Normal Curvature of a Surface

U ovom završnom radu upoznat ćemo se s normalnom zakrivljenošću plohe. Najprije ćemo definirati neke osnovne pojmove lokalne teorije krivulja, kao što su parametrizirana krivulja, Frenetov trobrid i Frenetove formule krivulje. Navest ćemo definiciju plohe, odnosno regularne plohe te ćemo definirati prvu, odnosno drugu fundamentalnu formu plohe, kao i operator oblika plohe. Normalna zakrivljenost u točki plohe u smjeru tangencijalnog vektora \(v_{p}\) se računa kao kvocijent skalarnog produkta vektora \(v_{p}\) i vektora dobivenog djelovanjem operatora oblika plohe na vektor \(v_{p}\) i kvadrata norme vektora \(v_{p}\). Geometrijski, takva zakrivljenost predstavlja zakrivljenost krivulje dobivene kao presjek plohe i ravnine koju razapinju vektor (v_{p}\) i vektor plošne normale. Rad ćemo završiti iskazom i dokazom dva važna teorema vezana uz normalnu zakrivljenost plohe.In this final paper we will analyze the normal curvature of a surface. First we will define the basic concepts of local theory of curves, i.e. a parametrized curve, Frenet frame field and Frenet formulas for parametrized curve. After that we will define what is surface, regular surface and we will say something more about the first and the second fundamental form, as well as about the shape operator of a surface. The normal curvature of a surface in a point on a surface in a direction of tangent vector \(v_{p}\) is calculated as the quotient of inner product of vectors \(v_{p}\) and vector obtained by the shape operator applied on a vector \(v_{p}\), and squared norm of vector \(v_{p}\). Geometrically, such curvature represents the curvature of a curve obtained as the intersection of a surface and plane determined by a tangent vector \(v_{p}\) and surface normal. We finish this work with two important theorems related to the normal curvature of a surface