General formula for an angled ellipsoid

  • Thread starter AntStrike
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Homework Statement


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.


Homework Equations


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

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?
 

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