How to find the z component of the parameterization of an ellipse that is the intersection of a vertical cylinder and a plane

Given the vertical cylinder with radius 1 ($x^2 + y^2 = 1$) oriented with an outward pointing normal and its intersection with the plane $z = xtan(phi)$

Consider the coordinates $tilde{x} = sqrt{x^2 + z^2}$ and $tilde{y} = y$

1. Show that this intersection is an ellipse
2. Find the curvature, geodesic curvature, and normal curvature of this ellipse (I do not actually need help solving these, but need help to find the parameterization so I can start solving them)

So, I did the following

$tilde{x} = sqrt{x^2 + (x tan(phi))^2} = sqrt{x^2 + x^2*tan^2(phi)} = sqrt{x^2 (1 + tan(phi))} = xsec(phi)$

and of course $tilde{y} = y$

and thus the intersction is $tilde{x}^2 + tilde{y}^2 = 1$

and to make it look like an ellipse you can do

$frac{x^2}{cos^2(phi)} + frac{y^2}{1^2} = 1$

where $ a = cos(phi)$ and $b = 1$

Now that I have found the ellipse, I need to work on finding the curvature, geodesic curvature, and the normal curvature. To do that the hint was given to parameterize the ellipse in terms of the roational angle $theta$ around the cylinder. So, I looked online and I saw that a general paraneterization for an ellipse is

$x = acos(t)$ and $y = bsin(t)$

However, upon running this information by my instructor I was told that this is only the ellipse, if the ellipse was planar. Therefore my paramterization needs a z-component. I was told that the parameterization is usually something along the lines of

$(x, y, z) = (cos(theta), sin(theta), 0)$

But for this I need to rotate through phi in the xz plane to get my parameterization and I’m not exactly sure what to do/what that looks like. Any advice? Thanks!