Hyperbolas

1
Hyperbolas

• Group III

2
Definition

• Two-branched open curve, a conic section,
  produced by the intersection of a circular cone
  and a plane that cuts both nappes of the cone. As
  a plane curve it may be defined as the path
  (locus) of a point moving so that the ratio of
  the distance from a fixed point (the focus) to
  the distance from a fixed line (the directrix) is
  a constant greater than one. The hyperbola,
  however, because of its symmetry, has two foci.
  Another definition is that of a point moving so
  that the difference of its distances from two
  fixed points, or foci, is a constant. A
  degenerate hyperbola (two intersecting lines) is
  formed by the intersection of a circular cone and
  a plane that cuts both nappes of the cone through
  the apex.

3
Definition

• A line drawn through the foci and prolonged
  beyond is the transverse axis of the hyperbola
  Perpendicular to that axis, and intersecting it
  at the geometric centre of the hyperbola, a point
  midway between the two foci, lies the conjugate
  axis. The hyperbola is symmetrical with respect
  to both axes.
• Two straight lines, the asymptotes of the curve,
  pass through the geometric centre. The hyperbola
  does not intersect the asymptotes, but its
  distance from them becomes arbitrarily small at
  great distances from the centre. The hyperbola
  when revolved about either axis forms a
  hyperboloid.

4
Definition

• For a hyperbola that has its centre at the origin
  of a Cartesian coordinate system and has its
  transverse axis lying on the x axis, the
  coordinates of its points satisfy the equation x/
  a - y/ b 1, in which a and b are constants.
• A hyperbola is a conic with an eccentricity
  greater than unity. It has the standard form
  shown in figure 26, in which F and F' are the
  foci. Its equation is one that involves a
  difference of squares and it cuts the x-axis at A
  and A', which have respective coordinates (a, 0)
  and (-a, 0). The segment AA', of length 2a, is
  called the transverse axis, and the segment of
  the y-axis, BB', with length 2b, is called the
  conjugate axis. The coordinates of the foci are
  (ae, 0) and (-ae, 0), and the directrices are the
  lines x a/e and x -a/e.

5
Definition

• It is not possible to draw tangents to the
  hyperbola through the origin The lines either
  intersect the curve or miss it altogether. A line
  y mx--that is, a straight line through the
  origin--cuts the hyperbola at points with
  x-coordinates given by the formula for the
  hyperbola with y replaced by mx. These meeting
  points coincide for the value of m for which y
  mx is a tangent. This occurs when a simple
  algebraic condition is met and x is infinity.
  Thus the tangent touches the curve at infinity
  and is called an asymptote. It has a value of m
  given by an algebraic solution showing two
  values. Thus there are two asymptotes with
  equations whose slopes differ by 1.

6
Definition

• The equation in which the roles of a and b are
  interchanged is that of the conjugate hyperbola
  of the one above. Conjugate hyperbolas have the
  same asymptotes, and the conjugate axis of one is
  the transverse axis of the other and vice versa.
• A hyperbola for which b a is called a
  rectangular hyperbola and has the equation in
  which the constant a is equal to the difference
  of the variables squared. The transverse and
  conjugate axes are then equal, and the asymptotes
  have equations y x and y -x. Clearly the
  asymptotes are perpendicular to each other and
  bisect the coordinate axes. If the asymptotes are
  used as the coordinate axes, a rectangular
  hyperbola has the simple equation xy c, in
  which c is a constant. Thus the rectangular
  hyperbola is a graph of two inversely
  proportional variable quantities related by the
  equation y c/x.

7
Application

• If a gun at position M in figure 6 were fired, a
  listener 1,100 feet (340 metres) away in any
  direction--that is, anywhere on the smallest
  circle centred at m--would hear the sound one
  second later A listener 2,200 feet away, on the
  second circle, two seconds later And so on. If
  guns at M and S were fired simultaneously, a
  listener anywhere on AB, equidistant from M and
  S, would hear them at the same time. On a craft
  closer to one gun than the other, the sound of
  the nearer gun would be detected first. If gun M
  were heard one second before gun S, the craft
  would lie on CD, one of the two branches of a
  hyperbola At a craft on C'D', the other branch
  of the same hyperbola, gun S would be heard one
  second earlier than gun M. At a craft 2,200 feet
  closer to gun M, that gun would be heard two
  seconds before gun S, and the craft would lie on
  EF. Hence, by timing the interval to the nearest
  second, it is possible to find nearest which
  hyperbola the observer is located Knowledge of
  which gun was fired first makes it possible to
  choose between the two branches.

8
Application

• For navigation, the firing of guns is replaced by
  radio transmissions. A family of hyperbolas as
  shown in Figure 1 may be printed on a chart. A
  second family of hyperbolas, referring to a
  second pair of stations, can be printed on the
  same chart the position of a craft is determined
  by the unique intersection of two curves. In
  radio systems, one of the two stations in a pair
  (the master) controls the other (the slave) to
  ensure accurate synchronization of the signals.
  In some systems, two or three slaves are
  distributed around a single master, and two or
  three families of hyperbolas are printed on the
  appropriate chart. Systems of this kind are
  called loran, an acronym for long-range
  navigation.

