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Math Help - Eclipses & Ellipsoids

  1. #1
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    Eclipses & Ellipsoids

    OK, it all started when my Geomotry teacher was curious about the area of an elipse, so I got thinking. Here are a few of my ides, tell me what you think, and any possible counter examples. I am curious what others have to say about my ideas.
    * = Multiply
    + = Add
    - = Subtract
    / = Divided
    = = Equals
    Pi = 3.14

    Area of an Elipse

    Pi (A) (B) is what I think the formula is.

    In a normal circle, it is
    Pi r, or Pi r r

    Since, all an eclipse is doing is stretching out the X and Y factors, you just change it from r to (r)(r), giving one the A and the other the B value. I am pretty sure I even read this somewhere, but just making sure.

    Volume of an Ellipsoid

    If you didn't already know, the area of a circle is 4/3 Pi r
    I took this, and figured about the same idea as i did with the Area.
    The volume of a circle is simply
    4/3 Pi (A) (B) (C)

    So I figure the same principles would work out, where the X, Y, and Z factors are just being stretched, does this seem right? So if you used this for a sphere with a radius of 5, it would still show as
    4/3 Pi (5) (5) (5), same as 4/3 Pi r

    Perimeter of an Elipse

    In case you didn't know the perimeter of a circle, it is
    2 Pi r

    So, let's say the radius was 5, it would look like
    2*Pi*5
    10*Pi
    31.4

    Now, I did research this, and only found HUGE equations! I tried many ways to shorten it, but nothing really worked, until I thought of a different approach. I was wondering if some of you would agree with something as crazy as this!

    I was thinking, maybe, just maybe, you could think of
    2Pi r As
    2*Pi*[(A+B)/2]

    Why like this?

    I thought since their was only 1 radius, it might be possible it was the mean of the combined radiuses. I don't know the plural of radius, so I'll use that word :P . Anyone, tell me what you think of this, cause I think it might be possible.
    Try to think of this hard, because it seems possible.

    Of course, just because
    2*Pi*r is equal to
    2*Pi*[(A+B)/2] with an equal radius, i will still probably be wrong, lol, since they are the same numbers i am averaging. But if somebody knows the real, long formula, can you compare them?

    And I didn't want to even imagine the Surface Area of an Ellipsoid, lol, too much for me to think about, for it is late right now, and I'm tired as it is! Thanks for reading this, and thanks you in advance for any replies!
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  2. #2
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    Your thinking works out correctly for the area of the ellipse and the volume of the ellipsoid.

    For the perimeter of the ellipse, your short formula \pi(A+B) is a rough approximation to the true value. This link has a good discussion. It gives the very good approximation as

    \pi (A+B)(1 + \frac{x^2}{4} + \frac{x^4}{64} + \frac{x^6}{256} +\frac{25x^8}{16384} )

    where x = \frac{A-B}{A+B} . The short formula drops all the terms with x in them.

    The link gives an example comparing the accuracy. When A = 15 and B = 6, the short formula gives 65.9734. The above approximation is 69.039336580. The true value is 69.03933778699452855....
    Last edited by JakeD; May 28th 2006 at 02:32 AM.
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  3. #3
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    I do not know if you taken calculus. But there is something called an Elliptic Integral (studied by Jacobi). Its name appears as a result of studing ellipses as the case here.

    The major problem is that this function cannot be computed the normal way. Basically meaning there is no formula. Hence we need to use an approximation.
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  4. #4
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    Thanks for looking over my work guys!

    BTW, I'm only in Geomotry.
    I'm taking Algebra 2 next year, then Pre-Cal, and then Cal/Trig in College (hopefully).

    BTW, I just have to say, this is an awsome site! Good job to whoever made it!
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  5. #5
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    Quote Originally Posted by Yuripro84
    Thanks for looking over my work guys!

    BTW, I'm only in Geomotry.
    I'm taking Algebra 2 next year, then Pre-Cal, and then Cal/Trig in College (hopefully).

    BTW, I just have to say, this is an awsome site! Good job to whoever made it!
    In that case, let me tell you that you have a good Math brain inside your coconut. You figured all those ideas?

    (Read that in reverse. Last sentence first.)
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  6. #6
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    lol, ya.

    I first read about the area of an ellipse, something about the X and Y being streched out. After this, i just guessed on the equations, according to the formulas for the circle and sphere.

    Thanks for that comment though

    But as the Formula for an allipsoid goes, I don't really want to even venture any further into that yet! Especially since I've been so tired lol. I will check that out later though, thanks again guys!
    Last edited by Yuripro84; May 28th 2006 at 04:09 PM.
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