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Equation of spheroid

  1. May 26, 2007 #1
    1. The problem statement, all variables and given/known data

    I have a doubt on spheroid equations. A prolate spheroid is obtained by rotating the ellipse,
    X^2/a^2 + Y^2/b^2 = 1 {Here a is major axis}
    about the semi-major axis a(i.e. X axis). Its equation is,
    X^2/a^2 + [Y^2+Z^2]/b^2 = 1
    An oblate spheroid is obtained by rotating the ellipse,
    X^2/a^2 + Y^2/b^2 = 1 {Here b is major axis}
    about the semi-minor axis a(i.e. X axis).Its equation is,
    X^2/a^2 + [Y^2+Z^2]/b^2 = 1
    The problem is that both equations are identical. What I have done is that I have taken ‘a’ always along X axis and ‘b’ always along Y axis. Is it necessary that the equations be distinguishable?

    2. Relevant equations

    X^2/a^2 + Y^2/b^2 = 1

    3. The attempt at a solution

    In order distinguish between the two, I will have to take ‘a’ along Y axis for one of them. Suppose I take ‘a’ along the Y axis for oblate spheroid case, the equation of the oblate spheroid is got by rotating the ellipse,
    X^2/b^2 + Y^2/a^2 = 1
    about the semi-minor axis ‘b’(i.e. X axis).Its equation is,
    X^2/b^2 + [Y^2+Z^2]/a^2 = 1
    Another way is to rotate the ellipse,
    X^2/a^2 + Y^2/b^2 = 1 {Here a is major axis}
    first along X axis(i.e. ‘a’) for prolate spheroid in which case the equation becomes,
    X^2/a^2 + [Y^2+Z^2]/b^2 = 1
    And then along Y axis(i.e. ‘b’) for oblate spheroid in which case the equation becomes,
    [X^2+Z^2]/a^2 + Y^2/b^2 = 1 {Here b is major axis}

    Is there a better way to this? Please help.
     
  2. jcsd
  3. May 26, 2007 #2

    jambaugh

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    Start with the general ellipsoid:
    [tex] \left(\frac{x}{a}\right)^2+\left(\frac{y}{b}\right)^2+\left(\frac{z}{c}\right)^2 = 1[/tex]

    if it is rotationally symmetric around any one principle axis the the (a,b,c) coefficients for the other two must be equal. Example: if you have rotational symmetry about x then b=c. Further in this example if a is much bigger than b=c then the ellipsoid is cigar shaped. If a is much smaller than b=c then it is "cow-pie" shaped.

    Regards,
    J. Baugh
     
  4. May 26, 2007 #3
    Could you please say what are the shapes of Cigar and Cow-pie in this context?
     
  5. May 27, 2007 #4

    jambaugh

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    Science Advisor
    Gold Member

    Sure, take a highly eccentric ellipse. Rotate about the long axis and you have an ellipsoid that is long and cylindrical like a tapered cigar.

    Take the same ellipse and rotate about the short axis and you have a tapered disk shaped ellipsoid, like the shape of a discus used in track-n-field events or like the pile of defecant a cow leaves behind.

    Regards,
    James Baugh
     
  6. May 27, 2007 #5
    So, the equation of prolate spheroid is,
    X^2/a^2 + [Y^2+Z^2]/b^2 = 1
    for a>b=c
    and the equation of oblate spheroid is also,
    X^2/a^2 + [Y^2+Z^2]/b^2 = 1
    but here a<b=c. I that what u meant?
     
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