TY  - CONF
AU  - Devetaković, Mirjana
AU  - Djordjević, Djordje
AU  - Popović, Nikola
AU  - Mitrović, Djordje
AU  - Djukanović, Gordana
AU  - Mišić, Slobodan
PY  - 2023
UR  - https://raf.arh.bg.ac.rs/handle/123456789/1728
AB  - The magic world of mirrors, their application, integration in interiors and exposure in exteriors, intrigues architectural designers and artists for a long time. Inspired with wide recent usages of mirrors in innovative architectural designs practice, as well as in arts, and grounded our understanding on well-known geometric/optic principles of mirroring, the authors conceptualized a Workshop that was offered to students of architecture, forestry and applied arts, as an extracurricular two-weeks activity. After briefly introducing a historical/chronological overview of mirrors application in architectural design practice and applied art, relevant theoretical aspects of mirroring were demonstrated on the Workshop. The participating students were offered two possibilities for integration of mirrors in their design-outcomes: to experiment with analogue models and digital/computing ones. For that reason, a large amount of small glass mirror parts was provided. Also, functional parametric families of digital mirrors (planar, concave and convex) were developed especially for the workshop and offered on-line. Particular foci of this paper are on both: the teaching methodology that was exclusively prepared for the Workshop, and specific usage of experimentation (to trigger unusual ideas and to utilize relevant familiar methods and procedural activities in the new – this context). In the discussion part of this study, a systematized overview of the Workshop results and its analysis from geometric and semantic point of view are given. The results are divided into: first interim critique, second interim critique and final submission. Through a generalization of conclusions, the lessons learned, namely, the level of subject-related geometric knowledge acquired from this unique teaching experience are reviewed, and an analysis of their possible applications in other pedagogic contexts related to geometry-teaching is presented.
PB  - Novi Sad : Faculty of Technical Sciences, University of Novi Sad
PB  - Belgrade : Serbian Society for Geometry and Graphics (SUGIG)
C3  - Proceedings / The 9th International Scientific Conference on Geometry, Graphics and Design in the Digital Age, MoNGeomatrija 2023, June 7-10, 2023, Novi Sad, Serbia
T1  - On the Other Side of Mirror – A Workshop on Incorporating Geometry of Mirroring in Architectural Practice and Applied Arts
SP  - 309
EP  - 322
UR  - https://hdl.handle.net/21.15107/rcub_raf_1728
ER  - 

@conference{
author = "Devetaković, Mirjana and Djordjević, Djordje and Popović, Nikola and Mitrović, Djordje and Djukanović, Gordana and Mišić, Slobodan",
year = "2023",
abstract = "The magic world of mirrors, their application, integration in interiors and exposure in exteriors, intrigues architectural designers and artists for a long time. Inspired with wide recent usages of mirrors in innovative architectural designs practice, as well as in arts, and grounded our understanding on well-known geometric/optic principles of mirroring, the authors conceptualized a Workshop that was offered to students of architecture, forestry and applied arts, as an extracurricular two-weeks activity. After briefly introducing a historical/chronological overview of mirrors application in architectural design practice and applied art, relevant theoretical aspects of mirroring were demonstrated on the Workshop. The participating students were offered two possibilities for integration of mirrors in their design-outcomes: to experiment with analogue models and digital/computing ones. For that reason, a large amount of small glass mirror parts was provided. Also, functional parametric families of digital mirrors (planar, concave and convex) were developed especially for the workshop and offered on-line. Particular foci of this paper are on both: the teaching methodology that was exclusively prepared for the Workshop, and specific usage of experimentation (to trigger unusual ideas and to utilize relevant familiar methods and procedural activities in the new – this context). In the discussion part of this study, a systematized overview of the Workshop results and its analysis from geometric and semantic point of view are given. The results are divided into: first interim critique, second interim critique and final submission. Through a generalization of conclusions, the lessons learned, namely, the level of subject-related geometric knowledge acquired from this unique teaching experience are reviewed, and an analysis of their possible applications in other pedagogic contexts related to geometry-teaching is presented.",
publisher = "Novi Sad : Faculty of Technical Sciences, University of Novi Sad, Belgrade : Serbian Society for Geometry and Graphics (SUGIG)",
journal = "Proceedings / The 9th International Scientific Conference on Geometry, Graphics and Design in the Digital Age, MoNGeomatrija 2023, June 7-10, 2023, Novi Sad, Serbia",
title = "On the Other Side of Mirror – A Workshop on Incorporating Geometry of Mirroring in Architectural Practice and Applied Arts",
pages = "309-322",
url = "https://hdl.handle.net/21.15107/rcub_raf_1728"
}

