<abstract xmlns="http://www.w3.org/1999/xhtml"><p>This article presents a simple analysis of cones which are used to generate a given conic curve by section by a plane. It was found that if the given curve is an ellipse, then the locus of vertices of the cones is a hyperbola. The hyperbola has foci which coincidence with the ellipse vertices. Similarly, if the given curve is the hyperbola, the locus of vertex of the cones is the ellipse. In the second case, the foci of the ellipse are located in the hyperbola’s vertices. These two relationships create a kind of conjunction between the ellipse and the hyperbola which originate from the cones used for generation of these curves. The presented conjunction of the ellipse and hyperbola is a perfect example of mathematical beauty which may be shown by the use of very simple geometry. As in the past the conic curves appear to be very interesting and fruitful mathematical beings.</p></abstract>

This article presents a simple analysis of cones which are used to generate a given conic curve by section by a plane. It was found that if the given curve is an ellipse, then the locus of vertices of the cones is a hyperbola. The hyperbola has foci which coincidence with the ellipse vertices. Similarly, if the given curve is the hyperbola, the locus of vertex of the cones is the ellipse. In the second case, the foci of the ellipse are located in the hyperbola’s vertices. These two relationships create a kind of conjunction between the ellipse and the hyperbola which originate from the cones used for generation of these curves. The presented conjunction of the ellipse and hyperbola is a perfect example of mathematical beauty which may be shown by the use of very simple geometry. As in the past the conic curves appear to be very interesting and fruitful mathematical beings.

Ellipse, hyperbola and their conjunction

Recreational Mathematics Magazine

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

{"article-title":"Ellipse, hyperbola and their conjunction"}

This article presents a simple analysis of cones which are used to generate a given conic curve by section by a plane. It was found that if the given curve is...

Ellipse, hyperbola and their conjunction

Pubblicato online: 31 dic 2018

Pagine: 29 - 38

DOI: https://doi.org/10.2478/rmm-2018-0006

Parole chiave
Geometry, conics, ellipse, hyperbola

© 2018 Arkadiusz Kobiera, published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

Ellipse, hyperbola and their conjunction

Pubblicato online: 31 dic 2018

Pagine: 29 - 38

DOI: https://doi.org/10.2478/rmm-2018-0006

Parole chiaveGeometry, conics, ellipse, hyperbola

© 2018 Arkadiusz Kobiera, published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

Parole chiave
Geometry, conics, ellipse, hyperbola