Earth stereographic projection line intersection

[manyhats]

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.

 

[Valerie Park]

The answer to your question is that it depends on the exact angle of the vector coming off the tip of your finger, as well as the shape of the Earth. The Earth is not a perfect sphere, but is an oblate spheroid with a radius of 6,378.1 km at the equator and 6,356.8 km at the poles. This means that the vector will travel different distances based on its angle and where it originates from. In order to accurately calculate the destination point of the vector, you would need to use a combination of stereographic projection and matrix algebra. Stereographic projection is a way of projecting points on the surface of a sphere onto a plane, and matrix algebra is used to represent rotations in three-dimensional space. Together, these two methods can be used to calculate the destination point of the vector given the starting point, the angle of the vector, and the shape of the Earth.