In the same way that the drawn sector with angle theta contains theta / 2pi of the circle's area, I would imagine that the drawn "solid sector" with solid angle omega contains omega / 4pi of the sphere's volume. Does this end up being the case? It passes the sanity check of being 100% at omega = 4pi and 0% at omega = 0.
lycorisradiatu
@zachtam that is a very interesting question. I searched online and it seems that for volume inside a solid angle, we can use equation V = ((4 * pi * r^3) / 3) * (omega / (4 * pi)) = (r^3 * omega) / 3, which indicates that the volume of "solid sector" as you said would be proportional to the volume of the sphere by 4pi.
tdkng
What exactly does the solid angle (omega) look like on the visual? I'm having a hard time grasping what omega represents since I've usually just worked with angles in 3D by using a 2D angle that lies on some 2D plane or axis.
spegeerino
@zachtam Yes, this is true. It can be proven by integrating over the relevant part of the surface of the sphere and dividing by the sphere's volume. Intuitively, as the area A of the sector approaches 0, the volume of the piece of the sphere subtended by the radius moving along the sector approaches that of a prism with height R (radius) and base A, or AR in other words, so the volume of the piece of the sphere should be proportional to the area of the sector on the surface.
GarciaEricS
When I learned about solid angles, I found myself wondering why we don't normalize these angle measurements. I was thinking, since we are basically wondering about the proportion of length to the circumference in the case of radians, or area compared to total surface area in the case of solid angles, why don't we normalize our angles so they are always between 0 or 1, ie a proportion. I believe the reason we don't do this, is because firstly, it doesn't really matter since the math with all work out in the end because solid angles is still proportional to the area measured, so we may always just divide or multiply by 4pi to keep everything in line. And secondly, it makes our calculus a lot worse. If we had for example that 1 radian is a full revolution around a circle, derivatives for things like sin(x) would look much worse with some adjustment factors. So we avoid this, and define angles and solid angles like we do here.
weszhuang
@tdkng
My personal visualization is just thinking of it as the size of A because there is no nice visualization for the solid angle associated with the center. As the solid angle could take some weird amorphous and arbitrary shape. Idk if this helps.
Refangs
I was very confused about what solid angles are (and still kind of am) but I think one way of thinking about a solid angle (that is slightly different from the slide) is that it is a measure of the proportion of the surface area of the sphere that some arbitrary piece of area takes up. If it is 4pi steradians then the area takes up the whole sphere. if it's 2pi steradians then it takes up half the sphere (of somewhat arbitrary shape as far as I know. but maybe it has to be contiguous). Kinda like for a circle how an angle of pi radians means the arc takes up half the circle.
sebzhao
Solid angles are also defined as a measure of the field of view that a given object covers [https://en.wikipedia.org/wiki/Solid_angle], and describes how the sun and moon project the same solid angle (which is why there can be eclipses).
In the same way that the drawn sector with angle theta contains theta / 2pi of the circle's area, I would imagine that the drawn "solid sector" with solid angle omega contains omega / 4pi of the sphere's volume. Does this end up being the case? It passes the sanity check of being 100% at omega = 4pi and 0% at omega = 0.
@zachtam that is a very interesting question. I searched online and it seems that for volume inside a solid angle, we can use equation V = ((4 * pi * r^3) / 3) * (omega / (4 * pi)) = (r^3 * omega) / 3, which indicates that the volume of "solid sector" as you said would be proportional to the volume of the sphere by 4pi.
What exactly does the solid angle (omega) look like on the visual? I'm having a hard time grasping what omega represents since I've usually just worked with angles in 3D by using a 2D angle that lies on some 2D plane or axis.
@zachtam Yes, this is true. It can be proven by integrating over the relevant part of the surface of the sphere and dividing by the sphere's volume. Intuitively, as the area A of the sector approaches 0, the volume of the piece of the sphere subtended by the radius moving along the sector approaches that of a prism with height R (radius) and base A, or AR in other words, so the volume of the piece of the sphere should be proportional to the area of the sector on the surface.
When I learned about solid angles, I found myself wondering why we don't normalize these angle measurements. I was thinking, since we are basically wondering about the proportion of length to the circumference in the case of radians, or area compared to total surface area in the case of solid angles, why don't we normalize our angles so they are always between 0 or 1, ie a proportion. I believe the reason we don't do this, is because firstly, it doesn't really matter since the math with all work out in the end because solid angles is still proportional to the area measured, so we may always just divide or multiply by 4pi to keep everything in line. And secondly, it makes our calculus a lot worse. If we had for example that 1 radian is a full revolution around a circle, derivatives for things like sin(x) would look much worse with some adjustment factors. So we avoid this, and define angles and solid angles like we do here.
@tdkng My personal visualization is just thinking of it as the size of A because there is no nice visualization for the solid angle associated with the center. As the solid angle could take some weird amorphous and arbitrary shape. Idk if this helps.
I was very confused about what solid angles are (and still kind of am) but I think one way of thinking about a solid angle (that is slightly different from the slide) is that it is a measure of the proportion of the surface area of the sphere that some arbitrary piece of area takes up. If it is 4pi steradians then the area takes up the whole sphere. if it's 2pi steradians then it takes up half the sphere (of somewhat arbitrary shape as far as I know. but maybe it has to be contiguous). Kinda like for a circle how an angle of pi radians means the arc takes up half the circle.
Solid angles are also defined as a measure of the field of view that a given object covers [https://en.wikipedia.org/wiki/Solid_angle], and describes how the sun and moon project the same solid angle (which is why there can be eclipses).