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General formula for an angled ellipsoid

  1. Jul 24, 2012 #1
    1. The problem statement, all variables and given/known data
    Hi, I'm writing a simple geophysics program in Fortran77.
    I'm trying to determine if a point (h,k,m) is within an angled ellipsoid.
    Theoretically I know the semi-axes of the ellipsoid (a,b,c), the value of the point (h,k,m), the azimuth (∅, +ve from the Y axis, 0≤∅<180°), the dip (β, +ve from the xy plane - for now, 0≤β≤90°) and the center of the ellipsoid (x,y,z).
    What I'm trying to determine is a formula which ties these all together. I've started developing one from what I know of angled ellipses.


    2. Relevant equations
    Let ∅ = 0, β = 0
    x^2/b^2 + y^2/a^2 + z^2/c^2 = 1

    3. The attempt at a solution
    When angled this means:
    Let β = 0
    ((x-h)sin∅ + (y-k)cos∅)^2 / a^2 + ((x-h)cos∅ + (y-k)sin∅)^2 / b^2 + z^2/c^2 = 1

    OR let ∅ = 0

    ((y-k)cosβ + (z-m)sinβ)^2 / a^2 + ((y-k)sinβ + (z-m)cosβ)^2 / c^2 + x^2/b^2 = 1

    ∴ ((y-k)sinβ + (z-m)cosβ)^2 / c^2 + ((x-h)cos∅ + (y-k)sin∅)^2 / b^2 + ((x-h)sin∅ + (y-k)cos∅ + (y-k)cosβ + (z-m)sinβ) ^2 / a^2 = 1 ?????

    I don't think the formula above is right as it doesn't seem to account for the change in x when the ellipsoid is angled and tilted. Am I on the right track?
     
  2. jcsd
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