geometry Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...

, a conjugate hyperbola to a given

hyperbola In mathematics, a hyperbola is a type of smooth function, smooth plane curve, curve lying in a plane, defined by its geometric properties or by equations for which it is the solution set. A hyperbola has two pieces, called connected component ( ...

shares the same

asymptote In analytic geometry, an asymptote () of a curve is a line such that the distance between the curve and the line approaches zero as one or both of the ''x'' or ''y'' coordinates tends to infinity. In projective geometry and related contexts, ...

s 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 section A conic section, conic or a quadratic curve is a curve obtained from a cone's surface intersecting a plane. The three types of conic section are the hyperbola, the parabola, and the ellipse; the circle is a special case of the ellipse, tho ...

s obtained from an intersecting plane that meets tangent double cones sharing the same

apex The apex is the highest point of something. The word may also refer to: Arts and media Fictional entities * Apex (comics) A-Bomb Abomination Absorbing Man Abraxas Abyss Abyss is the name of two characters appearing in Ameri ...

. Each cone has an axis, and the plane section is parallel to the plane formed by the axes. Using

analytic geometry In mathematics, analytic geometry, also known as coordinate geometry or Cartesian geometry, is the study of geometry using a coordinate system. This contrasts with synthetic geometry. Analytic geometry is used in physics and engineering, and als ...

, the hyperbolas satisfy the symmetric equations :

\frac - \frac = 1

, with vertices (''a'',0) and (–''a'',0), and :

\frac - \frac = -1

(which can also be written as

\frac - \frac = 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 In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine and standardize the position of the points or other geometric elements on a manifold such as Euclidean space. The coordinates are ...

form the asymptotes of the conjugate pair

xy = c^2

and

xy = -c^2

History

Apollonius of Perga Apollonius of Perga ( ; ) was an ancient Greek geometer and astronomer known for his work on conic sections. Beginning from the earlier contributions of Euclid and Archimedes on the topic, he brought them to the state prior to the invention o ...

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 In geometry, two diameters of a conic section are said to be conjugate if each chord (geometry), chord parallel (geometry), parallel to one diameter is bisection, bisected by the other diameter. For example, two diameters of a circle are conjugate ...

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 helast 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 In Euclidean geometry, a parallelogram is a simple polygon, simple (non-list of self-intersecting polygons, self-intersecting) quadrilateral with two pairs of Parallel (geometry), parallel sides. The opposite or facing sides of a parallelogram a ...

, and the diagonals of it, LM, L'M', pass through the center 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 ''Elements of Dynamic'' is a book published by William Kingdon Clifford in 1878. In 1887 it was supplemented by a fourth part and an appendix. The subtitle is "An introduction to motion and rest in solid and fluid bodies". It was reviewed positiv ...

'' (1878) by W. K. Clifford identifies the conjugate hyperbola. In 1894

Alexander Macfarlane Alexander Macfarlane FRSE LLD (21 April 1851 – 28 August 1913) was a Scottish logician, physicist, and mathematician. Life Macfarlane was born in Blairgowrie, Scotland, to Daniel MacFarlane (Shoemaker, Blairgowrie) and Ann Small. He s ...

used an illustration of conjugate right hyperbolas in his study "Principles of elliptic and hyperbolic analysis". In 1895 W. H. Besant noted conjugate hyperbolas in his book on conic sections. 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

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''." George Salmon illustrated a conjugate hyperbola as a dotted curve in this ''Treatise on Conic Sections'' (1900). In 1908 conjugate hyperbolas were used by

Hermann Minkowski Hermann Minkowski (22 June 1864 – 12 January 1909) was a mathematician and professor at the University of Königsberg, the University of Zürich, and the University of Göttingen, described variously as German, Polish, Lithuanian-German, o ...

to demarcate units of duration and distance in a

spacetime diagram A spacetime diagram is a graphical illustration of locations in space at various times, especially in the special theory of relativity. Spacetime diagrams can show the geometry underlying phenomena like time dilation and length contraction witho ...

illustrating a plane in his

Minkowski space In physics, Minkowski space (or Minkowski spacetime) () is the main mathematical description of spacetime in the absence of gravitation. It combines inertial space and time manifolds into a four-dimensional model. The model helps show how a ...

. The

principle of relativity In physics, the principle of relativity is the requirement that the equations describing the laws of physics have the same form in all admissible frames of reference. For example, in the framework of special relativity, the Maxwell equations ...

may be stated as "Any pair of conjugate diameters of conjugate hyperbolas can be taken for the axes of space and time". In 1957 Barry Spain illustrated conjugate rectangular hyperbolas.Barry Spain (1957
''Analytical Conics''
via HathiTrust

References

{{Reflist Conic sections