I'd like to describe the points on Earth's surface that are visible to a viewer who is 100 miles above the North Pole. I'm assuming a spherical Earth and radius of 3960 miles if it's needed. My thoughts are that you'll be able see a portion of the Earth, all the way to a "horizon", and this horizon can be described by a circle. Is there any way to find the equation of this circle? All I really need is the radius to do this. I just don't seem to have enough information to find the distance from the viewer to the horizon, which seems like the most direct way forward. Is there something else I can do? Thanks so much!
[Math] Describing points on Earth’s surface visible from space
geometry
Best Answer
As BeaumontTaz suggested, I've written an answer to my own question.
First, draw a triangle with vertices the viewer, the center of the Earth, and a point on the Earth's surface. This is a right triangle because a radius and tangent line form a right angle at the point of tangency. The hypotenuse is 4060 miles, the longer leg is 3960 miles, and the short leg is 895.545 miles (from the Pythagorean theorem). The angle at the viewer can be found by the law of cosines and is equal to $77.257^\circ$. Next the radius of the circle we are interested in is the length of the altitude drawn to the side with length 4060 miles. The length of this altitude is $895.545\sin 77.257=873.487$ miles. Then the equation for the circle we are looking for is $x^2+y^2=762979$ and it is $3862$ miles above the center of the Earth.