Surf area of sphere thru rectangular FOV

[Seriously]

I have a rectangular field of view (FOV) through which I am viewing a sphere at a distance such that I can only see small parts of the sphere through the FOV at a time. Usually my FOV contains part of the sphere and free space. My question is: how can I calculate the surface area of the sphere I am viewing in my FOV?

I think this will be solved with a double integral of the form

A = INTGRL INTGRL r^2 * sin(latitude) d(latitude) d(longitude)

but I'm not sure what my limits of integration should be, especially because 1) the edges of my FOV aren't necessarily parallel to the longitude or latitude lines (though maybe it doesn't matter, as the sphere is symmetric) and 2) the FOV goes off the edge of the sphere and I don't know how to handle this.

Or I could be going about this the completely wrong way, lol. Can anyone help? Or think of some references or key words I can search on? Thanks!

 

[Seriously]

Can anyone help with this?

 

[berkeman]

Don't you need to know the distance to the sphere? The surface area is going to vary a lot depending on how far away it is, even though the apparent size in your FOV stays the same...

 

[Seriously]

I have given neither the radius of the sphere nor the distance to it. Nor have I given the size of my field of view (FOV). All of these things are constants (I do have these values, but I don't think they're necessary - I'd like to do this symbolically). What is not constant is the position of the FOV with respect to the sphere.

I need a method for determining the surface area of the sphere as seen through a FOV that not only shifts but goes off the sphere. The easiest case is, I suppose, the symmetric one. I will attempt to attach an image...

 

[berkeman]

Well, since the sun and moon subtend about the same angle in the sky, how can you look through a small window and calculate their respective surface areas?

 

[Seriously]

I'm not sure that I understand your point/question.

 

[berkeman]

I'm not sure that I understand your point/question.

I may be misunderstanding your question, but it seems underconstrained as stated. If you only have a view of the sphere, but no information about how far away it is, how are you supposed to figure out how big it is? In my example, the sun and moon appear to have the same size to us. But the sun is actually much bigger and farther away. What am I missing in your problem statement?

 

[Seriously]

Ah, ok. I had hoped to solve this symbolically. Can't we just say that the radius of the sphere is R. The sphere is far enough away, at distance D, and the field of view small enough (say width W and height H) that we see some small portion of the sphere, as given in the picture.

If numbers are necessary... Here is an example to illustrate. Let R = radius of the moon = 1.738*10^3 km, and D = distance to the moon = 384.4*10^3 km. The FOV is determined by your telescope and is such that you can't see the entire moon, or even half of it -- you are constrained to viewing one small portion of the moon at a time. Capture an image of the moon with some recording device. Your image has width W and height H, and the moon fills up some portion of the image. The question is - how much of the lunar surface area is seen in the image?

Does that make sense?

 

[Seriously]

Can anyone help with this?