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corresponds to events a constant space-like interval from the origin event, the other hyperbola corresponds to events a constant time-like interval 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
381:: "If Q be any point on a hyperbola and CE be drawn from the centre parallel to the tangent at Q to meet the conjugate hyperbola in E, then (1) the tangent at E will be parallel to CQ and (2) CQ and CE will be conjugate diameters."
492:
286:
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distinguishes an ellipse from a hyperbola: In the ellipse every pair of conjugate diameters separates every other pair. In a hyperbola, one pair of conjugate diameters never separates another such pair.
547:. A diameter of one hyperbola is conjugate to its reflection in the asymptote, which is a diameter of the other hyperbola. As perpendicularity is the relation of conjugate diameters of a circle, so
853:
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are conjugate when each bisects all chords parallel to the other. In this case both the hyperbola and its conjugate are sources for the chords and diameters.
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to the ellipse at an endpoint of one diameter is parallel to the other diameter. Each pair of conjugate diameters of an ellipse has a corresponding
144:
622:
between points and lines. Using these concepts, "two diameters are conjugate when each is the polar of the figurative point of the other."
435:
229:
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gives a method for constructing the axes of an ellipse from a given pair of conjugate diameters. Another method is using
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656:. International series of monographs in pure and applied mathematics.v.3. New York: Pergamon Press. p. 49.
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an ellipse from any pair of conjugate diameters, or from any bounding parallelogram. For example, in
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for finding the directions and lengths of the major and minor axes of an ellipse regardless of its
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90:
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517:
859:
117:
798:
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proved by previous authors that all (bounding) parallelograms for a given ellipse have the same
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577:. The concept of relativity is first introduced in a plane consisting of a single dimension in
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563:
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179:
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47:
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Traité des sections coniques, Ie partie. faisant suite au traité de géométrie supérieure
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Perpendicular diameters of a circle or hyperbolic-orthogonal diameters of a hyperbola
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For any φ, the indicated diameters of the circles and hyperbolas are conjugate.
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Only one of the conjugate diameters of a hyperbola cuts the curve.
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Conjugate diameters of hyperbolas are also useful for stating the
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is the relation of conjugate diameters of rectangular hyperbolas.
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gave the following construction of conjugate diameters, given the
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in projective geometry, and each conic determines a relation of
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487:{\displaystyle \cosh \theta {\vec {a}}+\sinh \theta {\vec {b}}}
388:, if we let the vectors of the two conjugate half-diameters be
182:, if we let the vectors of the two conjugate half-diameters be
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is guided by the relation of conjugate diameters in a book on
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In the case of a rectangular hyperbola, its conjugate is the
281:{\displaystyle \cos \theta {\vec {a}}+\sin \theta {\vec {b}}}
858:. Historical Math Monographs. London; Ithaca, NY: G. Bell;
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time". This interpretation of relativity was enunciated by
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by the other diameter. For example, two diameters of a
759:(in French). Paris: Gauthier-Villars. pp. 116–23.
713:(1 ed.). Dublin: Longman, Green and Co. p.
614:. The ellipse, parabola, and hyperbola are viewed as
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A History of the
Theories of Aether and Electricity
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683:Osgood, William F.; Graustein, William C. (1921).
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101:, two diameters are conjugate if and only if the
671:Apollonius of Perga: Treatise on Conic Sections
852:(1895). "Properties of Conjugate Diameters".
367:Similar to the elliptic case, diameters of a
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689:. New York: The Macmillan Company. p.
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219:{\displaystyle {\vec {a}},{\vec {b}}}
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855:Conic sections treated geometrically
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686:Plane and solid analytic geometry
558:reinforcing a square assembly of
838:"Conjugate Diameters in Ellipse"
163:, which takes advantage of the
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81:Two conjugate diameters of an
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732:Synthetic Projective Geometry
581:, the second dimension being
814:A Treatise on Conic Sections
529:{\displaystyle \mathbb {R} }
85:. Each edge of the bounding
898:
795:Cambridge University Press
819:Longmans, Green & Co.
789:The Real Projective Plane
573:in the modern physics of
122:De motu corporum in gyrum
549:hyperbolic orthogonality
93:to one of the diameters.
571:principle of relativity
507:{\displaystyle \theta }
301:{\displaystyle \theta }
151:14 of Book VIII of his
598:In projective geometry
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120:). In his manuscript
116:(skewed compared to a
114:bounding parallelogram
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772:, page 90, link from
753:"Diamètres conjugués"
650:Spain, Barry (1957).
630:point-pair separation
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112:, sometimes called a
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157:Pappus of Alexandria
770:Elements of Dynamic
604:projective geometry
585:. In such a plane,
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375:Apollonius of Perga
161:Rytz's construction
53:to one diameter is
860:Cornell University
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143:It is possible to
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18:Conjugate diameter
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608:point at infinity
564:analytic geometry
554:The placement of
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16:(Redirected from
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862:. p. 109.
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783:Coxeter, HSM
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734:, #135, #141
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667:Thomas Heath
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347:Of hyperbola
308:varies over
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130:Isaac Newton
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103:tangent line
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43:
29:
797:. pp.
606:contains a
149:proposition
145:reconstruct
132:cites as a
871:Categories
817:. London:
774:HathiTrust
637:References
543:across an
541:reflection
153:Collection
73:Of ellipse
673:, page 64
594:in 1910.
575:spacetime
545:asymptote
502:θ
479:→
470:θ
467:
455:→
446:θ
443:
417:→
402:→
369:hyperbola
328:π
296:θ
273:→
264:θ
261:
249:→
240:θ
237:
211:→
196:→
126:Principia
65:they are
44:conjugate
36:diameters
821:p.
811:(1900).
785:(1955).
751:(1865).
707:(1910).
556:tie rods
173:shearing
169:rotation
107:tangent
91:parallel
55:bisected
51:parallel
46:if each
32:geometry
768:(1878)
730:(1906)
669:(1896)
560:girders
99:ellipse
97:For an
83:ellipse
616:conics
59:circle
34:, two
579:space
134:lemma
48:chord
38:of a
583:time
464:sinh
440:cosh
138:area
823:165
801:–5.
799:130
715:441
691:307
494:as
384:In
288:as
258:sin
234:cos
178:In
171:or
128:',
89:is
30:In
873::
840:.
755:.
566:.
536:.
343:.
175:.
155:,
140:.
69:.
844:.
825:.
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717:.
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523:R
476:b
461:+
452:a
414:b
408:,
399:a
331:]
325:2
322:,
319:0
316:[
270:b
255:+
246:a
208:b
202:,
193:a
20:)
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