Kubike konstantnog umnoška udaljenosti

The locus of points that determine a constant product of their distances to the sides of a triangle is a cubic curve in the projectively closed Euclidean triangle plane. In this paper, algebraic and geometric properties of these distance product cubics shall be studied. These cubics span a pencil of cubics that contains only one rational and non-degenerate cubic curve which is known as the Bataille acnodal cubic determined by the product of the actual trilinear coordinates of the centroid of the base triangle. Each triangle center defines a distance product cubic. It turns out that only a small number of triangle centers share their distance product cubic with other centers. All distance product cubics share the real points of inflection which lie on the line at infinity. The cubics\u27 dual curves, their Hessians, and especially those distance product cubics that are defined by particular triangle centers shall be studied.U projektivno zatvorenoj Euklidskoj ravnini trokuta geometrijsko mjesto točaka trokuta kojima je umnožak udaljenosti od stranica trokuta konstantan je jedna kubika. Proučavat će se algebarska i geometrijska svojstva tih kubika konstantnog umnoška udaljenosti. Takve kubike čine pramen kubika koje sadrže samo jednu racionalnu nedegeneriranu kubiku poznatu kao Batailleova kubika s izoliranom točkom, a koja je određena umnoškom pravih trilinearnih koordinata težišta temeljnog trokuta. Svaka točka trokuta određuje jednu kubiku konstantnog umnoška udaljenosti. Ispostavlja se da mali broj točaka trokuta međusobno dijele kubiku konstantnog umnoška udaljenosti. Sve kubike konstantnog umnoška udaljenosti dijele realne točke infleksije koje leže na pravcu u beskonačnosti. Proučavat će se dualne krivulje kubike, njihove Hessianove matrice i posebno one kubike konstantnog umnoška udaljenosti koje su određene poznatim točkama trokuta

O algebarskim minimalnim plohama

We give an overiew on various constructions of algebraic minimal surfaces in Euclidean three-space. Especially low degree examples shall be studied. For that purpose, we use the different representations given by WEIERSTRASS including the so-called Bjorling formula. An old result by LIE dealing with the evolutes of space curves can also be used to construct minimal surfaces with rational parametrizations. We describe a one-parameter family of rational minimal surfaces which touch orthogonal hyperbolic paraboloids along their curves of constant Gaussian curvature. Furthermore, we find a new class of algebraic and even rationally parametrizable minimal surfaces and call them cycloidal minimal surfaces.Dajemo pregled različitih konstrukcija algebarskih minimalnih ploha u euklidskom trodimenzionalnom prostoru. Posebice se promatraju primjeri niskog stupnja. U tu svrhu koristimo različite prikaze koje daje WEIERSTRASS, uključujući takozvanu Bjorlingovu formulu. LIJEV stari rezultat pokazuje da se evolute prostornih krivulja mogu koristiti za konstruiranje minimalnih ploha s racionalnim parametrizacijama. Mi opisujemo jednoparametarsku familiju racionalnih minimalnih ploha koje diraju ortogonalne hiperboličke paraboloide duž njihovih krivulja s konstantnom Gaussovom zakrivljenosću. Štoviše, nalazimo novu klasu algebarskih i čak racionalno parametrizirajućih minimalnih ploha i nazivamo ih cikloidnim minimalnim plohama

Udaljenosti i centralna projekcija

Given a point P in Euclidean space R^3 we look for all points Q such that the length PQ of the line segments PQ from P to Q equals the length of the central image of the segment. It turns out that for any fixed point P the set of all points Q is a quartic surface Φ. The quartic Φ carries a one-parameter family of circles, has two conical nodes, and intersects the image plane Pi along a proper line and the three-fold ideal line p of Pi if we perform the projective closure of the Euclidean three-space. In the following we shall describe and analyze the surface Φ.Za danu točku P u euklidskom prostoru R^3 traže se sve točke Q takve da je duljina PQ dužine PQ jednaka duljini njezine centralne projekcije. Pokazuje se da je za čvrstu točku P skup svih točaka Q kvartika Φ. Kvartika Φ sadrži jednoparametarsku familiju kružnica, ima dvije dvostruke točke, te siječe ravninu slike Pi po jednom pravom pravcu i tri puta brojanom idealnom pravcu p ravnine Pi (promatra se projektivno proširenje trodimenzionalnog euklidskog prostora). U radu se opisuje i istražuje ploha Φ

