To reiterate, angle is ratio of subtended arc length on circle to radius (the length of the arc created by the angle over the length of the entire circle, which is 2pi). On the other hand, solid angle is the ratio of subtended area on sphere to radius squared (the area created by the solid angle / the entire area of the sphere, which is 4pi steradians). As a note, steradian is radians squared, and is also the SI unit of solid angle!

Is there a way to visualize $\Omega$ like we can visualize $\theta$ ? I know $\Omega$ is just a unitless ratio but so is $\theta$ - except $\theta$ can be intuitively presented as an "angle".

To anyone confused about how to get the subtended area on a sphere, its equation is A = 2·pi·r·h. I wasn't familiar with the formula myself so I thought putting it here might be helpful.

I'm still confused how d(Omega) becomes a vector in the following slides but is non-representable here.

What shape is the "subtended area" A supposed to be? Is it meant to be an weird blob-ish shape as in the slides, or is it supposed to be round, or something else?

Kind of following the above questions - we have a way of defining $\theta$ to be an angle which can be standardized by setting 0 to be the positive axis. Is there a standard definition of the steradian?

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@julialuo The steradian cuts out an area of r^2. Makes sense that the sphere has 4pi steradians then, since the surface area is 4pi*r^2.

Does anybody have a more intuitive explanation for what the radius squared represents in the solid angle equation?

@Carpetfizz: I could be wrong here as I've mostly just thought about it, but I believe that there isn't really some well-defined way to visualize the steradian. I think this is because, while we can unambigously visually represent a 1D length $l$ along the circumference of the circle, we can't do the same for an area $A$ on the surface of the sphere. (I.e., we can unambiguously define a segment of length $l$ on the circle by its endpoints (up to rotation of course), but there are infinitely many ways to define a shape on the sphere with area $A$ — for example, think about all the squares you can create that have area 1, and consider the rational numbers). Perhaps there is a canonical way to think about them by just fixing the shape to be a circular projection, but I'm not sure how that would necessarily be useful.

For anyone looking for more context to @knguyen0811's comment, this is referring to the area subtended by a cone with angle $\theta$. (Of course, we can obtain the area from the steradian definition as $A = \Omega r^2$.) For the derivation and more reading, the Wikipedia page on spherical caps is pretty helpful.

@ellenluo I think the two symbols are referring to different things. Note that here we use capital $\Omega$ and the symbol in slide 18 is lower case $\omega$. Although, $\omega$ could maybe be considered as an infinitesimal solid angle. Note that from slide 17 it seems like we have $\Omega = \int_A d\omega$.

@sandykzhang If you read my previous comment, I provide some thought regarding why there might not be a canonical way to visualize the subtended area. In other words, it's maybe not really "meant" or "supposed" to be any shape in particular, because there are many to choose from. (Although, perhaps the projected circle is the easiest to think about.)

@julialuo Not sure, but I think the closest thing to a canonical visualization would be the spherical cap with area $A$ centered at the north pole. Again, not sure how this would necessarily be useful because I'm pretty sure solid angles are typically used with regard to objects of varying geometry.

@wangcynthia Forgive the somewhat circular definitions (haha puns, right?) I use in this comment, but I believe one intuition behind the $r^2$ in the definition for steradians is the same as one intuition behind the $r$ in the definition for radians. One of the classical definitions for $\pi$ is the ratio of circumference to diameter. That is, $\pi = \frac{C} {d} = \frac{C} {2r} \rightarrow \frac{C} {r} = 2\pi$. From this definition the relationship with radians becomes clear: the maximum value for $l$ is $C$, so the radians $\frac{l} {r}$ simply give us the fraction of the full circumference that we cover (as we're used to). To make this relationship even clearer, consider the equality $\frac{l} {r} = \frac{k \cdot C} {r} = k \cdot 2 \pi$ radians, where $k$ is the fractional value.

Then, if we take on faith (for the moment) that the surface area of a sphere is $4 \pi r^2$ (which is also then the maximum value for $A$), we see that the steradians similarly give us the fraction of the full surface area that we cover.

I guess in some sense this shows that the radius $r$ doesn't necessarily represent anything insightful, as in the circular case it seems like it's kinda just there as an artifact from the definition of $\pi$.

I thought a solid angle was not a unit-less quantity but rather an analog to angle in 3D space. If we recall from calculus the subtended arc length is r x theta, so dividing the arc-length by theta gives the angle. This formula for arc-length comes from the circumference of a circle. In 3D, the analog to the circumference is surface area of a sphere. From calculus, we know the surface area is 4 x pi x r^2, so we need to divide by r^2 as solid angle should not be dependent on radius of a sphere (This is because it's the analog of angle which is independent of radius). We can gain further intuition for the r^2, by thinking of A as a square formed by two subtended arcs in 2D (Just intersect the sphere with 2 planes). This is the same idea as Bezier surface patches from project 2. Then, if the two arc lengths are r x theta and r x phi, we can approximate the area of A by multiplying the two sides to get r^2 x phi x theta. So, A is proportional to r^2 and we need to divide by r^2 to get a solid angle that is independent of radius.