TY  - JOUR
AU  - Mišić, Slobodan
AU  - Obradović, Marija
AU  - Đukanović, Gordana
PY  - 2015
UR  - https://grafar.grf.bg.ac.rs/handle/123456789/1036
AB  - In this paper, the geometry of concave cupolae has been the starting
point for the generation of composite polyhedral structures, usable as formative
patterns for architectural purposes. Obtained by linking paper folding geometry
with the geometry of polyhedra, concave cupolae are polyhedra that follow the
method of generating cupolae (Johnson’s solids: J3, J4 and J5); but we removed
the convexity criterion and omitted squares in the lateral surface. Instead of alter-
nating triangles and squares there are now two or more paired series of equilateral
triangles. The criterion of face regularity is respected, as well as the criterion of
multiple axial symmetry. The distribution of the triangles is based on strictly
determined and mathematically defined parameters, which allows the creation of
such structures in a way that qualifies them as an autonomous group of polyhedra
— concave cupolae of sorts II, IV, VI (2N). If we want to see these structures as
polyhedral surfaces (not as solids) connecting the concept of the cupola (dome) in
the architectural sense with the geometrical meaning of (concave) cupola, we re-
move the faces of the base polygons. Thus we get a deltahedral structure — a shell
made entirely from equilateral triangles, which is advantageous for the purpose
of prefabrication. Due to the congruence of the major 2n-sided bases of concave
cupolae of sort II with the minor bases of the corresponding concave cupolae of
sort IV, it is possible to combine these polyhedra in composite polyhedra. But
also their elongation with concave antiprisms of sort II or the augmentation with
concave pyramids of sort II could be performed. Based on the foregoing, we exam-
ine the possibilities of combining the considered polyhedra into unified composite
structures.
T2  - Journal for Geometry and Graphics
T1  - Composite Concave Cupolae as Geometric and Architectural Forms
EP  - 91
EP  - 
EP  - 
IS  - 1
SP  - 79
VL  - 19
UR  - https://hdl.handle.net/21.15107/rcub_grafar_1036
ER  - 

@article{
author = "Mišić, Slobodan and Obradović, Marija and Đukanović, Gordana",
year = "2015",
abstract = "In this paper, the geometry of concave cupolae has been the starting
point for the generation of composite polyhedral structures, usable as formative
patterns for architectural purposes. Obtained by linking paper folding geometry
with the geometry of polyhedra, concave cupolae are polyhedra that follow the
method of generating cupolae (Johnson’s solids: J3, J4 and J5); but we removed
the convexity criterion and omitted squares in the lateral surface. Instead of alter-
nating triangles and squares there are now two or more paired series of equilateral
triangles. The criterion of face regularity is respected, as well as the criterion of
multiple axial symmetry. The distribution of the triangles is based on strictly
determined and mathematically defined parameters, which allows the creation of
such structures in a way that qualifies them as an autonomous group of polyhedra
— concave cupolae of sorts II, IV, VI (2N). If we want to see these structures as
polyhedral surfaces (not as solids) connecting the concept of the cupola (dome) in
the architectural sense with the geometrical meaning of (concave) cupola, we re-
move the faces of the base polygons. Thus we get a deltahedral structure — a shell
made entirely from equilateral triangles, which is advantageous for the purpose
of prefabrication. Due to the congruence of the major 2n-sided bases of concave
cupolae of sort II with the minor bases of the corresponding concave cupolae of
sort IV, it is possible to combine these polyhedra in composite polyhedra. But
also their elongation with concave antiprisms of sort II or the augmentation with
concave pyramids of sort II could be performed. Based on the foregoing, we exam-
ine the possibilities of combining the considered polyhedra into unified composite
structures.",
journal = "Journal for Geometry and Graphics",
title = "Composite Concave Cupolae as Geometric and Architectural Forms",
pages = "91---79",
number = "1",
volume = "19",
url = "https://hdl.handle.net/21.15107/rcub_grafar_1036"
}

Mišić, S., Obradović, M.,& Đukanović, G.. (2015). Composite Concave Cupolae as Geometric and Architectural Forms. in Journal for Geometry and Graphics, 19(1), 79-91.
https://hdl.handle.net/21.15107/rcub_grafar_1036

Mišić S, Obradović M, Đukanović G. Composite Concave Cupolae as Geometric and Architectural Forms. in Journal for Geometry and Graphics. 2015;19(1):79-91.
https://hdl.handle.net/21.15107/rcub_grafar_1036 .

Mišić, Slobodan, Obradović, Marija, Đukanović, Gordana, "Composite Concave Cupolae as Geometric and Architectural Forms" in Journal for Geometry and Graphics, 19, no. 1 (2015):79-91,
https://hdl.handle.net/21.15107/rcub_grafar_1036 .

