Surf area of sphere thru rectangular FOV

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Homework Help Overview

The discussion revolves around calculating the surface area of a sphere as viewed through a rectangular field of view (FOV). The original poster is exploring the mathematical approach to determine this area, particularly through the use of double integrals, while considering the complexities introduced by the FOV's position and orientation relative to the sphere.

Discussion Character

  • Exploratory, Assumption checking, Problem interpretation

Approaches and Questions Raised

  • The original poster suggests using a double integral to calculate the surface area but is uncertain about the limits of integration due to the FOV's alignment and its potential to extend beyond the sphere. Other participants question the necessity of knowing the distance to the sphere and the implications of not having specific values for the radius or FOV dimensions. They also raise concerns about the underconstrained nature of the problem.

Discussion Status

Participants are actively engaging with the original poster's question, raising important considerations about the variables involved and the assumptions being made. Some guidance has been offered regarding the need for specific measurements, but there is no explicit consensus on how to proceed with the calculations.

Contextual Notes

The original poster has not provided specific values for the radius of the sphere, the distance to it, or the dimensions of the FOV, indicating a preference to approach the problem symbolically. This lack of information is noted as a constraint in the discussion.

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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!
 
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Can anyone help with this?
 
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...
 
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...
 

Attachments

  • SphereFOV.GIF
    SphereFOV.GIF
    6.9 KB · Views: 537
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?
 
I'm not sure that I understand your point/question.
 
Seriously said:
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?
 
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?
 
Can anyone help with this?
 

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