Mathematische Annalen

, Volume 103, Issue 1, pp 516–531 | Cite as

Some theorems on the generators of a hyperboloid

  • Charles H. Rowe


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  1. 1).
    We shall find it convenient to speak of the distance of one line from another instead of the shortest distance between the two lines.Google Scholar
  2. 2).
    An argument of a nature similar to this allows us to establish the following extension of the property of the focal distances of a point on a central conic:The distances of a point on a central conic from two fixed parallel generators of a hyperboloid of the confocal system determined by the conic have a constant sum or difference according as the conic is an ellipse or a hyperbola.Google Scholar
  3. 3).
    Fokalachse is the term used by Reye,Die Geometrie der Lage (Leipzig, 1892), vol. 2 (1926), p. 153.Google Scholar
  4. 4).
    Unlike the fixed planes that present themselves in connexion with the analogous property of the modular foci of a quadric, these fixed planes are different for different focal axes. It will be found that they cut the plane at infinity in the real pair of lines that contains the four points where the imaginary tangent planes throughl′ cut the circle at infinity. If we wish to define the directions of these planes without using imaginary elements, we can verify without difficulty that they are planes of circular section of the quadric cylinder withl′ as axis that will be found to pass through the curve in whichS is cut by any right circular cylinder withl as axis.Google Scholar
  5. 5).
    See H. Schröter,Über ein einfaches Hyperboloid von besonderer Art, Crelle85 (1878), p. 26, or Salmon-Fiedler,Analytische Geometrie des Raumes 1 (1922), p. 170. A hyperboloid is said to be orthogonal if it has a generator that is perpendicular to the planes of a system of circular sections. If a generator satisfies this condition, it is one of the four generators that meet the major axis; and the remaining three of these four also satisfy the condition.Google Scholar
  6. 6).
    The necessity of this condition follows at once when we allow a vertex of a variable inscribed-circumscribed quadrilateral to tend to a point of intersection of the conics. Its sufficiency may be proved perhaps most rapidly by projecting the two conics into two circles, one of which passes through the centre of the other.Google Scholar
  7. 7).
    This may be verified by projecting in the manner indicated in the preceding foot-note, or by observing that the points of contact with Ω of pairs of opposite sides of the variable quadrilateralQ form an involution on Ω.Google Scholar
  8. 8).
    The distances of a point onS fromg′ andg″ have a constant ratio, and therefore the minima of the distances of a point on a generator ofS fromg′ andg″ are attained simultaneously and have the same constant ratio.Google Scholar
  9. 9).
    The asymptotes of the focal hyperbola are the axes of the unique pair of right circular cylinders that can be circumscribed to the hyperboloid. They are consequently the only lines whose distances from a variable generator are constant.Google Scholar
  10. 11).
    It may be shown that the principal focal axes ofS are the polar lines of the four generators ofS that meet the major axis with respect to the unique orthogonal hyperboloid of the confocal system.Google Scholar
  11. 12).
    This locus is discussed by Schoenflies in the special case where the hyperboloid is orthogonal and the fixed line is one of the focal axes that cut the major axis at right-angles. (Zeitschrift für Math. u. Phys.23 (1878), p. 276) He states inaccurately that the locus is a rational quartic in this case.Google Scholar
  12. 13).
    We may prove these statements in another way. The pointP lies on the line in which the plane throughg parallel tod is cut by the perpendicular plane throughd. In case (I) the former plane passes through the generator of the second system that is parallel tod, and thereforeP lies on a right circular cylinder that contains a generator ofS. In case (II) the former plane touches a cylinder circumscribed toS of whichd is a focal line, and thereforeP lies on a right circular cylinder that has double contact withS.Google Scholar
  13. 14).
    Wiener Berichte125 (1916), p. 280.Google Scholar

Copyright information

© Springer-Verlag 1930

Authors and Affiliations

  • Charles H. Rowe
    • 1
  1. 1.DublinIrland

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