9
Application
Figure 1
10
Application

• Loran in its original form (now called loran A)
  was introduced during World War II it operated
  at frequencies near two megahertz, but
  interference with and by other services and
  unreliable performance at night and over land led
  to its replacement by loran C. Loran C
  transmitters operate at the frequency of 100
  kilohertz, and the signals are useful at
  distances of 1,800 nautical miles or more.

11
Application

• Decca, named for the British company that
  introduced it in 1946, is a hyperbolic system
  related to loran. Its master and slave
  transmitters broadcast different harmonics of a
  common frequency as continuous waves, rather than
  pulses. The hyperbolic position lines for any
  pair of transmitters are determined by the phase
  difference between the signals received, rather
  than the difference in arrival times of pulses.
  This arrangement provides a remarkably accurate
  and reliable system covering a range of 100-300
  miles (160-480 kilometres) from the master
  station. Decca equipment is widely installed on
  ships and enjoys particular favour among
  fishermen, who can use it to return to specific
  shoals with great precision. Aircraft
  installations are less common than those of
  VOR/DME, the internationally accepted system for
  position finding. Decca is very well suited to
  navigation of helicopters, however, which usually
  operate at altitudes well below those at which
  VOR/DME is most effective.

12
Application

• Omega is a very-long-wave hyperbolic system that
  probably has the longest range of all radio
  navigation aids. Eight permanent transmitters,
  mutually synchronized, provide worldwide service.
  There is little redundancy in the Omega system,
  however if a single transmitter becomes
  inoperative, a considerable gap in the coverage
  results. Its signals penetrate seawater to a
  substantial extent, so that submarines can
  receive them. Like those of Decca, transmissions
  are continuous, rather than pulsed, and are
  encoded to prevent ambiguities.
• During World War II, Germany produced the Sonne
  beacon, which has been developed into the consol
  and consolan systems, in which the masters and
  slaves are so close together that the hyperbolas
  become practically straight lines radiating out
  from the centre. By changing the phase of
  transmissions of the slaves, the hyperbolas can
  be made to swing to and fro. Morse signals are
  superimposed so that a craft can tell its
  location in a particular lane or sector.

13
Application

• Decca covers a wide area and is inherently a
  flexible navigation system. Before Decca's
  invention, however, aviation policy had been
  committed to signpost navigation using VOR
  beacons and subsequently VOR/DME. Consequently,
  the Decca system was adapted to work with a
  computer that would make the area system work as
  if there were beacons placed anywhere the
  navigator wished. Thus Decca could work as a
  signpost or point navigation system.

14
Application

• In the absence of planetary perturbations and
  nongravitational forces, a comet will orbit the
  Sun on a trajectory that is a conic section with
  the Sun at one focus. The total energy E of the
  comet, which is a constant of motion, will
  determine whether the orbit is an ellipse, a
  parabola, or a hyperbola. The total energy E is
  the sum of the kinetic energy of the comet and of
  its gravitational potential energy in the
  gravitational field of the Sun. Per unit mass, it
  is given by E 1/2 v - GMrsup - 1, where v is
  the comet's velocity and r its distance to the
  Sun, with M denoting the mass of the Sun and G
  the gravitational constant. If E is negative, the
  comet is bound to the Sun and moves in an
  ellipse. If E is positive, the comet is unbound
  and moves in a hyperbola. If E 0, the comet is
  unbound and moves in a parabola.

15
Application

• In polar coordinates written in the plane of the
  orbit, the general equation for a conic section
  is
•  
•  
•  
• where r is the distance from the comet to the
  Sun, q the perihelion distance, e the
  eccentricity of the orbit, and an angle measured
  from perihelion. When 0 e
  orbit is an ellipse (the case e 0 is a circle,
  which constitutes a particular ellipse) when e
  1, E 0 and the orbit is a parabola and when e
  1, E 0 and the orbit is a hyperbola.

16
Application

• In space a comet's orbit is completely specified
  by six quantities called its orbital elements.
  Among these are three angles that define the
  spatial orientation of the orbit i, the
  inclination of the orbital plane to the plane of
  the ecliptic , the longitude of the ascending
  node measured eastward from the vernal equinox
  and , the angular distance of perihelion from
  the ascending node (also called the argument of
  perihelion). The three most frequently used
  orbital elements within the plane of the orbit
  are q, the perihelion distance in astronomical
  units e, the eccentricity and T, the epoch of
  perihelion passage.

17
Hyperboloid

• The open surface generated by revolving a
  hyperbola (q.v.) about either of its axes. If the
  tranverse axis of the surface lies along the x
  axis and its centre lies at the origin and if a,
  b, and c are the principal semi-axes, then the
  general equation of the surface is expressed as x
  /a /- y /b - z /c 1.
• Revolution of the hyperbola about its conjugate
  axis generates a surface of one sheet, an
  hourglass-like shape for which the second term of
  the above equation is positive. The intersections
  of the surface with planes parallel to the xz and
  yz planes are hyperbolas. Intersections with
  planes parallel to the xy plane are circles or
  ellipses.

18
Hyperboloid

• Revolution of the hyperbola about its transverse
  axis generates a surface of two sheets, two
  separate surfaces for which the second term of
  the general equation is negative. Intersections
  of the surface(s) with planes parallel to the xy
  and xz planes produce hyperbolas. Cutting planes
  parallel to the yz plane and at a distance
  greater than the absolute value of a,a, from
  the origin produce circles or ellipses of
  intersection, respectively, as a equals b or a is
  not equal to b.

19
Hyperboloid