earth stereographic projection line intersection

Hi,

Consider you are standing upright and pointing your finger at the ground. Where does the vector coming off the tip of your finger arrive when it hits ground level on the other side of the Earth?

..Think as if you were going to imperviously dig a hole through the Earth and could travel only on a line.. how would you know where you would arrive on the other side?

Here are some thoughts and research I've gathered:

longitude/latitude

oblate spheroid(6,378.1 km, 6,356.8 km) Earth

pitch
roll
yaw

I want to use these variables to get a longitude and latitude of where the line would intersect on the Earth. I suppose it would intersect twice and the first time would be your longitude and latitude. I think stereographic projection and matrix algebra should be used here but I am stumped.


I want to use a phone to make such a function given longitude/latitude and pitch roll and yaw.