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B Spheroids and elliptoids

  1. Jul 3, 2016 #1

    DaveC426913

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    A story I wrote depends on some geometry, and I want to get it straight.

    Assumptions:
    1] The 2D math that applies to circles and ellipses is analagous to the 3D math for spheres and ellipsoids in the ways relevant to the rest of my post.
    2] An ellipse is a circle whose two foci are co-incident. Thus, a sphere is an ellipsoid whose two foci are co-incient.

    What I want to figure out is what happens to the surface of an ellipse/ellipsoid when the two co-incident foci are moved apart.

    Say I have a sphere of unit radius. I move its two foci two units apart. What happens to the surface?
    What is the length of its semi-minor axis?
    What is the length of its semi-major axis?

    And, more specifically, what happens to the distance from focus-to-surface near the "ends"?
    Does the distance from focus to surface decrease as the ellipsoid "stretches" thinner? (i.e. is the focus now closer to the surface than one unit?)


    [ EDIT ] The correct term for my shape is a prolate spheroid - a cigar shape**. Equatorial radii a and b are the same. Polar radius c is longer.

    ** by magical coincidence, I am writing this while smoking a cigar.
     
    Last edited: Jul 3, 2016
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  3. Jul 3, 2016 #2

    mfb

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    Same as in the two-dimensional case. The ellipsoid you get in that way is the ellipse rotated around the symmetry axis. Note that not all ellipsoids can be produced that way - you are limited to those with one large axis and two identical smaller axes.



    I wonder what shape you get if you require the sum of distances to three (instead of two) points to be constant. The two-dimensional shape can be egg-like. Another egg.
     
  4. Jul 3, 2016 #3

    fresh_42

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  5. Jul 3, 2016 #4

    DaveC426913

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    Right. Which is what?

    (The only reason I mentioned the 2D case is so that, when I set up some details in my story, I can sketch the simpler 2D geometry. I don't have to wory about whether 3D geometry has different behavior).
     
  6. Jul 3, 2016 #5

    DaveC426913

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    The end-result of my question is this: as the two foci separate, does the focus-to-surface distance (what was a moment ago, the radius) at the pole decrease to less than one unit?
     
    Last edited: Jul 3, 2016
  7. Jul 3, 2016 #6

    fresh_42

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    ... as long as the NFL is off season ...
    I have a question. Why isn't the equation for the eccentricity ##e^2 = 1 - \frac{a^2}{c^2}## from Wiki (https://en.wikipedia.org/wiki/Spheroid) not the answer because the planar foci are ##(0,±e)##?
     
  8. Jul 3, 2016 #7

    DaveC426913

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    Ah. OK. Once I drew it out, it was pretty simple. Should have started with that.

    elliptoid.png
    So, the distance I was looking for was BP = .5
     
  9. Jul 3, 2016 #8

    DaveC426913

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    Shoot. So THAT's a problem. The above ellipse assumes that it is the radii that remain constant. It can not grow larger than a major axis of 2. At that point, it degenerates to a line. That's useless to me.

    I'm going to need to redo it using 3D volume as a constant.

    So, a sphere of one unit radius has a volume of 4/3π. I need to find the major and minor axes of a prolate elliptoid with a volume of 4/3π whose foci are one unit apart.


    Volume of sphere s = 4/3π*r3
    Volume of prolate elliptoid e = 4/3π*a2c

    e=s
    4/3π*r3 = 4/3π*a2c
    r=a2c
    a2c = 1

    Hm. Missing something here...
    I need to specify a as a ratio of c.

    How does the focal separation relate to the major and minor axes?
    More specifically, how can I specify a and c so that the foci are one unit apart?
     
    Last edited: Jul 3, 2016
  10. Jul 3, 2016 #9

    DaveC426913

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