TY  - CONF
AU  - Obradović, Marija
AU  - Malešević, Branko
AU  - Petrović, Maja
AU  - Đukanović, Gordana
PY  - 2013
UR  - https://grafar.grf.bg.ac.rs/handle/123456789/1983
AB  - The starting constructions is the well known ellipse construction using the concentric circles c1 and c2. By eccentricity of the center C2 for some value w, and for the preserved center of the transformation in the center C1, the degree of the obtained curve rises from two to three, as done by mathematician Fritz Hügelschäffer. If we also displace the center of the transformation from C1, we obtain a variety of higher order curves using the same principle in the constructive procedure.
PB  - Timişoara: Editura Politehnica
C3  - Scientific Bulletin of the "POLITEHNICA" University of Timişoara, Romania TRANSACTIONS on HYDROTECHNICS
T1  - Generating Curves of Higher Order Using the Generalisation of Hügelschäffer’ Egg Curve construction
EP  - 114
IS  - 1
SP  - 110
VL  - 58(72)
UR  - https://hdl.handle.net/21.15107/rcub_grafar_1983
ER  - 

@conference{
author = "Obradović, Marija and Malešević, Branko and Petrović, Maja and Đukanović, Gordana",
year = "2013",
abstract = "The starting constructions is the well known ellipse construction using the concentric circles c1 and c2. By eccentricity of the center C2 for some value w, and for the preserved center of the transformation in the center C1, the degree of the obtained curve rises from two to three, as done by mathematician Fritz Hügelschäffer. If we also displace the center of the transformation from C1, we obtain a variety of higher order curves using the same principle in the constructive procedure.",
publisher = "Timişoara: Editura Politehnica",
journal = "Scientific Bulletin of the "POLITEHNICA" University of Timişoara, Romania TRANSACTIONS on HYDROTECHNICS",
title = "Generating Curves of Higher Order Using the Generalisation of Hügelschäffer’ Egg Curve construction",
pages = "114-110",
number = "1",
volume = "58(72)",
url = "https://hdl.handle.net/21.15107/rcub_grafar_1983"
}

Obradović, M., Malešević, B., Petrović, M.,& Đukanović, G.. (2013). Generating Curves of Higher Order Using the Generalisation of Hügelschäffer’ Egg Curve construction. in Scientific Bulletin of the "POLITEHNICA" University of Timişoara, Romania TRANSACTIONS on HYDROTECHNICS
Timişoara: Editura Politehnica., 58(72)(1), 110-114.
https://hdl.handle.net/21.15107/rcub_grafar_1983

Obradović M, Malešević B, Petrović M, Đukanović G. Generating Curves of Higher Order Using the Generalisation of Hügelschäffer’ Egg Curve construction. in Scientific Bulletin of the "POLITEHNICA" University of Timişoara, Romania TRANSACTIONS on HYDROTECHNICS. 2013;58(72)(1):110-114.
https://hdl.handle.net/21.15107/rcub_grafar_1983 .

Obradović, Marija, Malešević, Branko, Petrović, Maja, Đukanović, Gordana, "Generating Curves of Higher Order Using the Generalisation of Hügelschäffer’ Egg Curve construction" in Scientific Bulletin of the "POLITEHNICA" University of Timişoara, Romania TRANSACTIONS on HYDROTECHNICS, 58(72), no. 1 (2013):110-114,
https://hdl.handle.net/21.15107/rcub_grafar_1983 .

TY  - JOUR
AU  - Đukanović, Gordana
AU  - Obradović, Marija
PY  - 2012
UR  - https://grafar.grf.bg.ac.rs/handle/123456789/462
AB  - This paper shows the process of inverting the 4th ordered space curve of the first category with a self-intersecting point (with two planes of symmetry) and determining its harmonic equivalent. There are harmonic equivalents for five groups of surfaces obtained through the 4th order space curve of the 1st category. Mapping was done through a system of circular cross-sections. Both classical and relativistic geometry interpretations are presented. We also designed spatial models - a spatial model of the pencil of quadrics and a spatial model of the pencil of equivalent quadrics. Besides the boundary surfaces, one surface of the 3rd order, which is an equivalent to a triaxial ellipsoid, passes through this pencil of surface of the 4th order. The center of inversion is located on the contour of the ellipsoid. The parabolic cylinder is mapped into its equivalent, by mapping the contour parabola of the cylinder, in the frontal projection, in relation to the center and the sphere of inversion into a contour curve of the 4th order surface. The generating lines of the parabolic cylinder, which are in a projecting position and pass through the antipode, are mapped into circles (also in a projecting position) whose diameters are from the center of inversion to the contour line. The application of the 4th order surfaces in architectural practice is also presented.
AB  - U radu je inverzijom preslikana prostorna kriva 4. reda prve vrste sa samopresečnom tačkom (sa dve ravni simetrije) i određen je njen harmonijski ekvivalent. Prikazani su harmonijski ekvivalenti za pet grupa površi koje su dobijene kroz prostornu krivu 4 reda 1 vrste. Preslikavanje je rađeno preko sistema kružnih preseka. Dato je klasično i tumačenje u relativističkooj geometriji. Takođe su urađeni i prostorni modeli - prostorni model pramena kvadrika i pramena ekvivalentnih kvadrika. Kroz ovaj pramen površi 4. reda, osim graničnih površi, prolazi i jedna površ 3. reda koja je ekvivalent troosnom elipsoidu. Centar inverzije nalazi se na konturi elipsoida. Parabolički cilindar se preslikava u svoj ekvivalent, tako što se konturna parabola cilindra, za drugu projekciju, preslika u odnosu na centar i sferu inverzije u konturnu krivu površi 4. reda. Izvodnice paraboličkog cilindra, koje su u projicirajućem položaju i prolaze kroz antipod, preslikavaju se u krugove (takođe u projicirajućem položaju) čiji su prečnici od centra inverzije do konturne linije. Prikazana je i primena površi 4. reda u arhitektonskoj praksi.
PB  - Univerzitet u Nišu, Niš
T2  - Facta universitatis - series: Architecture and Civil Engineering
T1  - The pencil of the 4th and 3rd order surfaces obtained as a harmonic equivalent of the pencil of quadrics through a 4th order space curve of the 1st category
T1  - Pramen površi 4. i 3. reda dobijen kao harmonijski ekvivalent pramena kvadrika kroz prostornu krivu 4. reda 1. vrste
EP  - 207
IS  - 2
SP  - 193
VL  - 10
DO  - 10.2298/FUACE1202193D
ER  - 

