Conjugate diameters

Conjugate diameters
Two conjugate diameters of an ellipse. Each edge of the bounding parallelogram is parallel to one of the diameters.

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

For an ellipse, two diameters are conjugate if and only if the tangent line to the ellipse at one endpoint of a diameter is parallel to the tangent at the second endpoint. Each pair of conjugate diameters of an ellipse has a corresponding tangent parallelogram, sometimes called the bounding parallelogram. In his manuscript De motu corporum in gyrum, and in the 'Principia', Isaac Newton cites as a lemma proved by previous authors that all (bounding) parallelograms for a given ellipse have the same area.

It is possible to reconstruct an ellipse from any pair of conjugate diameters, or from any bounding parallelogram. For example, in proposition 14 of Book VIII of his Collection, Pappus of Alexandria gives a method for constructing the axes of an ellipse from a given pair of conjugate diameters.

Blue and green hyperbolas are conjugate. A diameter from (x,y) to (−x,−y) is conjugate to the one from (y,x) to (−y,−x).

Two hyperbolas are conjugate if they are images of each other in a reflection across an asymptote. A diameter of one hyperbola is conjugate to its reflection in the asymptote, which is a diameter of the other hyperbola. They are hyperbolic-orthogonal to each other.

Conjugate diameters of hyperbolas are useful for stating the principle of relativity in the modern physics of spacetime. The concept of relativity is first introduced in a plane consisting of a single dimension in space, the second dimension being 'time'. In such a plane, one hyperbola corresponds to events a constant distance from the origin event, the other hyperbola corresponds to events a constant proper time from it. The principle of relativity can be formulated "Any pair of conjugate diameters of conjugate hyperbolas can be taken for the axes of space and time". This interpretation of relativity was enunciated by E. T. Whittaker in 1910.


References


Wikimedia Foundation. 2010.

Игры ⚽ Поможем сделать НИР

Look at other dictionaries:

  • Conjugate diameters — Conjugate Con ju*gate, a. [L. conjugatus, p. p. or conjugare to unite; con + jugare to join, yoke, marry, jugum yoke; akin to jungere to join. See {Join}.] 1. United in pairs; yoked together; coupled. [1913 Webster] 2. (Bot.) In single pairs;… …   The Collaborative International Dictionary of English

  • conjugate diameters — plural noun Two diameters in a conic section, such that each is parallel to the tangent at the extremity of the other • • • Main Entry: ↑conjugate …   Useful english dictionary

  • Conjugate diameters — Diameter Di*am e*ter, n. [F. diam[ e]tre, L. diametros, fr. Gr. ?; dia through + ? measure. See {Meter}.] 1. (Geom.) (a) Any right line passing through the center of a figure or body, as a circle, conic section, sphere, cube, etc., and terminated …   The Collaborative International Dictionary of English

  • Conjugate — Con ju*gate, a. [L. conjugatus, p. p. or conjugare to unite; con + jugare to join, yoke, marry, jugum yoke; akin to jungere to join. See {Join}.] 1. United in pairs; yoked together; coupled. [1913 Webster] 2. (Bot.) In single pairs; coupled.… …   The Collaborative International Dictionary of English

  • Conjugate axis of a hyperbola — Conjugate Con ju*gate, a. [L. conjugatus, p. p. or conjugare to unite; con + jugare to join, yoke, marry, jugum yoke; akin to jungere to join. See {Join}.] 1. United in pairs; yoked together; coupled. [1913 Webster] 2. (Bot.) In single pairs;… …   The Collaborative International Dictionary of English

  • Conjugate focus — Conjugate Con ju*gate, a. [L. conjugatus, p. p. or conjugare to unite; con + jugare to join, yoke, marry, jugum yoke; akin to jungere to join. See {Join}.] 1. United in pairs; yoked together; coupled. [1913 Webster] 2. (Bot.) In single pairs;… …   The Collaborative International Dictionary of English

  • Conjugate mirrors — Conjugate Con ju*gate, a. [L. conjugatus, p. p. or conjugare to unite; con + jugare to join, yoke, marry, jugum yoke; akin to jungere to join. See {Join}.] 1. United in pairs; yoked together; coupled. [1913 Webster] 2. (Bot.) In single pairs;… …   The Collaborative International Dictionary of English

  • Conjugate point — Conjugate Con ju*gate, a. [L. conjugatus, p. p. or conjugare to unite; con + jugare to join, yoke, marry, jugum yoke; akin to jungere to join. See {Join}.] 1. United in pairs; yoked together; coupled. [1913 Webster] 2. (Bot.) In single pairs;… …   The Collaborative International Dictionary of English

  • Self-conjugate triangle — Conjugate Con ju*gate, a. [L. conjugatus, p. p. or conjugare to unite; con + jugare to join, yoke, marry, jugum yoke; akin to jungere to join. See {Join}.] 1. United in pairs; yoked together; coupled. [1913 Webster] 2. (Bot.) In single pairs;… …   The Collaborative International Dictionary of English

  • conjugate — 1. Joined or paired. SYN: conjugated. 2. C. diameters of the pelvis. The distance between any two specified points on the periphery of the pelvic canal. [L. conjugatus, joined together. See conjugata] anatomical c. [TA] measure of pelvic …   Medical dictionary

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”