# Determining if a point is inside an infinite 3d elliptical tilted cone

I have an infinite 3d elliptical, tilted cone that is defined by a vertex point P(x,y,z) and by 4 angles: the first pair of angles represent the spatial orientation of the cone: θ is the polar angle and φ is the azimuthal angle (in a similar way to the way spherical coordinates are defined). the second pair of angles, α and β, represent the vertical and horizontal "aperture" of the cone (I'm not sure if it's the proper terminology), respectively.

I want to check whether a point U(x2,y2,z1) is inside this cone. How can I do that?

I've found similar questions here, but in those questions the cone was defined differently, and was not tilted. therefore I'm not sure how to apply the previously offered solutions to this problem.

1. Translate $U$ by $-(x,y,z)$
2. Convert the translated point to spherical coordinates - call the resulting point $(r, t, p)$.
3. Check if $({t - \theta \over \alpha})^2 + ({p - \phi \over \beta})^2 < 1$