General formula for an angled ellipsoid

[AntStrike]

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?

 

[mighty2000]

Thank you for sharing your progress and seeking input from the community. Your approach seems to be on the right track, but there are a few things that you may want to consider.

Firstly, when dealing with angled ellipsoids, it is important to take into account the orientation of the ellipsoid with respect to the coordinate system. This means that the semi-axes of the ellipsoid may not necessarily be aligned with the x, y, and z axes. Therefore, it may be helpful to define the semi-axes in terms of their components along the x, y, and z axes (e.g. a = a_x, b = b_y, c = c_z).

Additionally, when determining if a point is within an angled ellipsoid, it may be easier to first transform the coordinates of the point and the center of the ellipsoid into a new coordinate system where the ellipsoid is aligned with the x, y, and z axes. This can be done using a rotation matrix, which can be constructed using the azimuth and dip angles. Once the coordinates are transformed, the formula for a non-angled ellipsoid (x^2/a_x^2 + y^2/b_y^2 + z^2/c_z^2 = 1) can be used to determine if the point is within the transformed ellipsoid.

I hope this helps and good luck with your program!