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Re: Solved Circumference of Oval and Ellipse, both simultaneously

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Archimedes Plutonium <plutonium.archimedes@gmail.com> writes:

>Solved the problem of Circumference for Ellipse, Oval. See no 2.71... 
>involved, but see an easy trig function solution

Solved, the problem of whether the ellipse is a conic section.

Some preliminaries:

Top view of the conic section and depiction of the coordinate system used 
in the proof:

              ^ x
              |
             -+- <= x=h
         .'   |   `.
        .     |     .
        |     |     |
        '     |     '      
         `.   |   .'
 y <----------+ <= x=0
             
Cone (side view):
                 .
                /|\
               / | \
              /b |  \
             /---+---' <= x = h
            /    |'   \
           /   ' |     \
          / '    |      \
x = 0 => '-------+-------\
        /    a   |        \

Proof:

r(x) = a - ((a-b)/h)x  and  d(x) = a - ((a+b)/h)x,  hence

y(x)^2 = r(x)^2 - d(x)^2 = ab - ab(2x/h - 1)^2 = ab(1 - 4(x - h/2)^2/h^2.

Hence (1/ab)y(x)^2 + (4/h^2)(x - h/2)^2 = 1  ...equation of an ellipse

qed

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Solved Circumference of Oval and Ellipse, both  simultaneously Archimedes Plutonium <plutonium.archimedes@gmail.com> - 2018-01-15 12:05 -0800
  Re: Solved Circumference of Oval and Ellipse, both simultaneously moroney@world.std.spaamtrap.com (Michael Moroney) - 2018-01-16 05:29 +0000
    Re: Solved Circumference of Oval and Ellipse, both simultaneously Serg io <invalid@invalid.com> - 2018-01-22 19:38 -0600
      Re: Solved Circumference of Oval and Ellipse, both simultaneously benj <benj@nobody.net> - 2018-01-23 12:56 -0500
        Re: Solved Circumference of Oval and Ellipse, both simultaneously Archimedes Plutonium <plutonium.archimedes@gmail.com> - 2018-01-23 11:30 -0800
  Solved Circumference of Oval and Ellipse, both  simultaneously Archimedes Plutonium <plutonium.archimedes@gmail.com> - 2018-01-16 07:26 -0800

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