For an ellipsoid of the form

with orientation vector and centre at point , how to find whether a point is inside the ellipsoid or not?

An additional note that the geometry actually is with a=b (spheroid) and therefore one axis is sufficient to define orientation

Note: I see a similar question asked in the forum. But, it is about an ellipsoid at origin and without any arbitrary orientation and here both arbitrary position and orientation are considered.

Answer:

Find affine transform M that translates this ellipse in axis-oriented one (translation by -p and rotation to align orientation vector r and proper coordinate axis).

Then apply this transform to point p and check that p' lies inside axis-oriented ellipsoid, i.e.

`x^2/a^2+ y^2/b^2+z^2/c^2 <= 1`

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