The shape of the catenary, y = cosh(x) looks very much like that of the parabolic curve, y = x2. But is it close enough to do a decent job at focusing incomming light rays to a single focus?
The simple answer, as shown by the plots below, is that the catenary is not as efficient, but it is very close, especially when the bottom of the curve is used.
A reflector sheet hung symmetrically in this manner must track the sun's movement to maximize the ability for the reflector to focus.
I plot the two curves shown using Microsoft Excel. The curves are laid sideway to ease the calculation of the incident rays and reflecting rays (in red). This means that the parabolic curve is y = x1/2 and the catenary curve is y = arccosh(x).
For the mathematically inclined, to do such calculation, one must find the tangent to a given point in the curve by taking the derivative of the curves at that point, create a line there, then compare it against the slope of the incident line to calculate the slope of the reflecting line. Probably a good mid-term question for a college freshman math class. By the way, the derivative of an arccosh(x) curve is dy/dx = 1/(x2-1)1/2
The result from this calculation suggests that for the catenary to emmulate the parabolic reflector, it should have an aspect ratio of about 1 to 4, that is, it should be 1 unit deep by 4 unit wide. The following graph shows how the deviation ("DELTA") increases sharply when this 1:4 ratio is exceeded. [NOTE: a multiplier constant is used with the parabolic curve to "spread" it out to about the same level as the catenary curve]
One interesting phenomenon is that for a very simple implementation of this curve, at the price of lower efficiency, one can build a very large stationary reflector to accommodate the movement of the sun's changing angle. Such a reflector should be aligned east-west, and is asymmetrical, essentially, one half of the normal symmetrical reflector.
It is well known that the parabolic reflector quickly loses its ability to focus the incoming light as soon as the incindent angle deviates from dead straight (0 degrees).
By comparison, the catenary half-reflector will continue to be able to focus significant amount of light energy up to 30-40 degrees incident angle. By slightly adjusting the target instead of trying to move the reflector, one can still make use of some of the focusing ability of the catenary half-reflector. Sure, it's not as efficient as pointing the parabolic reflector directly at the sun; but you can make a huge reflector at a pittance, so who cares?
The set of graphs below show the movement of path of the reflected beams, in red, as the source of light changes from direct vertical to 10, 20, 30, 45, and 60 degrees from vertical. As expected, the focus point changes with the changing incident beams. However, by slight movements of the target, we can still make effective use of some of the incident beams.
If you would like a copy of the calculators that I made for these calculations, drop me an email. They are Microsoft Excel files.
Update: A very clever programmer participated in the project! Mr. Ben Schaffhausen created a very clever calculator that can be downloaded and run on the desk top. Ben expanded the model to calculate circular, parabolic, and catenary curves. He is also very generous in allowing me to host the current version of it here.