- #1
CraigH
- 222
- 1
Hi,
It's surprising how little information is available on this topic, considering it seems like such a fundamental problem. The only tutorial I have found is http://www.umich.edu/~ners312/Course%20Library/SolidAngleOfADiskOffAxis.pdf, and my university does not have access to the other papers on the topic.
I'm finding it really difficult to understand this tutorial, so I would really appreciate it if someone could help explain how to do this calculation. It's probably best explaining this question with an example. See the picture bellow:
http://imageshack.us/a/img824/2072/tfw0.jpg
In the first example the source is directly above the disk. The solid angle is the area intersected by the cone and a unit sphere centered around the source, so for this example the calculation is easy. The area intersected (and hence the solid angle) will be the area of the disk divided by the square of the distance between the source and the disk. So if the source is 20m above a 5m radius disk, it will subtend an angle of ∏/16
[itex]\frac{\pi*5^{2}}{20^{2}} = \frac{\pi}{16}[/itex]
In the second example the source lies on the plane of the disk. This is also easy to calculate; it subtends an angle of 0∏.
The third example is where I am having difficulties. Because it is off axis it will have a smaller angle subtended, even if the distance is the same. So if the source is 20m away from the center of the 5m radius disk, and it is ∏/4 radians counterclockwise from the y-axis (the vertical axis coming out of the plane) , and 3∏/8 radians counterclockwise from the z axis (the horizontal axis going into the screen), what angle does it subtend the disk?
Here's my attempt so far
If I try and follow the tutorial I posted it seems like there is a lot of steps. First I have to check whether or not the source lies outside the perimeter of the disk. It does, so the formula to find the solid angle is:
[itex]Ω = - \frac{2L}{R_{max}}K(k) + \pi\Lambda_{0}(\xi,k)[/itex]
where:
L = the perpendicular height of the source above the plane of the disk.
R max = the maximum distance between the source and a point on the perimiter of the disk.
K(k) = Legendre's form of a complete elliptic integral of the first kind. K is a function and k is the argument.
k = square root of (1 - (R1^2)/(R max ^2)).
R1 = ? I can't work this out
lambda 0 (xi,k) = Heuman lambda function
xi = arcsin (L/R1)
This is all I can deduce from the tutorial. However I can not work out what R1 is, and I do not know how to do the elliptic integrals. The tutorial references the handbook "elliptic integrals for engineers and scientists" which I have a copy of, however this is even more difficult than the original tutorial.
I have looked online and found that
[itex]K(k) = \int^{\pi/2}_{0}\frac{1}{\sqrt{1-k^{2}sin^{2}(t)}}dt[/itex]
But I do not know what t is, or how to calculate this integral.
I have also found that
[itex]\Lambda_{0}(\xi,k) = \frac{2}{\pi} (E(k) F(\xi,k') - K(k) E(\xi,k') + K(k) F(\xi,k'))[/itex]
where:
F(xi,k') = incomplete elliptic integral of the first kind
E(k) = complete elliptic integral of the second kind
I also have formulas for these which I found online, but like the first elliptic integral, I do not know how to calculate them.Even though I have posted a lot of information here I understand very little of it, it may be completlely wrong. It also looks very complicated, is there an easier way? I just want to know how to find the solid angle subtended by an off axis disk. I don't even care that much if I do not understand where the calculations have come from, at this point I just want to be able to do them. I'd also like to point out that once I know how to do them I plan on coding a MATLAB function that will calculate them for me.
So please, can somebody tell me how I find this angle?
Thankyou very much!
It's surprising how little information is available on this topic, considering it seems like such a fundamental problem. The only tutorial I have found is http://www.umich.edu/~ners312/Course%20Library/SolidAngleOfADiskOffAxis.pdf, and my university does not have access to the other papers on the topic.
I'm finding it really difficult to understand this tutorial, so I would really appreciate it if someone could help explain how to do this calculation. It's probably best explaining this question with an example. See the picture bellow:
http://imageshack.us/a/img824/2072/tfw0.jpg
In the first example the source is directly above the disk. The solid angle is the area intersected by the cone and a unit sphere centered around the source, so for this example the calculation is easy. The area intersected (and hence the solid angle) will be the area of the disk divided by the square of the distance between the source and the disk. So if the source is 20m above a 5m radius disk, it will subtend an angle of ∏/16
[itex]\frac{\pi*5^{2}}{20^{2}} = \frac{\pi}{16}[/itex]
In the second example the source lies on the plane of the disk. This is also easy to calculate; it subtends an angle of 0∏.
The third example is where I am having difficulties. Because it is off axis it will have a smaller angle subtended, even if the distance is the same. So if the source is 20m away from the center of the 5m radius disk, and it is ∏/4 radians counterclockwise from the y-axis (the vertical axis coming out of the plane) , and 3∏/8 radians counterclockwise from the z axis (the horizontal axis going into the screen), what angle does it subtend the disk?
Here's my attempt so far
If I try and follow the tutorial I posted it seems like there is a lot of steps. First I have to check whether or not the source lies outside the perimeter of the disk. It does, so the formula to find the solid angle is:
[itex]Ω = - \frac{2L}{R_{max}}K(k) + \pi\Lambda_{0}(\xi,k)[/itex]
where:
L = the perpendicular height of the source above the plane of the disk.
R max = the maximum distance between the source and a point on the perimiter of the disk.
K(k) = Legendre's form of a complete elliptic integral of the first kind. K is a function and k is the argument.
k = square root of (1 - (R1^2)/(R max ^2)).
R1 = ? I can't work this out
lambda 0 (xi,k) = Heuman lambda function
xi = arcsin (L/R1)
This is all I can deduce from the tutorial. However I can not work out what R1 is, and I do not know how to do the elliptic integrals. The tutorial references the handbook "elliptic integrals for engineers and scientists" which I have a copy of, however this is even more difficult than the original tutorial.
I have looked online and found that
[itex]K(k) = \int^{\pi/2}_{0}\frac{1}{\sqrt{1-k^{2}sin^{2}(t)}}dt[/itex]
But I do not know what t is, or how to calculate this integral.
I have also found that
[itex]\Lambda_{0}(\xi,k) = \frac{2}{\pi} (E(k) F(\xi,k') - K(k) E(\xi,k') + K(k) F(\xi,k'))[/itex]
where:
F(xi,k') = incomplete elliptic integral of the first kind
E(k) = complete elliptic integral of the second kind
I also have formulas for these which I found online, but like the first elliptic integral, I do not know how to calculate them.Even though I have posted a lot of information here I understand very little of it, it may be completlely wrong. It also looks very complicated, is there an easier way? I just want to know how to find the solid angle subtended by an off axis disk. I don't even care that much if I do not understand where the calculations have come from, at this point I just want to be able to do them. I'd also like to point out that once I know how to do them I plan on coding a MATLAB function that will calculate them for me.
So please, can somebody tell me how I find this angle?
Thankyou very much!
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