Devetaković, M., Djordjević, D., Popović, N., Mitrović, D., Djukanović, G.,& Mišić, S.. (2023). On the Other Side of Mirror – A Workshop on Incorporating Geometry of Mirroring in Architectural Practice and Applied Arts. in Proceedings / The 9th International Scientific Conference on Geometry, Graphics and Design in the Digital Age, MoNGeomatrija 2023, June 7-10, 2023, Novi Sad, Serbia
Novi Sad : Faculty of Technical Sciences, University of Novi Sad., 309-322.
https://hdl.handle.net/21.15107/rcub_raf_1728

Devetaković M, Djordjević D, Popović N, Mitrović D, Djukanović G, Mišić S. On the Other Side of Mirror – A Workshop on Incorporating Geometry of Mirroring in Architectural Practice and Applied Arts. in Proceedings / The 9th International Scientific Conference on Geometry, Graphics and Design in the Digital Age, MoNGeomatrija 2023, June 7-10, 2023, Novi Sad, Serbia. 2023;:309-322.
https://hdl.handle.net/21.15107/rcub_raf_1728 .

Devetaković, Mirjana, Djordjević, Djordje, Popović, Nikola, Mitrović, Djordje, Djukanović, Gordana, Mišić, Slobodan, "On the Other Side of Mirror – A Workshop on Incorporating Geometry of Mirroring in Architectural Practice and Applied Arts" in Proceedings / The 9th International Scientific Conference on Geometry, Graphics and Design in the Digital Age, MoNGeomatrija 2023, June 7-10, 2023, Novi Sad, Serbia (2023):309-322,
https://hdl.handle.net/21.15107/rcub_raf_1728 .

TY  - CONF
AU  - Djukanovic, Gordana
AU  - Djordjevic, Djordje
AU  - Devetakovic, Mirjana
AU  - Mitrovic, Djordje
PY  - 2023
UR  - https://raf.arh.bg.ac.rs/handle/123456789/1729
AB  - In this paper, an elliptical pencil of circles is mapped by homology (perspective collineation) into a parabolic-parabolic pencil of hyperbolas because it has two by two common points 1 = 2 and 3 = 4 in infinity, i.e., four overlapping points in the antipode. The mapping also includes the mapping of degenerated conics decomposed into corresponding pairs of straight lines. The elliptical pencil of circles (the ellipses on plane 1) is mapped into a hyperbolic pencil (on plane 2) by perspective collineation for pole S1. The vanishing line v1 intersects all the circles of the pencil so that by homology all the circles are mapped into hyperbolas. To apply supersymmetry to the obtained pencils, the angle between plane 1 and plane 2 must be 45°. The pencil of hyperbolas is mapped by supersymmetry into a pencil of higher-order curves. The obtained pencils are further mapped by inversion and then again by supersymmetry to obtain higher-order curves rich in different shapes. In the second example, the elliptical pencil of circles is placed to the vanishing line v1 so that by homology, it is mapped into a pencil of conics containing an ellipse (circle 1 does not intersect the vanishing line), a parabola (circle 2 touches the vanishing line), a hyperbola (circle 3 intersects the vanishing line). The pencil of conics is elliptical-parabolic because it has two real and separate points and two infinite, i.e., two overlapping points in the antipode. This pencil of conic is also mapped by supersymmetry into a pencil of higher-order curves that intersect at the same number and type of base points as the starting pencil. Higher order curves obtained by mapping are rational line curves.
PB  - Novi Sad : Faculty of Technical Sciences, University of Novi Sad
PB  - Belgrade : Serbian Society for Geometry and Graphics (SUGIG)
C3  - Proceedings / The 9th International Scientific Conference on Geometry, Graphics and Design in the Digital Age, MoNGeomatrija 2023, June 7-10, 2023, Novi Sad, Serbia
T1  - Transformation of Pencils of Circles into Pencils of Conics and These into Pencils of Higher-Order Curves
SP  - 133
EP  - 141
UR  - https://hdl.handle.net/21.15107/rcub_raf_1729
ER  - 

