Re: Solid angle


qwerty wrote:
I'm struggling a bit with the concept of the solid angle. I
understand that the measurement of that solid angle in steradians is
the the area of some object's projection to a sphere, divided by the
square of the sphere's radius.

My question is this: Let's say there's an object some distance from a
point. We can measure the angle it subtends by measuring the area it
projects to on a sphere. Will any sphere do? If the object lies 10m
away, will the angle measure the same if we measure it with a 2m
radius sphere or with a 5m radius? How about a 20m radius sphere? My
first conlusion is that if the larger sphere we take the less area
that object will project to on its surface. I know I'm wrong.

Seems that when you divide the area of the projection of the object
onto a spehere of radius R by the square of that sphere's radius (R^2),
the result will not be dependent on the sphere's radius anymore(since
the surface area due to the projection has a R^2 dependence), and so
you should be able to use any sphere. I believe this is the reason why
one way of computing the solid angle is to consider the unit sphere,
since then you don't have to do any division (although, for the reasons
mentioned above, it seems that any sphere will do: the unit sphere is
just used for the sake of convenience).

Using the unit sphere, then you can think about solid angle as just the
integral of the unit sphere's surface of the projection wrt phi and
theta (i.e. the two spherical coords)

Matt

.

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