Second order surfaces and applications

Abstract

U ovom radu proučavamo plohe drugog reda. Glavni cilj rada je uz pomoć linearne algebre i kvadratne forme klasificirati plohe drugog reda u realnom afinom prostoru. Iskazan je Sylvesterov zakon inericije te je pomoću njega na primjerima ilustirano kako odrediti vrstu plohe drugog reda. Takoder, opisan je geometrijski pristup određivanja vrsta ploha drugog reda. Drugi dio bavi se osnovnim definicijama i teoremima iz diferencijalne geometrije i teorije ploha. Dane su definicije ploha drugog reda i njihovi presjeci koordinatnim ravninama. Opisali smo rotacijske i translacijske plohe, a u zasebnom poglavlju definirali zakrivljenost plohe te ju pročili na pravčastim plohama. Na kraju dajemo primjere veličanstvenih gradevina moderne arhitekture u kojima vidimo primjenu ploha drugog reda.In this thesis our main concern will be surfaces of second order. Using linear algebra and quadratic form theory our main goal is to provide classification of quadric surfaces in real affine space. In examples we illustrate how to determine the surface using the Sylvester’s law of inertia. Furthermore, we describe geometrical approach of determinating quadric surfaces. The second chapter deals with definitions and theorems in differential geometry and surface theory. We defined second-order surfaces and presented their intersections with coordinate planes. Rotational and translational surfaces are described and in a separate section we defined surface curvature which we study on ruled surfaces. Finally, we provide examples of magnificent buildings in modern architecture in which we can see the implementation of this quadric surfaces