Booth's Lemniscate:

- The inversion in a circle of conic sections:

  x^2       y^2            ( + : ellipse)
  ---  +/-  ---  =  1      ( - : hyperbola)
  a^2       b^2

  or in polar coordinates

                      a^2 b^2
  r^2 = ---------------------------------
        b^2 cos^2 phi  +/-  a^2 sin^2 phi

  Inversion in a circle at the origin with radius K gives:

        K^4 cos^2 phi       K^4 sin^2 phi
  r^2 = -------------  +/-  -------------
            a^2                 b^2

                    K^4           K^4       ( + : Booth's elliptic Lemniscate)
  (x^2 + y^2)^2  =  --- x^2  +/-  --- y^2   ( - : Booth's hyperbolic Lemniscate)
                    a^2           b^2


- Toric sections - hippopede of Proclus: analyzed by Perseus 

  The toric with two radii 'c' and 'd'.

  4 c^2 (x^2 + z^2) = (x^2 + y^2 + z^2 + c^2 - d^2)^2

  Now make the section at z = c - d. This gives

  (x^2 + y^2)^2  = 4cd x^2 - 4c(c - d) y^2

  Letting a^2 = 4cd and b^2 = 4c(c - d) we again get 
  Booth's Lemniscate

  (x^2 + y^2)^2  =  a^2 x^2  -  b^2 y^2

  For d>c we get an elliptic Lemniscate which has no double point.


- Cassinian Oval: Section of a 'Cassinian Surface'

  The Cassinian oval is the location of all points P in the x-y-plane
  with PF1 * PF2 = a^2 with F1 = (c,0) and F2 = (-c,0). This yields
    (x^2 + y^2)^2 - 2c^2(x^2 - y^2) - (a^4 - c^4) = 0
  The 'Cassinian Surface' is the location of all points P in the x-y-z-space.
  In cylinder coordinates (r, phi, x)
    (x^2 + r^2)^2 - 2c^2(x^2 - r^2) - (a^4 - c^4) = 0
  In cartesian cordinates we get
    (x^2 + y^2 + z^2)^2 - 2c^2(x^2 - y^2 - z^2) - (a^4 - c^4) = 0   (*)
  Now cut this surface with a plane z^2 = constant.
  Expand (*) by z^2 and we get
    (x^2 + y^2)^2 + 2z^2 (x^2 + y^2) - 2c^2(x^2 - y^2) + p(z,a,c) = 0

  To get the Booth's Lemniscate p(z,a,c) must be zero. Therefore
  0 = p(z,a,c) = z^4 + 2c^2 z^2 - (a^4-c^4) = (z^2 + c^2 + a^2)(z^2 + c^2 - a^2)

  We get Booth's hyperbolic Lemniscate for z^2 = a^2 - c^2
    (x^2 + y^2)^2  =  2(2c^2 - a^2) x^2  -  2a^2 y^2
  We must have 2c^2 >= a^2 to get more than the point (0,0).

  We get Booth's elliptic Lemniscate for z^2 = - a^2 - c^2.
  This time we are cutting with an imaginary plane.

References:

- J. Dennis Lawrence;
  A Catalog of Special Plane Curves,
  Dover Publ., New York 1972
  - p120: Lemniscate of Bernoulli  r^2 = cos(2 phi)
  - p144: Hippopede r^2 = 4b(a - b sin^2(phi))

- Hermann Schmidt;
  Die Inversion und Ihre Anwendungen,
  (engl: The Inversion and Its Application)
  Oldenbourg Verlag, M\"unchen 1950

- Hans Schupp, Heinz Dabrock;
  H\"ohere Kurven,
  Situative, mathematische, historische und didaktische Aspekte,
  BI Wissenschaftsverlag 1995, ISBN 3-411-17221-5

- Robert Ferreol, Jacques Mandonnet;
  http://www.mathcurve.com/courbes2d/booth/booth.shtml
  "Courbes, Ovales, Lemniscates de Booth"
  - J. Booth (1810-1878), hippopede de Proclus

- Jan Wassenaar;
  http://www.2dcurves.com/

- Xah Leh;
  http://xahlee.org/SpecialPlaneCurves_dir/LemniscateOfBernoulli_dir/lemniscateOfBernoulli.html

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