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The VERY, VERY general equation of an ellipsoid - Who knows it?

  1. Feb 18, 2010 #1
    I have been asking/looking around for the general equation of an ellipsoid and I am unable to find it anywhere.

    Does anyone know what it is?

    BTW: What I mean by the general equation of an ellipsoid, one that can be rotated in any way, that is 2 angles of rotation and one that does not have to be centered at the origin. - I know the one for a general ellipse moved from the center. So I don't want this!
    [tex]\frac{(x-x_c)^2}{a^2} + \frac{(y-y_c)^2}{b^2} + \frac{(z-z_c)^2}{c^2} = 1[/tex]

    If possible could it be in Implicit Cartesian or Spherical Polars form?

    Thanks for any help
  2. jcsd
  3. Feb 27, 2010 #2
    A polynomial of degree 2 in variables x,y,z is ellipsoid or hyperboloid or paraboloid.


    There are some inequalities on the coefficients that determine which of the three types it is.
  4. Feb 27, 2010 #3
    Wow dude thanks for the help!

    However, I was wondering if you would not happen to know the inequalities for an ellipsoid? Also, if possible you would not happen to have the equations relating your constants:

    a, b, c, d, f, g, h, i, j to the:

    xc: center of ellipse in the x-direction
    yc: center of ellipse in the y-direction
    zc: center of ellipse in the z-direction

    xr: equatorial radius in the x-direction
    yr: equatorial radius in the y-direction
    zr: polar radius in the z-direction

    [tex]\gamma[/tex]: the angle of rotation in the xy plane (starting from the positive x-axis, where [tex]0 \leq \gamma < 2\pi[/tex])
    [tex]\eta[/tex]: the angle of rotation from the positive z-axis ([tex]0 \leq \eta \leq \pi[/tex])

    Though these last two could be expressed as the components of vectors in the x, y, z direction.

    At any rate thanks a lot for the equation! You have helped a lot! The stuff above is not such a big deal: though if you happened to have it on hand it would really useful!
  5. Feb 27, 2010 #4

    Ben Niehoff

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    Do you know how to use rotation matrices? The equation of an ellipse can be written as a matrix equation:

    [tex](\mathbf{x - c})^T \mathbf{A} (\mathbf{x - c}) = 1[/tex]

    where x is a column vector (x, y, z), c is a column vector representing the center of the ellipse (xc, yc, zc), and A is a square, symmetric matrix. In your equation in the OP, your matrix A is diagonal, with entries (1/a^2, 1/b^2, 1/c^2).

    Now, to rotate your equation to an ellipse in a general orientation, you just need to apply a rotation matrix R as follows:

    [tex](\mathbf{x - c})^T \mathbf{R}^T \mathbf{A} \mathbf{R} (\mathbf{x - c}) = 1[/tex]

    For an ellipse, the matrix A is always positive definite. For a hyperboloid, it may have signature (1, -1, -1) or (-1, 1, 1).
  6. Feb 27, 2010 #5
    The matrix form would actually be really cool to use; but I am ignorant of using matrix geometry. If someone could provide explanations and examples of the following, I might be able to understand what to do:

    [tex](\mathbf{x - c})^T \mathbf{R}^T \mathbf{A} \mathbf{R} (\mathbf{x - c}) = 1[/tex]

    What does the [tex]^T [/tex] mean in the [tex](\mathbf{x - c})^T[/tex] - inverted matrix?
    What does a rotation matrix,[tex]\mathbf{R}[/tex], look like - say I wanted to rotate it [tex]\theta[/tex] degrees in the xy-plane and [tex]\phi[/tex] from the z-axis, what would it look like?
    Finally what is a square, symmetric matrix, [tex]\mathbf{A}[/tex], what does it do?

    Thanks once again to anyone who helps out!
  7. Feb 27, 2010 #6

    Ben Niehoff

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    T means Transpose. For rotation matrices and symmetric matrices, try typing those phrases into Google.
  8. Feb 27, 2010 #7
    Thanks dude!
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