorem which doesn't have an analogue in Euclidean geometry. In the fifth chapter we studied the isometry group of the sphere. We defined the isometry of sphere, orthogonal operator on \(\mathbb{R}^3\) and proved that there is a isomorphism between the isometry group of sphere and the orthogonal group of \(\mathbb{R}^3\): Then, we defined reactions and proved that every isometry of the sphere can be represented as the composition of at most three reactions. At the end we proved that besides ordinary congruence criteria for triangles, on the sphere the \( AAA\) criteria also holds