@conference{
author = "Djukanovic, Gordana and Djordjevic, Djordje and Devetakovic, Mirjana and Mitrovic, Djordje",
year = "2023",
abstract = "In this paper, an elliptical pencil of circles is mapped by homology (perspective collineation) into a parabolic-parabolic pencil of hyperbolas because it has two by two common points 1 = 2 and 3 = 4 in infinity, i.e., four overlapping points in the antipode. The mapping also includes the mapping of degenerated conics decomposed into corresponding pairs of straight lines. The elliptical pencil of circles (the ellipses on plane 1) is mapped into a hyperbolic pencil (on plane 2) by perspective collineation for pole S1. The vanishing line v1 intersects all the circles of the pencil so that by homology all the circles are mapped into hyperbolas. To apply supersymmetry to the obtained pencils, the angle between plane 1 and plane 2 must be 45°. The pencil of hyperbolas is mapped by supersymmetry into a pencil of higher-order curves. The obtained pencils are further mapped by inversion and then again by supersymmetry to obtain higher-order curves rich in different shapes. In the second example, the elliptical pencil of circles is placed to the vanishing line v1 so that by homology, it is mapped into a pencil of conics containing an ellipse (circle 1 does not intersect the vanishing line), a parabola (circle 2 touches the vanishing line), a hyperbola (circle 3 intersects the vanishing line). The pencil of conics is elliptical-parabolic because it has two real and separate points and two infinite, i.e., two overlapping points in the antipode. This pencil of conic is also mapped by supersymmetry into a pencil of higher-order curves that intersect at the same number and type of base points as the starting pencil. Higher order curves obtained by mapping are rational line curves.",
publisher = "Novi Sad : Faculty of Technical Sciences, University of Novi Sad, Belgrade : Serbian Society for Geometry and Graphics (SUGIG)",
journal = "Proceedings / The 9th International Scientific Conference on Geometry, Graphics and Design in the Digital Age, MoNGeomatrija 2023, June 7-10, 2023, Novi Sad, Serbia",
title = "Transformation of Pencils of Circles into Pencils of Conics and These into Pencils of Higher-Order Curves",
pages = "133-141",
url = "https://hdl.handle.net/21.15107/rcub_raf_1729"
}

Djukanovic, G., Djordjevic, D., Devetakovic, M.,& Mitrovic, D.. (2023). Transformation of Pencils of Circles into Pencils of Conics and These into Pencils of Higher-Order Curves. in Proceedings / The 9th International Scientific Conference on Geometry, Graphics and Design in the Digital Age, MoNGeomatrija 2023, June 7-10, 2023, Novi Sad, Serbia
Novi Sad : Faculty of Technical Sciences, University of Novi Sad., 133-141.
https://hdl.handle.net/21.15107/rcub_raf_1729

Djukanovic G, Djordjevic D, Devetakovic M, Mitrovic D. Transformation of Pencils of Circles into Pencils of Conics and These into Pencils of Higher-Order Curves. in Proceedings / The 9th International Scientific Conference on Geometry, Graphics and Design in the Digital Age, MoNGeomatrija 2023, June 7-10, 2023, Novi Sad, Serbia. 2023;:133-141.
https://hdl.handle.net/21.15107/rcub_raf_1729 .

Djukanovic, Gordana, Djordjevic, Djordje, Devetakovic, Mirjana, Mitrovic, Djordje, "Transformation of Pencils of Circles into Pencils of Conics and These into Pencils of Higher-Order Curves" in Proceedings / The 9th International Scientific Conference on Geometry, Graphics and Design in the Digital Age, MoNGeomatrija 2023, June 7-10, 2023, Novi Sad, Serbia (2023):133-141,
https://hdl.handle.net/21.15107/rcub_raf_1729 .