Ellipse Hyperbola and Their Conjunction

TL;DR

This paper explores the geometric relationship between ellipses and hyperbolas, showing how the vertices and foci of cones generating these conics are interconnected through simple geometric analysis.

Contribution

It reveals a novel geometric conjunction between ellipses and hyperbolas based on cone sections, highlighting their interconnected properties.

Findings

Locus of cone vertices for an ellipse forms a hyperbola.

Locus of cone vertices for a hyperbola forms an ellipse.

Foci of the ellipse coincide with hyperbola vertices and vice versa.

Abstract

The article presents 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 vertexes of the cones is a hyperbola. The hyperbola has focuses which coincidence with the ellipse vertexes. Similarly, if the given curve is the hyperbola, the locus of vertex of the cones is the ellipse. In the second case, the focuses of the ellipse are located in the hyperbola's vertexes. These two relationships create 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 perfect example of mathematical beauty which may be shown by use of very simple geometry. As in the past the conic curves appear to be very interesting and fruitful mathematical beings.