Surf area of sphere thru rectangular FOV

In summary: I don't understand what you're asking.In summary, the sphere viewed in the FOV has a surface area that is proportional to the integral of the radius over the surface of the sphere.
  • #1
Seriously
11
0
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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  • #2
Can anyone help with this?
 
  • #3
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...
 
  • #4
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...
 

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  • #5
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?
 
  • #6
I'm not sure that I understand your point/question.
 
  • #7
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?
 
  • #8
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?
 
  • #9
Can anyone help with this?
 

Related to Surf area of sphere thru rectangular FOV

1. What is the surf area of a sphere through a rectangular field of view?

The surf area of a sphere through a rectangular field of view is the amount of surface area that is visible when looking at the sphere from a specific angle or perspective. It is influenced by the size and shape of the field of view, as well as the radius of the sphere.

2. How is the surf area of a sphere through a rectangular field of view calculated?

The surf area of a sphere through a rectangular field of view can be calculated using the formula A = 4πr²sin(θ/2)², where A is the surf area, r is the radius of the sphere, and θ is the angle of the field of view. This formula assumes that the center of the sphere is aligned with the center of the field of view.

3. How does the surf area of a sphere change with different field of view angles?

The surf area of a sphere will increase as the angle of the field of view increases. This is because a larger field of view allows for more of the sphere's surface area to be visible. However, the increase in surf area will not be proportional to the increase in field of view angle, as it is affected by the sine of the angle in the calculation formula.

4. Can the surf area of a sphere through a rectangular field of view be greater than the surface area of the sphere itself?

No, the surf area of a sphere through a rectangular field of view cannot be greater than the surface area of the sphere. The maximum possible surf area of a sphere through a rectangular field of view is equivalent to the total surface area of the sphere, which is 4πr².

5. How is the surf area of a sphere through a rectangular field of view used in scientific research?

The surf area of a sphere through a rectangular field of view is often used in fields such as astronomy and optics to calculate the amount of light or energy that is captured by a telescope or lens. It is also used in computer graphics to create realistic 3D images and in simulations for virtual reality. Additionally, it can be used in engineering and architecture to determine the visibility and coverage of structures or objects from a specific viewpoint.

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