Poopćene konhoide

We adapt the classical definition of conchoids as known from the Euclidean plane to geometries that can be modeled within quadrics. Based on a construction by means of cross ratios, a generalized conchoid transformation is obtained. Basic properties of the generalized conchoid transformation are worked out. At hand of some prominent examples - line geometry and sphere geometry - the actions of these conchoid transformations are studied. Linear and also non-linear transformations are presented and relations to well-known transformations are disclosed.Prilagođavamo klasičnu definiciju konhoida iz euklidske ravnine geometrijama definiranim kvadrikama. Postiže se poopćena konhoidna transformacija koja se temelji na konstrukciji pomoću dvoomjera. Proučavaju se osnovna svojstva ovakve transformacije. Djelovanje poopćene konhoidne transformacije se proučava na nekim istaknutim primjerima kao što su pravčasta i sferna geometrija. Prikazuju se linearne i nelinearne transformacije te su opisane veze s dobro poznatim transformacijama

A point model for the free cyclic submodules over ternions

We show that the set of all (unimodular and non-unimodular) free cyclic submodules of T^2, where T is the ring of ternions over a commutative field, admits a point model in terms of a smooth algebraic variety

Dva konvergentna niza trokuta

A semi-orthogonal path is a polygon inscribed into a given polygon such that the i-th side of the path is orthogonal to the i-th side of the given polygon. Especially in the case of triangles, the closed semi-orthogonal paths are triangles which turn out to be similar to the given triangle. The iteration of the construction of semi-orthogonal paths in triangles yields infinite sequences of nested and similar triangles. We show that these two different sequences converge towards the bicentric pair of the triangle\u27s Brocard points. Furthermore, the relation to discrete logarithmic spirals allows us to give a very simple, elementary, and new constructions of the sequences\u27 limits, the Brocard points. We also add some remarks on semi-orthogonal paths in non-Euclidean geometries and in n-gons.Poluortogonalan put je poligonalna linija upisana u dani mnogokut takva da je i-ta stranica poligonalne linije okomita na i-tu stranicu danog mnogokuta. U slučaju trokuta, zatvoreni poluortogonalni putovi su trokuti slični danom trokutu. Iteracijom konstrukcije poluortogonalnih putova u trokutima dobivaju se beskonačni nizovi upisanih sličnih trokuta. Pokazujemo da ova dva različita niza konvergiraju prema bicentričnom paru Brocardovih točaka trokuta. Nadalje, veza s diskretnim logaritamskim spiralama omogućuje vrlo jednostavnu, elementarnu novu konstrukciju limesa ovih nizova, Brocardovih točaka. Iznosimo i neke napomene o poluortogonalnim putovima kako u neeuklidskim geometrijama i tako i za n-kute

Factor-Group-Generated Polar Spaces and (Multi-)Qudits

Recently, a number of interesting relations have been discovered between generalised Pauli/Dirac groups and certain finite geometries. Here, we succeeded in finding a general unifying framework for all these relations. We introduce gradually necessary and sufficient conditions to be met in order to carry out the following programme: Given a group \vG, we first construct vector spaces over \GF(p), $p$ a prime, by factorising \vG over appropriate normal subgroups. Then, by expressing \GF(p) in terms of the commutator subgroup of \vG, we construct alternating bilinear forms, which reflect whether or not two elements of \vG commute. Restricting to $p=2$, we search for ``refinements'' in terms of quadratic forms, which capture the fact whether or not the order of an element of \vG is $\leq 2$. Such factor-group-generated vector spaces admit a natural reinterpretation in the language of symplectic and orthogonal polar spaces, where each point becomes a ``condensation'' of several distinct elements of \vG. Finally, several well-known physical examples (single- and two-qubit Pauli groups, both the real and complex case) are worked out in detail to illustrate the fine traits of the formalism.Comment: 20 pages, 6 figures, 1 table; Version 2 - slightly polished, updated references; Version 3 - published version in SIGM

