The problem concerns visible area based on a field of view from the center of a sphere.
I was never taught spherical trigonometry so even basic terminology is hard. After trying to figure out the problem for a few hours by myself I know some terms but don't fully understand them.
From the center of a sphere, based on two central angles (in degrees) that fall along two circular, perpendicular planes (x and y in what would be spherical coordinates I think).
How large is the visible surface area based on one horizontal and one vertical angle? As in, for a person who can turn left-right/up-down 360°?
This interactive Wolfram Demonstration has solutions for spherical triangles but those need 3 input parameters and I don't know enough about them to tell if they're even the right shape to cover all cases. It glitches on some inputs, too.
- hemisphere: would result from a full 360° horizontal/left-right angle and a 90° vertical/up-down angle; this could be represented by a "spherical triangle" which itself has all 3 angles at 180°.
- semi-hemisphere: 180° horizontal, 90° vertical or a spherical triangle that has 90°, 90° and 180° angles.
Those are the simplest subsections of a sphere where the area are whole multiples of
What's the formula that would allow the calculation of this surface area from these two central angles, for cases like 45°/270°, 180°/10°, any other number?