@article{
author = "Đukanović, Gordana and Obradović, Marija",
year = "2012",
abstract = "This paper shows the process of inverting the 4th ordered space curve of the first category with a self-intersecting point (with two planes of symmetry) and determining its harmonic equivalent. There are harmonic equivalents for five groups of surfaces obtained through the 4th order space curve of the 1st category. Mapping was done through a system of circular cross-sections. Both classical and relativistic geometry interpretations are presented. We also designed spatial models - a spatial model of the pencil of quadrics and a spatial model of the pencil of equivalent quadrics. Besides the boundary surfaces, one surface of the 3rd order, which is an equivalent to a triaxial ellipsoid, passes through this pencil of surface of the 4th order. The center of inversion is located on the contour of the ellipsoid. The parabolic cylinder is mapped into its equivalent, by mapping the contour parabola of the cylinder, in the frontal projection, in relation to the center and the sphere of inversion into a contour curve of the 4th order surface. The generating lines of the parabolic cylinder, which are in a projecting position and pass through the antipode, are mapped into circles (also in a projecting position) whose diameters are from the center of inversion to the contour line. The application of the 4th order surfaces in architectural practice is also presented., U radu je inverzijom preslikana prostorna kriva 4. reda prve vrste sa samopresečnom tačkom (sa dve ravni simetrije) i određen je njen harmonijski ekvivalent. Prikazani su harmonijski ekvivalenti za pet grupa površi koje su dobijene kroz prostornu krivu 4 reda 1 vrste. Preslikavanje je rađeno preko sistema kružnih preseka. Dato je klasično i tumačenje u relativističkooj geometriji. Takođe su urađeni i prostorni modeli - prostorni model pramena kvadrika i pramena ekvivalentnih kvadrika. Kroz ovaj pramen površi 4. reda, osim graničnih površi, prolazi i jedna površ 3. reda koja je ekvivalent troosnom elipsoidu. Centar inverzije nalazi se na konturi elipsoida. Parabolički cilindar se preslikava u svoj ekvivalent, tako što se konturna parabola cilindra, za drugu projekciju, preslika u odnosu na centar i sferu inverzije u konturnu krivu površi 4. reda. Izvodnice paraboličkog cilindra, koje su u projicirajućem položaju i prolaze kroz antipod, preslikavaju se u krugove (takođe u projicirajućem položaju) čiji su prečnici od centra inverzije do konturne linije. Prikazana je i primena površi 4. reda u arhitektonskoj praksi.",
publisher = "Univerzitet u Nišu, Niš",
journal = "Facta universitatis - series: Architecture and Civil Engineering",
title = "The pencil of the 4th and 3rd order surfaces obtained as a harmonic equivalent of the pencil of quadrics through a 4th order space curve of the 1st category, Pramen površi 4. i 3. reda dobijen kao harmonijski ekvivalent pramena kvadrika kroz prostornu krivu 4. reda 1. vrste",
pages = "207-193",
number = "2",
volume = "10",
doi = "10.2298/FUACE1202193D"
}

Đukanović, G.,& Obradović, M.. (2012). The pencil of the 4th and 3rd order surfaces obtained as a harmonic equivalent of the pencil of quadrics through a 4th order space curve of the 1st category. in Facta universitatis - series: Architecture and Civil Engineering
Univerzitet u Nišu, Niš., 10(2), 193-207.
https://doi.org/10.2298/FUACE1202193D

Đukanović G, Obradović M. The pencil of the 4th and 3rd order surfaces obtained as a harmonic equivalent of the pencil of quadrics through a 4th order space curve of the 1st category. in Facta universitatis - series: Architecture and Civil Engineering. 2012;10(2):193-207.
doi:10.2298/FUACE1202193D .

Đukanović, Gordana, Obradović, Marija, "The pencil of the 4th and 3rd order surfaces obtained as a harmonic equivalent of the pencil of quadrics through a 4th order space curve of the 1st category" in Facta universitatis - series: Architecture and Civil Engineering, 10, no. 2 (2012):193-207,
https://doi.org/10.2298/FUACE1202193D . .