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Math Help - Eccentricity of Lemniscate of Bernoulli

  1. #1
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    Eccentricity of Lemniscate of Bernoulli

    Hi,

    For the ellipse and hyperbola there are simple equations for finding the eccentricity.

    Does anyone know of an analogous equation for finding the eccentricity of the Lemniscate of Bernoulli (or any Cassini oval for that matter)?

    Also, does the lemniscate have a semi-minor axis like the ellipse does?

    Thanks
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  2. #2
    Senior Member pacman's Avatar
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    http://xahlee.org/SpecialPlaneCurves...niscate_b.html

    i have little ideea about this lemniscate, but it is a beautiful curve. i can not find properties fro the NET.




    The lemniscate is a lesser known cousin of the circle with quite interesting properties. the Lemniscate OP = RS relation below is an intuitive relationship.
    A short history of the lemniscate.
    In an article titled "Acta Eruditorumon", Jacob Bernoulli named the lemniscate after the latin term meaning "shaped like a figure 8, or a knot, or the bow of a ribbon". The lemniscate is a special case of the Cassinian Oval which was described earlier by Cassini in 1680.
    In 1750, Giovanni Fagnano, discovered general properties of the lemniscate and in 1951 Euler investigated the lemniscate arc length leading to elliptic functions.
    Ways to generate a lemniscate.
    Create then connect the dots with math functions.
    1) The easiest way to create a Lemniscate is through the parametric form by applying expression



    and substituting values in t to get <x,y> pairs. Beginning with the value t=0, the first point is
    <x, y> = <a, 0> and migrates to from the initial rightmost location upwards and curving towards
    the origin. You can plot and connect the dots for discrete values of t. The origin is reached when
    t is or 90 degrees.
    2) Using polar equation


    substitute degrees and convert to radians using


    then find the radius values. The <x,y> values can be found with the polar-cartesian conversion,

    Then plot and connect the dots.
    3) To appreciate the previous two math plot and connect functions, ask yourself how difficult it
    would be to find the <x,y> points from the cartesian equation,


    Are you a dyed-in-wool programmer?
    Create the lemniscate as a Java applet. Visit the BrainyPage Lemniscate applet. This applet is great for segregating graphical content and a control panel. It has a flexible placement for the panels. The "connect the dots" polar equation above is iterated from 0 to 360 degrees with steps of 1 degree for a complete lemniscate curve generation. Finding the intersection points involved solving x and y in two equations,

    For teachers, here are a few lemniscate manipulative's.
    1) Link sticks are a fun way to create a lemniscate. Visit the Xahlee website for the link stick construction.
    2) You can generate a Lemniscate curve by drawing a circle, selecting a point a distance of the radius times the square root of 2 from the center, then drawing a line from this external point to all points on the circle, and use the circle chord distance as the distance from the external point to the lemniscate intercept.
    3) The lemniscate curve is the envelope of a rectangular hyperbola generated by the family of circles with each circle center on the hyperbola and the circle curve intersecting the hyperbola center. You can first draw the hyperbola then use a compass to create the curves.
    Finding the area of a lemniscate lobe.








    Visual Dictionary of Special Plane Curves

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  3. #3
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    Yeah I had seen that website before. It's the best info I've found yet, especially the picture at the top, but as far as I can see it offers no advice on how to derive the eccentricity.

    Perhaps I should emphasize that I would be happy with an eccentricity equation for any cassini oval, of which the lemniscate is a special case.
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