This helpful video also explains how to calculate the solid angle with animations and visualizations: https://www.youtube.com/watch?v=VmnkkWLwVsc
LeslieTrue
It's a common and useful representation in spherical coordinates, i.e. intergration of the cone represented by dA is the total volumn of the sphere. It can also be useful when calculating moment of Inertia for any shape in spherical coordinates.
LeslieTrue
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edithllontop1
What if I wanted to find how much light was projected onto a wall, does this equation still apply? The wall would have a flat surface but the area that we are calculating here is curved
jonathanlu31
I think worst case scenario, you could integrate the irradiance over each point on the wall, but otherwise, maybe there could be someway to project the wall on to the surface of the sphere? I'm not sure, but adding on this, how do steradians work for nonconvex areas?
chetan-khanna
I was wondering what the generalisation of angles in n dimensions were (since we just moved “up a dimension” to get solid angles, which I found really interesting). This paper (https://link.springer.com/content/pdf/10.1007/s00454-006-1253-4.pdf) delves into the derivation of solid angles (in n-dimensions!) among other things, but the main thing I saw in it (and took away from it) was that is no closed form formula as one might otherwise be reflexively be inclined to think (i.e. one cannot just integrate again).
This helpful video also explains how to calculate the solid angle with animations and visualizations: https://www.youtube.com/watch?v=VmnkkWLwVsc
It's a common and useful representation in spherical coordinates, i.e. intergration of the cone represented by dA is the total volumn of the sphere. It can also be useful when calculating moment of Inertia for any shape in spherical coordinates.
[deleted]
What if I wanted to find how much light was projected onto a wall, does this equation still apply? The wall would have a flat surface but the area that we are calculating here is curved
I think worst case scenario, you could integrate the irradiance over each point on the wall, but otherwise, maybe there could be someway to project the wall on to the surface of the sphere? I'm not sure, but adding on this, how do steradians work for nonconvex areas?
I was wondering what the generalisation of angles in n dimensions were (since we just moved “up a dimension” to get solid angles, which I found really interesting). This paper (https://link.springer.com/content/pdf/10.1007/s00454-006-1253-4.pdf) delves into the derivation of solid angles (in n-dimensions!) among other things, but the main thing I saw in it (and took away from it) was that is no closed form formula as one might otherwise be reflexively be inclined to think (i.e. one cannot just integrate again).