Pencils of Frégier Conics

Za svaku točku P na konici c, involucija pravih kutova u točki P inducira eliptičnu involuciju na konici c čije se središte F zove Frégierova točka od P. Zamjena pravih kutova u točki P između označenih krakova s proizvoljnim kutom fi vodi ka projektivnom preslikavanju u pramenu točke P, a tako i na konici c. Pravci koji povezuju odgovarajuće točke na konici c više ne prolaze kroz jednu točku nego omataju koniku f koja se vidi kao generalizacija Frégierove točke i zvat će se generalizirana Frégierova konika. Mijenjajući kut, dobivamo pramen generaliziranih Frégierovih konika koji je pramen treće vrste. Proučavat ćemo tako definirane konike i otkriti među ostalim i generalizirane familije Ponceletovih trokuta.For each point P on a conic c, the involution of right angles at P induces an elliptic involution on c whose center F is called the Frégier point of P. Replacing the right angles at P between assigned pairs of lines with an arbitrary angle fi yields a projective mapping of lines in the pencil about P, and thus, on c. The lines joining corresponding points on c do no longer pass through a single point and envelop a conic f which can be seen as the generalization of the Frégier point and shall be called a generalized Frégier conic. By varying the angle, we obtain a pencil of generalized Frégier conics which is a pencil of the third kind. We shall study the thus defined conics and discover, among other objects, general Poncelet triangle families

Konhoide na sferi

The construction of planar conchoids can be carried over to the Euclidean unit sphere. We study the case of conchoids of (spherical) lines and circles. Some elementary constructions of tangents and osculating circles are stil valid on the sphere. Further, we aim at the illustration and a precise description of the algebraic properties of the principal views of spherical conchoids, i.e., the conchoid’s images under orthogonal projections onto their symmetry planes.Konstrukcija ravninskih konhoida može se prenijeti na euklidsku jediničnu sferu. Promatramo slučaj konhoida generiranih sfernim pravcima i kružnicama. Neke elementarne konstrukcije tangenata i kružnica zakrivljenosti vrijede i za sferne konhoide. Nadalje, naš je cilj ilustracija i precizan opis algebarskih svojstava glavnih pogleda sfernih konhoida, tj. slika konhoida pri ortogonalnom projiciranju na njihove ravnine simetrije

Udaljenosti i centralna projekcija

Given a point P in Euclidean space R^3 we look for all points Q such that the length PQ of the line segments PQ from P to Q equals the length of the central image of the segment. It turns out that for any fixed point P the set of all points Q is a quartic surface Φ. The quartic Φ carries a one-parameter family of circles, has two conical nodes, and intersects the image plane Pi along a proper line and the three-fold ideal line p of Pi if we perform the projective closure of the Euclidean three-space. In the following we shall describe and analyze the surface Φ.Za danu točku P u euklidskom prostoru R^3 traže se sve točke Q takve da je duljina PQ dužine PQ jednaka duljini njezine centralne projekcije. Pokazuje se da je za čvrstu točku P skup svih točaka Q kvartika Φ. Kvartika Φ sadrži jednoparametarsku familiju kružnica, ima dvije dvostruke točke, te siječe ravninu slike Pi po jednom pravom pravcu i tri puta brojanom idealnom pravcu p ravnine Pi (promatra se projektivno proširenje trodimenzionalnog euklidskog prostora). U radu se opisuje i istražuje ploha Φ