Leptiri u izotropnoj ravnini

A real affine plane A2 is called an isotropic plane I2, if in A2 a metric is induced by an absolute {f, F}, consisting of the line at infinity f of A2 and a point F ∈ f. In this paper the well-known Butterfly theorem has been adapted for the isotropic plane. For the theorem that we will further-on call an Isotropic butterfly theorem, four proofs are given.Realna afina ravnina A2 se naziva izotropnom ravninom I2 ako je metrika u A2 inducirana apsolutnom figurom {f, F}, koja se sastoji od neizmjerno dalekog pravca f ravnine A2 i točke F∈ f. U ovom je radu poznati Leptirov teorem smješten u izotropnu ravninu. Za taj teorem, kojeg od sada nazivamo Izotropnim leptirovim teoremom, dana su četiri dokaza

O krivulji fokusa pramena konika u I2

Within the classification of conic pencils in the isotropic plane, which has been carried out using methods of analytical geometry and linear algebra, it is specially interesting to observe the curve of isotropic focuses, which is shown to be a 3rd order curve. The properties of this very curve for the discussed, most common subtype of conic pencils, are determined. It is shown that, referring to the selection of the fundamental points, it is possible to determine its shape, and to classify it according to the Newton\u27s principle. The discussed cases of conic pencils with its focal curves are illustrated with the figures drawn by Mathematica®.Pri klasifikaciji pramenova konika u izotropnoj ravnini, provedenoj metodama analitičke geometrije i linearne algebre, posebno je interesantno promatrati krivulju izotropnih fokusa za koju se pokazuje da je krivulja 3. reda. Određena su svojstva krivulje fokusa za promatrani, najopćenitiji, podtip pramena. Pokazano je da već prema odabiru temeljnih točaka možemo odrediti njen oblik i klasifikaciju prema Newtonovom principu. Promatrani slučajevi su prikazani crtežima izrađenim pomoću programa Mathematica®

On pencil of quadrics in I3(2)

An affine space A3 is called a double isotropic space I3 (2) , if in A3 a metric is induced by an absolute , f , F, consisting of the line f in the plane of infinity of A3, and a point F in f . The pencil of quadrics is a set of 1 2nd order surfaces having common 4th order space curve. Intersecting a pencil of quadrics by a general plane we obtain a pencil of 2nd order curves. In this paper pencils of quadrics in a double isotropic space I3 (2) are analysed whereby the pencil of surfaces is observed as the pencil associated with the pencil of second order curves (conics) belonging to isotropic absolute plane . In this process we use the classification of pencils of conics in the isotropic plane given in 2, the classification of 2nd order surfaces in I3 (2) 4, and the projective properties of the pencils of second order surfaces [9, 16]. In order to obtain a more complete classification, the fundamental curve of the pencil, the curve of the centres, and the focal surface of the pencil of quadrics are analysed

Classification of conic sections in PE_2(R)

This paper gives a complete classification of conics in PE_2(R). The classification has been made earlier (Reveruk [5]), but it showed to be incomplete and not possible to cite and use in further studies of properties of conics, pencil of conics, and of quadratic forms in pseudo-Euclidean spaces. This paper provides that. A pseudo-orthogonal matrix, pseudo-Euclidean values of a matrix, diagonalization of a matrix in a pseudo-Euclidean way are introduced. Conics are divided in families and by types, giving both of them geometrical meaning. The invariants of a conic with respect to the group of motions in PE_2(R) are determined, making it possible to determine a conic without reducing its equation to canonical form. An overview table is given

Pogreška kompenzacije nivelira s automatskim horizontiranjem i njeno ispitivanje

U radu je izložena jedna metoda određivanja pogreške kompenzacije nivelira s automatskim horizontiranjem. Posebna pažnja posvećena je računanju korekcije rezultata mjerenja. U literaturi, gdje se obrađuje ova problematika, ta korekcija ili nije uzeta u obzir, ili nije dovoljno razrađena. Prikazani su i rezultati ispitivanja pogreške kompenzacije nekih nivelira, određeni ovom metodom

Diagonal triangle of a non-tangential quadrilateral in the isotropic plane

Properties of the non-tangential quadrilateral $lijepoa lijepob lijepoc lijepod$ in the isotropic plane concerning its diagonal triangle are given in this paper. A quadrilateral is called standard if a parabola with the equation $x=y^2$ is inscribed in it. Every non-tangential quadrilateral can be represented in the standard position. First, the vertices and the equations of the sides of the diagonal triangle are introduced. It is shown that the midlines of the diagonal triangle touch the inscribed parabola of the quadrilateral. Furthermore, quadrilaterals formed by two diagonals and some two sides of the non-tangential quadrilateral $lijepoa lijepob lijepoc lijepod$ are studied and a few theorems on its foci are presented

O problemu bojanja grafova s primjenom u kartografiji

The problem of colouring geographical political maps has historically been associated with the theory of graph colouring. In the middle of the 19th century the following question was posed: how many colours are needed to colour a map in a way that countries sharing a border are coloured differently. The solution has been reached by linking maps and graphs. It took more than a century to prove that 4 colours are sufficient to create a map in which neighbouring countries have different colours.Problem bojanja geografskih političkih karata povijesno je vezan uz teoriju bojanja grafova. Polovicom 19. stoljeća nametnulo se pitanje koliko je boja potrebno da bi se dana geografska karta obojila tako da zemlje koje graniče budu obojane različitim bojama. Do rješenja se došlo povezivanjem karata i grafova. Bilo je potrebno više od jednog stoljeća kako bi se dokazalo da su četiri boje dovoljne za obojiti (geografsku) kartu na takav način da susjedna područja (države) imaju različitu boju