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Re: Solved Circumference of Oval and Ellipse, both simultaneously

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On 1/15/2018 11:29 PM, Michael Moroney wrote:
> 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
> 

Bonus Question:

who solved these first ?
(we have their solution in writings passed down through history)



Bonus Problem:

solve using spherical coordinates

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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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