Lesson #2 - Directions and Solid Angles
Reading Assignment: Intro. to Ch. 2, Section 2.1This is one of the shortest readings in the course, but it is one of the hardest ones to master (so these notes will be more complete than usual). The determination of solid angle draws on calculus that you may not have used in a while and requires a little thought.
DirectionAt first glance, the direction part looks like it would be easy; we all know what "that way" means. But the idea of direction depends on comparison with a set of reference directions. You are usually free to pick the reference directions to fit the problem at hand.
The first choice of direction references that occurs to us is the 3 Cartesian directions - ,, and -- which we will recall are unit length vectors in the directions of the x, y, and z axes, respectively:
Consider a vector that is also unit length and points in the 1st quadrant (i.e., +x,+y,+z):
The simplest way to characterize its direction is to "drop" perpendiculars to each of the three Cartesian axes and denote the direction from the lengths (u,v,w) of these three projections:
so that we have:
This 3-coordinate directional approach is intuitive, logical, and easy to understand. Unfortunately, though, we seldom use it for two principal reasons:
so, if you know two of them, the third can be deduced from those two.
This gives us one dimension, what about the other? Well, in following our Earth analogy, that first angle gave us a latitude-like variable (although Earth latitude is measured from the Equator, not from the North Pole), so we follow with a longitude-like variable by projecting onto the x-y plane, call the new (flat) direction , and use the angle between this projected vector and the (arbitrarily chosen) x axis as the second angle, which we will denote as :
This gives us a 2-dimensional representation of direction that is not only more concise than the (u,v,w) representation, but also turns out to be more useful (if the polar axis is properly chosen).
Solid Angle of a section of a sphereThis gives a representation of the direction of a vector, ,in terms of the angles and , but since we are going to want to integrate over all directions, we must relate the differentials as well. For this, we return to our graph and put in a differential area element, , bounded by latitude and longitude lines:
From this figure, we see that the "north-to-south" lines that border the element have length , but the "east-to-west" lines have a length equal to (since the distance you must travel "around the world" on a give latitude line gets shorter as you get closer to the North Pole). Using these two differentials allows us to express the differential solid angle as:
This representation of is most useful for situations in which we want to determine the solid angle associated with a section on the surface of a sphere -- especially a section that is bordered by constant and lines.
Example:The solid angle associated with a region on a sphere (not necessarily a unit sphere) bordered by , , , and is given by:
Solid Angle of a Cartesian Surface ElementA second way of attacking solid angle that is equally valid, and better in some situations, is to recognize that solid angle subtended by a differential area (from a given point) is equal to the projection of the area (i.e., the area as seen from the point) divided by the square of the distance from the point to the differential area. For example, in the following figure:
the distance from Point P to the differential area is given by R and the projected area of dA from the point P is:
the solid angle is the (slightly unwieldy):
This representation is most useful for determining the solid angle of a rectangular surface, although the integrals tend to be difficult to work out.
Example:The solid angle subtended by a rectangular region of width W and length L, as seen from a point a distance z perpendicularly above the center is given by:
Homework problem 2.6 gives a solution for this in closed form.
Solid Angle of a Cylindrical Surface ElementNote that the same problem with the volume element expressed in cylindrical coordinates in the x-y plane would not be so messy:
In this case, the solid angle works out to be:
and z is a constant, we can differentiate both sides to get:
Substituting this gives us:
This representation is most useful for determining the solid angle of planar surfaces that are sections of disks. You may find this useful in doing Homework problem 2.1.
Reduction of more complicated shapes to one of the threeNow that we know how to attack solid angle determinations for three situations:
Flat rectangular surfaces
Flat disk-section surfaces,
Therefore, the solid angle of a given 2D or 3D object (as measured from a Point P) can be found by finding the solid angle of the object's shadow cast onto either a flat surface or an enclosing sphere, whichever is most convenient.
We will work some examples in class.
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