Measuring Angles Using Steradians

Date: 2/8/96 at 17:35:55
From: "Kathleen M. Scaffidi"
Subject: question

We are from Divine Savior Holy Angels High School in Milwaukee Wi.  It's 
an all girls, college prep h.s.   We came across a problem in one of our 
math books and we need help.

How do you measure a solid angle by using steradians?

Your reply would be greatly appreciated. 
The students from Mrs. Scaffidi's Honors Math III class
   Thanks so much!

Date: 2/14/96 at 10:1:27
From: Doctor Ethan
Subject: Re: question

Much like radian measure, which is a unitless description of the size 
of an angle in two dimensions, steradians are a similar measure of 
'solid angles' in three dimensions.  An angle in radians compares that 
angle to a complete circle (2*Pi).  Similarly, steradians relate solid 
angles (which you can imagine as cones radiating out from a point) to an 
angles that subtend an entire sphere (4*Pi).  The two are not 
interchangable, however.

But that's not a precise enough definition with which to do 
calculations.  

Consider some surface S enclosing a point P.  Now imagine a small cone
which intersects an area, da, on S.  The cone defines the solid angle
'subtended' by that area at point P.  By definition, the solid angle
is the area da projected on a plane perpendicular to the radius vector
from P to the area (R), and divided by the magnitude of R squared (r^2).

If dU is the solid angle subtended by da, we therefore have:

      R * da
dU = -------
       r^2

where * is being used as the _dot product_ (not multiplication).

The expression can be integrated over a region of S to find the total
solid angle subtended.

        /   R * da
U   =   |  -------- 
        |    r^2
       /       

If the angle between R and the vector normal to da is a constant, A, 
this expression can be simplified to:

        /  da(cos A)
U   =   |  -------- 
        |    r^2
       /       
I hope you followed my rather crude attempt to draw an integral symbol.
Keep up the good work!
         
-Doctor Ethan,  The Math Forum