Konstruktivni postupak za transformaciju kolinearnih prostora

This paper offers the constructive solution for transformation of points between two general collinear spaces S and S1 by using Monge\u27s projection. The constructive procedure is based on the fact that each straight line of space S which is parallel to vanishing plane R∈S and one side of autocollinear tetrahedron D1D2D3D4, is transformed into the straight line of space S1 which is parallel to vanishing plane Q1∈ S1 and the same side of autocollinear tetrahedron.U radu se daje konstruktivno rješenje za preslikavanje točaka između dva opće kolinearna prostora S i S1, u Mongeovoj projekciji. Konstrukcija se temelji na činjenici da se svaki pravac prostora S koji je paralelan s izbježnom ravninom R∈S i jednom od stranica autokolinearnog tetraedra D1D2D3D4, preslikava u pravac prostora S1 koji je paralelan s izbježnom ravninom Q1∈ S1 i istom stranicom autokolinearnog tetraedra

Modelling shape of architectural structure: Elliptic hyperboloid of one sheet

The combination of straight lines of constructive elements and curved contours of structural shapes is always challenging in the engineering practice. The 2nd order surface - elliptic hyperboloid of one sheet (ELHY) provides such combination. Given that in the architectural surroundings, arts, or other scientific fields ELHY is less common, than the other representatives of the same family of ruled surfaces, it is worth of attention and research. Here presented constructive geometry approach resulted in Auto CAD application for generating 3D wire-frame and triangulated net model of ELHY surface and some examples of structures - prototypes of structural shapes, designed by using ELHY fragments. [Projekat Ministarstva nauke Republike Srbije, br. TR36008: Development and application of scientific methods in design and building of high economic structural systems by application of new technologies, the part of which is the present study

An algebraic approach to Harder-Narasimhan filtrations

It is a well-known fact that a stability condition $\phi: Obj^* \mathcal{A} \to \mathcal{I}$ over an abelian length category $\mathcal{A}$ induces a chain of torsion classes $\eta_\phi$ indexed by the totally ordered set $\mathcal{I}$. Inspired by this fact, in this paper we study all chains of torsion classes $\eta$ indexed by a totally ordered set $\mathcal{I}$ in $\mathcal{A}$. Our first theorem says that every chain of torsion classes $\eta$ indexed by $\mathcal{I}$ induces a Harder-Narasimhan filtration to every nonzero object of $\mathcal{A}$. Building on this, we are able to generalise several of the results showed by Rudakov in \cite{Rudakov1997}. Moreover we adapt the definition of slicing introduced by Bridgeland in \cite{Bridgeland2007} and we characterise them in terms of indexed chain of torsion classes. Finally, we follow ideas of Bridgeland to show that all chains of torsion classes of $\mathcal{A}$ indexed by the set $[0,1]$ form a metric space with a natural wall and chamber structure. Moreover we show a one to one correspondence between the chambers of this metric space and the maximal green sequences of $\mathcal{A}$.Comment: Typos and minor corrections were mad

Augmented reality application in engineering students spatial abilities assessment

Expanding technology, augmented reality (AR), where the customer/user can obtain various information about 3D digital model of an object in real time virtual presentation, has a wide range of applications. One of the fields of interest is cultural heritage preservation. Combined with 3D modeling, mostly based on 3D laser scanning data, AR enables a common user to visualize the object, i.e. cultural heritage monument in its' real surroundings. This is especially important in the cases when the monument is in ruined or devastated state. If reconstructed 3D model would give the clear visual impression about the monument and its historical value. This topic is especially interesting for engineering students, whose further professional activities would be involved in the tasks of cultural heritage protection. Augmented reality applications, among other, are available through android phones, i-phones and tablets by 3D model upload and its real time presentation. AR applications aimed for architectural objects require certain manipulation skills regarding object's positioning on the mobile device's screen, as well as, geometry knowledge for obtaining correctness. In this research, a testing was performed regarding spatial abilities of the engineering students, which had a task to incorporate the 3D model of the cultural heritage monument into several variants of 2D and 3D scene, as well as decomposition and recognition of elements of the structure. 3D digital Auto CAD model (geometric model) of reconstructed one nave church in Serbia, in monastery complex Kastaljan, on the mountain Kosmaj, was used for the testing purposes

3D animation applications in descriptive geometry teaching

The use of digital technology includes an interdisciplinary approach. The focus is on fusion of different media and genres into new forms of artistic expression as well as transcending the boundaries between art and education, the arts and research. Teaching descriptive geometry using animation softwares enables development of the ability of spatial representation, perception and understanding of space. The specific contribution is in the education by working with 3D animation for students of art and engineering field of technical - technological group of students. We emphasize the importance of dealing with design dynamically- generated form. Experimental design (freeform) presented on DVD includes: generating a surface with the two profiles as guidelines, generating free form using lattice deformers and generate free-form by the duplicating along curve tool. The key concept of dynamic geometry is to select the item which is then changed position and simutaneously you can see changes in the structure. This ability of movement is fundamentally improvement regard to the drawing on the paper or static CAD models. The movement in animation is captured by sequence of static pictures. If we have the form which changes during the time, we could observe these changes also sequentially. On this way, the form cloud be observed during the morphing in time, and each particular frame could be the base for further development.be observed during the morphing in time, and each particular frame could be the base for further development

Geometrija spiralnih površi primenjena na horizontalne komunikacije u arhitekturi

