26 research outputs found
One Application of Perspective Collineation
Cilj je ovog rada da, podsjeÄanjem na neke stare geometrijske konstrukcije izvedene pomoÄu perpektivne kolineacije i afiniteta, ponovno aktualizira crtež u geometriji, Äija je izrada danas olakÅ”ana upotrebom raÄunala. Zadaci se rjeÅ”avaju metodom perspektivno kolinearno pridruženih figura, tj. svaki se problem u vezi s konikom perspektivnom kolineacijom preslika u elementarno rjeÅ”iv zadatak u vezi s kružnicom, Å”to rezultira geometrijski toÄnim rjeÅ”enjem poÄetnog problema.The aim of this paper is to remind us of some previous geometrical constructions derived by the means of a perspective collineation and an affinity. It should refresh the drawing in geometry. Nowdays it is much easier by using the computer. The tasks are solved with the method of the collinear corresponded figures. Each problem connected with a conic by using the method of a perspective collineation can be transformed into the elementary problem connected with a circle. It results with the geometricaly correct solution of the initial problem
Pencil of circles in isotropic plane
Promatraju se izotropne kružnice u modelu izotropne ravnine s apsolutnom figurom (f,F)u konaÄnosti. U usporedbi s euklidskom ravninom, u kojoj je pramen kružnica moguÄe zadati samo na pet razliÄitih naÄina, pokazuje se da je u izotropnoj ravnini to moguÄe uÄiniti na devet naÄina. Konstruira se po jedna kružnica za svaki od zadanih pramenova.The model of an isotropic plane with absolute figure (f,F) in finiteness is studied. In comparation with Euclidean plane, where pencil of circles can be set only on five different ways, it is shown that in an isotropic plane it can be done on nine ways. A circle is constructed for every given pencil
Oskulacijske kružnice konika u Cayley-Klein-ovim ravninama
In the Euclidean plane there are several well-known methods of constructing an osculating (Euclidean) circle to a conic. We show that at least one of these methods can be ātranslatedā into a construction scheme of finding the osculating non-Euclidean circle to a given conic in a hyperbolic or elliptic plane. As an example we will deal with the
Klein-model of these non-Euclidean planes, as the projective geometric point of view is common to the Euclidean as well as to the non-Euclidean cases.U euklidskoj ravnini postoji nekoliko dobro poznatih metoda konstrukcija oskulacijske kružnice konike. Cilj je te konstrukcije ātranslatiratiā u neke od neeuklidskih ravnina. U Älanku se daje opÄa konstrukcija oskulacijske kružnice konike zadane s pet elemenata u euklidskoj ravnini. Pokazuje se da je konstruktivna metoda primjenjiva u hiperboliÄkoj i eliptiÄkoj ravnini. BuduÄi da je
projektivno geometrijsko glediÅ”te zajedniÄko euklidskom i neeuklidskim sluÄajevima, analogne se konstrukcije koriste na Klein-ovim modelima neeuklidskih ravnina
Steinerova krivulja u pramenu parabola
Using the facts from the theory of conics, two theorems that are analogous to the theorems in triangle geometry are proved. If the pencil of parabolas is given by three lines a, b, c, it is proved that, the vertex tangents of all the parabolas in the pencil, envelop the Steiner deltoid curve Ī“, and the axes of all parabolas in the same pencil envelop further deltoid curve Ī±. Furthermore, the deltoid curves are homeothetic. It is proved that all the vertices in the same pencil of parabolas are located at the 4th degree curve. The above mentioned curves are constructed and treated by synthetic methods.KoristeÄi Äinjenice teorije konika, dokazuju se dva teorema koji su analogoni klasiÄnih teorema geometrije trokuta. Za pramen parabola zadan trima temeljnim tangentama a, b, c dokazuje se da tjemene tangente svih parabola omataju deltoidu Ī“, a osi parabola u istom pramenu deltoidu Ī±. Pokazuje se da su deltoide homotetiÄne. JoÅ” se dokazuje da sva tjemena parabola u istom pramenu leže na krivulji 4. reda. Spomenute krivulje se konstruiraju i istražuju metodama sintetiÄke geometrije
Some Planimetric Constructions in the H-plane
Na Kleinovom modelu hiperboliÄke ravnine uspostavlja se centralna kolineacija izmeÄu apsolute i H-kružnice. Pokazuje se kako je moguÄe uz pomoÄ ove centralne kolineacije, sredstvima euklidske geometrije izvoditi elementarne geometrijske konstrukcije u H-ravnini. U tom su smislu rijeÅ”ena dva zadatka.On the Klein\u27s model of the hyperbolic plane we can define the central collineation between absolute and H-circle. Using that central collineation it\u27s possible to make the elementary geometric constructions in the H-plane by the instruments of the euclidean geometry. Two problems have been solved in this way
Perspektivna kolineacija i oskulacijska kružnica konike u PE-ravnini i I-ravnini
All perspective collineations in a real affine plane are classified according to a constant cross-ratio and the position of the center and axis. A special attention will be given to the conditions which basic elements of perspective collineation have to fulfill in order to obtain the touch or osculation or hyperosculation of two conics. On the affine models of an isotropic and pseudo-Euclidean plane the osculating circle of a conic is constructed by using perspective collineations.Sve perspektivne kolineacije u realnoj afinoj ravnini klasificiraju se s obzirom na karakteristiÄnu konstantu te položaj srediÅ”ta i osi. Pokazuje se, kako odabrati temeljne elemente perspektivne kolineacije kako bi se neka konika i njezina slika dodirivale u jednoj ili dvije toÄke, oskulirale se ili hiperoskulirale. Na afinim se modelima izotropne i pseudoeuklidske ravnine pomoÄu perspektivne kolineacije konstruiraju oskulacijske kružnice konika