Conic Section of a Type of Egg Curve Based Conoid
2010
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Faculty of Architecture in Belgrade; Serbian Society for Geometry and Graphics (SUGIG)
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A cubic egg curve obtained by Hügelschäffer’s construction,
can be spatially interpreted as a plane section of a type of a conoid
set through a specially chosen 4-th order intersecting curve of two
quadrics: right cylinder and cone. That implies that the apex of a
cone must lay on the axis of a cylinder in order to obtain one sheet
surface. This type of conoid will be of 4-th order, and will exclude
plane sections by conics. We consider a special case of forming an akin conoid that would include also conic sections. If the apex of the cone is set off the cylinder axis, there would appear a double conoid, as a surface set through the intersection curve of the quadrics. Its plane section will be a double egg curve obtained by generalized Hügelschäffer’s construction. In case that cylinder and cone would intersect by a degenerated 4-th degree space curve on two conics (circle and ellipse), there would emerge double egg curve, as a plane section of the double conoid. The curve degene...rates onto ellipse and a quartic curve - Granville’s egg. We also gave a mathematical condition of degeneration of the base double egg curve.
Keywords:
conoid / Hügelschäffer’s construction / ellipse / egg curveSource:
PROCEEDINGS | BILTEN of 25th National and 2nd International Scientific Conference moNGeometrija 2010, 2010, 447-466Publisher:
- Belgrade: Faculty of Architecture in Belgrade; Serbian Society for Geometry and Graphics (SUGIG)
Funding / projects:
- Primena rezultata naprednog razvoja prostornih struktura u oblasti 3D transformacija, konstruisanja, novih materijala - Simprolita i tehnologija (RS-MESTD-MPN2006-2010-16009)
Note:
- http://mongeometrija.com/media/mongeometrija/2010/moNGeometrija%202010%20-%20PAGINACIJA.pdf
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GraFarTY - CONF AU - Obradović, Marija AU - Malešević, Branko AU - Petrović, Maja PY - 2010 UR - https://grafar.grf.bg.ac.rs/handle/123456789/2052 AB - A cubic egg curve obtained by Hügelschäffer’s construction, can be spatially interpreted as a plane section of a type of a conoid set through a specially chosen 4-th order intersecting curve of two quadrics: right cylinder and cone. That implies that the apex of a cone must lay on the axis of a cylinder in order to obtain one sheet surface. This type of conoid will be of 4-th order, and will exclude plane sections by conics. We consider a special case of forming an akin conoid that would include also conic sections. If the apex of the cone is set off the cylinder axis, there would appear a double conoid, as a surface set through the intersection curve of the quadrics. Its plane section will be a double egg curve obtained by generalized Hügelschäffer’s construction. In case that cylinder and cone would intersect by a degenerated 4-th degree space curve on two conics (circle and ellipse), there would emerge double egg curve, as a plane section of the double conoid. The curve degenerates onto ellipse and a quartic curve - Granville’s egg. We also gave a mathematical condition of degeneration of the base double egg curve. PB - Belgrade: Faculty of Architecture in Belgrade; Serbian Society for Geometry and Graphics (SUGIG) C3 - PROCEEDINGS | BILTEN of 25th National and 2nd International Scientific Conference moNGeometrija 2010 T1 - Conic Section of a Type of Egg Curve Based Conoid EP - 466 SP - 447 UR - https://hdl.handle.net/21.15107/rcub_grafar_2052 ER -
@conference{ author = "Obradović, Marija and Malešević, Branko and Petrović, Maja", year = "2010", abstract = "A cubic egg curve obtained by Hügelschäffer’s construction, can be spatially interpreted as a plane section of a type of a conoid set through a specially chosen 4-th order intersecting curve of two quadrics: right cylinder and cone. That implies that the apex of a cone must lay on the axis of a cylinder in order to obtain one sheet surface. This type of conoid will be of 4-th order, and will exclude plane sections by conics. We consider a special case of forming an akin conoid that would include also conic sections. If the apex of the cone is set off the cylinder axis, there would appear a double conoid, as a surface set through the intersection curve of the quadrics. Its plane section will be a double egg curve obtained by generalized Hügelschäffer’s construction. In case that cylinder and cone would intersect by a degenerated 4-th degree space curve on two conics (circle and ellipse), there would emerge double egg curve, as a plane section of the double conoid. The curve degenerates onto ellipse and a quartic curve - Granville’s egg. We also gave a mathematical condition of degeneration of the base double egg curve.", publisher = "Belgrade: Faculty of Architecture in Belgrade; Serbian Society for Geometry and Graphics (SUGIG)", journal = "PROCEEDINGS | BILTEN of 25th National and 2nd International Scientific Conference moNGeometrija 2010", title = "Conic Section of a Type of Egg Curve Based Conoid", pages = "466-447", url = "https://hdl.handle.net/21.15107/rcub_grafar_2052" }
Obradović, M., Malešević, B.,& Petrović, M.. (2010). Conic Section of a Type of Egg Curve Based Conoid. in PROCEEDINGS | BILTEN of 25th National and 2nd International Scientific Conference moNGeometrija 2010 Belgrade: Faculty of Architecture in Belgrade; Serbian Society for Geometry and Graphics (SUGIG)., 447-466. https://hdl.handle.net/21.15107/rcub_grafar_2052
Obradović M, Malešević B, Petrović M. Conic Section of a Type of Egg Curve Based Conoid. in PROCEEDINGS | BILTEN of 25th National and 2nd International Scientific Conference moNGeometrija 2010. 2010;:447-466. https://hdl.handle.net/21.15107/rcub_grafar_2052 .
Obradović, Marija, Malešević, Branko, Petrović, Maja, "Conic Section of a Type of Egg Curve Based Conoid" in PROCEEDINGS | BILTEN of 25th National and 2nd International Scientific Conference moNGeometrija 2010 (2010):447-466, https://hdl.handle.net/21.15107/rcub_grafar_2052 .