Contemporary technologies and materials continuously challenge structural designers, as well as architects, to achieve unique applications of various geometric surfaces. Twisted structures are actual and numerous in building of extremely tall towers. However, rare examples can be found as connecting elements of two buildings, i.e. horizontal communications. Authors' aim was to contribute in area of application such geometry in architecture. Term 'twist' is here related to ruled surface - elliptic hyperboloid of one sheet. The insertion of regular polygons (triangle and square) in circular sections of elliptic hyperboloid made possible 'step by step' presentation of two varieties of twisted surfaces. 3D wireframe and surface models are created in engineering software Auto CAD.Savremene tehnologije i materijali, u kontinuitetu postavljaju izazove pred projektante u smislu originalnosti primene raznovrsnih geometrijskih površi spiralne strukture (engl. 'twisted structures '), danas aktuelne i brojne u gradnji izuzetno visokih objekata. Retki su primeri primene ovakvih površi na veznim elementima između objekata - horizontalnim komunikacijama. Autori rada su želeli da daju doprinos u domenu geometrije površi koja bi imala mogućnost ovakve primene u arhitekturi. Pojam 'uvrtanja' je u ovom radu povezan sa pravoizvodnom površi - jednogranim eliptičkim hiperboloidom. Uvodeći u sisteme kružnih preseka eliptičkog hiperboloida pravilne poligone (trougao i kvadrat) autori postupno prikazuju tako nastala dva varijeteta površi. 3D žičani i površinski modeli su kreirani u inženjerskom softveru Auto CAD

Geometrije konstrukcija putem savremenih tehnologija

Contemporary technological development (CAD/CAM/CAE, VR, AR, MR) made conventional methods of Descriptive and Constructive Geometry uncompleted. Application and use of new technologies in Constructive geometry requires educational process with the aim to have complete knowledge of all fields that belong to this area. The aspect and results research shows that knowledge acquired in this way by using new technology, develops students' skills that are very important in senior years of studies, particularly in the field of engineering design. Interactive dynamic 3D geometry could not be achieved by conventional ways of studying. The use of modern technology should enable expanding the fields of research as well as preservation of the theoretical knowledge of descriptive geometry.Razvoj savremenih tehnologija (CAD/CAM/CAE, VR, AR, MR) je učinio tradicionalne metode nacrtne geometrije i geometrije konstrukcija nezavršenim. Primena i upotreba novih tehnologija u geometriji konstrukcija zahteva obrazovni proces sa ciljem da se ima kompletno znanje svih polja koje pripadaju ovoj oblasti. Aspekt i rezultati istraživanja pokazuju da znanje stečeno na ovaj način korišćenjem nove tehnologije razvija veštine studenata što je veoma važno u višim godinama studija, posebno u oblasti tehničkog dizajna. Interaktivna dinamična 3D geometrija ne bi mogla da se postigne tradicionalnim načinima studiranja. Upotreba moderne tehnologije bi trebalo da omogući proširenje polja istraživanja kao i očuvanje teoretskog znanja nacrtne geometrije

Cylindrical Mirror Anamorphosis and Urban-Architectural Ambience

Cylindrical mirror surfaces fall into the group of reflecting surfaces that give a distorted image of an object. However, if the object is designed according to the laws of optical geometry, in a way that its mirror image is conceived in advance, then this is anamorphosis. The objective of the present study is to emphasise the potential of cylindrical mirror anamorphosis, in the context of change in the urban-architectural ambience. In this respect it is necessary to obtain a constructive, geometrically correct solution of the 3D model of cylindrical-mirror anamorphosis, whereby the mirror surface is a vertical rotating cylinder. This topic is the primary focus of the present research. In addition, the conditions for change in the anamorphic form were analysed, and its possible functions in architecture were identified. Various examples of existing buildings with cylindrical mirror elements, in respect of which it was possible to construct and apply these types of anamorphoses, were used. The method of constructive perspective and the laws of optical geometry were applied. Analyses were made on the basis of experiments and using AutoCAD 3D methods to analyse the mirror anamorphosis of a cube and octahedron

Geometrija spiralnih površi primenjena na horizontalne komunikacije u arhitekturi

Contemporary technologies and materials continuously challenge structural designers, as well as architects, to achieve unique applications of various geometric surfaces. Twisted structures are actual and numerous in building of extremely tall towers. However, rare examples can be found as connecting elements of two buildings, i.e. horizontal communications. Authors' aim was to contribute in area of application such geometry in architecture. Term 'twist' is here related to ruled surface - elliptic hyperboloid of one sheet. The insertion of regular polygons (triangle and square) in circular sections of elliptic hyperboloid made possible 'step by step' presentation of two varieties of twisted surfaces. 3D wireframe and surface models are created in engineering software Auto CAD.Savremene tehnologije i materijali, u kontinuitetu postavljaju izazove pred projektante u smislu originalnosti primene raznovrsnih geometrijskih površi spiralne strukture (engl. 'twisted structures '), danas aktuelne i brojne u gradnji izuzetno visokih objekata. Retki su primeri primene ovakvih površi na veznim elementima između objekata - horizontalnim komunikacijama. Autori rada su želeli da daju doprinos u domenu geometrije površi koja bi imala mogućnost ovakve primene u arhitekturi. Pojam 'uvrtanja' je u ovom radu povezan sa pravoizvodnom površi - jednogranim eliptičkim hiperboloidom. Uvodeći u sisteme kružnih preseka eliptičkog hiperboloida pravilne poligone (trougao i kvadrat) autori postupno prikazuju tako nastala dva varijeteta površi. 3D žičani i površinski modeli su kreirani u inženjerskom softveru Auto CAD