Drawing a line, a plane, and a sphere, and their points of intersection

Question

The following code renders a sphere and a plane. How do I get the plane to be at the south pole of the sphere - and centered? On this diagram, I would like to draw a ray from the north pole N through the point P = (-\sqrt{3}, -1/\sqrt{2},-1/\sqrt{2}) and through the point Q = ((-\sqrt{3}/7)(-8+2\sqrt{2}), (-\sqrt{2}/14)(-8+2\sqrt{2}), -2) of intersection between the ray and the plane, and I would like to mark N, P, and Q with a dot. (I want to keep the code in a TikZ environment.)

\documentclass{amsart}

\usepackage{mathtools,array}

\usepackage{tikz}
\usetikzlibrary{calc,intersections}


\begin{document}

\begin{center}
\begin{tikzpicture}

\shade[ball color=blue!25] (0,0,0) circle (2);
\fill[opacity=0.25] (-3,0,4) -- (-1,2,4) -- (5,2,4) -- (3,0,4) -- cycle;

\end{tikzpicture}
\end{center}

\end{document}

Answer 1

This example could be useful to you: http://texample.net/tikz/examples/map-projections/ .