conjugate hyperbola

File:Hyperbler_konjugerte.png

In geometry, a conjugate hyperbola to a given hyperbola shares the same asymptotes but lies in the opposite two sectors of the plane compared to the original hyperbola.

A hyperbola and its conjugate may be constructed as conic sections obtained from an intersecting plane that meets tangent double cones sharing the same apex. Each cone has an axis, and the plane section is parallel to the plane formed by the axes.

Using analytic geometry, the hyperbolas satisfy the symmetric equations

: $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ , with vertices (a,0) and (–a,0), and

: $\frac{x^2}{a^2} - \frac{y^2}{b^2} = -1$ (which can also be written as $\frac{y^2}{b^2} - \frac{x^2}{a^2} = 1$ ), with vertices (0,b) and (0,–b).

In case a = b they are rectangular hyperbolas, and a reflection of the plane in an asymptote exchanges the conjugates.

Similarly, for a non-zero constant c, the coordinate axes form the asymptotes of the conjugate pair $xy = c^2$ and $xy = -c^2$ .

History

File:Minkowski2.png

Apollonius of Perga introduced the conjugate hyperbola through a geometric construction: "Given two straight lines bisecting one another at any angle, to describe two hyperbolas each with two branches such that the straight lines are conjugate diameters of both hyperbolas."Thomas Heath (1896) Apollonius of Perga: Treatise on Conic Sections, pages 47, 48, 54 "The two hyperbolas so constructed are called conjugate hyperbolas, and [the] last drawn is the hyperbola conjugate to the first."

The following property was described by Apollonius: let PP', DD' be conjugate diameters of two conjugate hyperbolas, Draw the tangents at P, P', D, D'. Then ... the tangents form a parallelogram, and the diagonals of it, LM, L'M', pass through the center [C]. Also PL = PL' = P'M = P'M' = CD. It is noted that the diagonals of the parallelogram are the asymptotes common to both hyperbolas. Either PP' or DD' is a transverse diameter, with the opposite one being the conjugate diameter.

Elements of Dynamic (1878) by W. K. Clifford identifies the conjugate hyperbola.W. K. Clifford (1878) [https://archive.org/details/elementsofdynami01clifiala/page/90/mode/2up?q=%22conjugate+hyperbola%22 Elements of Dynamic], page 90, via Internet Archive

In 1894 Alexander Macfarlane used an illustration of conjugate right hyperbolas in his study "Principles of elliptic and hyperbolic analysis".Alexander Macfarlane (1894) [https://archive.org/details/principlesalgeb01macfgoog/page/n212/mode/2up Principles of Space Analysis] via Internet Archive

In 1895 W. H. Besant noted conjugate hyperbolas in his book on conic sections.W. H. Besant (1895) [https://babel.hathitrust.org/cgi/pt?id=coo.31924059322481&seq=11 Conic Sections Treated Geometrically], page 25 via HathiTrust

In his 1896 book on Appolonius cited above, Thomas Heath introduced the configuration of conjugate hyperbolas and their various diameters as follows: "the determination of the conjugate hyperbola with two branches as the complete hyperbola which has a pair of conjugate diameters common with the original hyperbola, with the difference that the secondary diameter of the original hyperbola is the transverse diameter of the conjugate hyperbola, and vice versa."Heath (1896) "Introduction to Appolonius", page lxxxv

George Salmon illustrated a conjugate hyperbola as a dotted curve in this Treatise on Conic Sections (1900).George Salmon (1900) [https://archive.org/details/dli.bengal.10689.19743/page/n181/mode/2up A Treatise on Conic Sections] via Internet Archive

In 1908 conjugate hyperbolas were used by Hermann Minkowski to demarcate units of duration and distance in a spacetime diagram illustrating a plane in his Minkowski space.{{citation|last=Minkowski|first=Hermann|author-link=Hermann Minkowski|year=1907–1908|title=Die Grundgleichungen für die elektromagnetischen Vorgänge in bewegten Körpern|trans-title=The Fundamental Equations for Electromagnetic Processes in Moving Bodies|journal=Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse|pages=53–111|title-link=s:de:Die Grundgleichungen für die elektromagnetischen Vorgänge in bewegten Körpern}}

Published translation: {{cite journal |author-last=Carus |author-first=Edward H. |title=Space and Time |journal=The Monist |year=1918 |volume=28 |issue=288 |pages=288–302 |doi=10.5840/monist19182826 |url=https://zenodo.org/records/1812623/files/article.pdf}}
Wikisource translation: The Fundamental Equations for Electromagnetic Processes in Moving Bodies
{{citation|last=Minkowski|first=Hermann|year=1908–1909|title=Raum und Zeit|trans-title=Space and Time|journal=Physikalische Zeitschrift|volume=10|pages=75–88|title-link=s:de:Raum und Zeit (Minkowski)}} Various English translations on Wikisource: Space and Time.

The principle of relativity may be stated as "Any pair of conjugate diameters of conjugate hyperbolas can be taken for the axes of space and time".{{cite book |author=Whittaker, E.T. |authorlink=E. T. Whittaker |year=1910 |edition=1 |title=A History of the Theories of Aether and Electricity |page=[https://archive.org/details/historyoftheorie00whitrich/page/441 441] |location=Dublin |publisher=Longman, Green and Co.}}

In 1957 Barry Spain illustrated conjugate rectangular hyperbolas.Barry Spain (1957) [https://babel.hathitrust.org/cgi/pt?id=mdp.39015017338461&seq=66 Analytical Conics] via HathiTrust

References

Category:Conic sections