Hi Amin,
Since it appears that you know the rotation matrix, you can rotate the point p in question in the opposite direction to how you rotated the ellipsoid, keeping in mind that you have to rotate about the same rotation axis as before. For point p, the coordinates (x,y,z) --> (x1,y1,z1). Then use the new coordinates in the equations that you show above. Except that the ellipse is centered at xc,yc,zc so correct equation is actually
(x-xc)^2 (y-yc)^2 (z-zc)^2
-------- + -------- + -------- <1 inside or >1 outside
xr^2 yr^2 zr^2
Best Answer