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[SOLAR] Math Note: Radius of Curvature of a Parabola

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To: <solar-concentrator@cichlid.com> 
Subject: [SOLAR] Math Note: Radius of Curvature of a Parabola 
From: "David Wells" <laseraxe@erols.com> 
Date: Mon, 11 Dec 2000 06:47:32 -0500 
Reply-To: solar-concentrator@cichlid.com 
Sender: solar-concentrator-errors@cichlid.com 

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For those of you interested, mathematics describing the parabola has been
brought up recently.  As you have seen, the general equation for a parabola
is:

    y=(1/4f) x^2

where y= the depth of the parabola
            x= the distance from the start or vertex of the parabola
            f= the focal length

So, for example, for a parabola with a focal length of four meters, at a
distance of 1 meter from the vertex, the depth is:

    y = (1/4*4) (1)^2
        = 1/16 meters

Now, if you wanted to construct this parabola out of smaller sections or
facets, you need to consider another thing.  There is a neat function in
math called the radius of curvature, which is:

             ( 1 + y'^2 )^3/2
   R(x) =    -----------------
                       y"

Those of you who've taken calculus will recognize y' as the derivative of a
function, and y'' as the second derivative.

For the  parabola,

    y' = x/2f

    y'' = 1/2f

For fun, let's evaluate the radius of curvature at two different points: at
the vertex (x=0), and at the point where x=1.

For our parabola with a four-meter focal length,

    y' = x/2f = x/8

    y'' = 1/2f = 1/8

Substituting into the equation for radius of curvature,

             [1 + (x/8)^2]^3/2
R(x)= -------------------------
                        (1/8)

Now, when you plug in for the vertex, x=0, then you get R(x))= 8 meters.
Great! This checks with your general knowledge that the focal point of a
cylinder or sphere is about 1/2 the radius AT THE VERTEX.

Now, when you plug in for the point one meter from the vertex, i.e. plug in
x=1, you get

             [1 + (1/8)^2]^1.5
 R(x)= ------------------------
                    (1/8)

        = 8.188

Similarly, you can evaluate for any point.  For x=2, you get

            [1 + (2/8)^2]^1.5
            ----------------------
                    (1/8)

= 8.761 meters.

So, you see, there is a need to change the radius of curvature of facets to
approach a parabola, but the amount of change of the radius of curvature is
very small.


Back in the early '80s, the Test Bed Concentrator, designed and built by
engineers at the Jet Propulsion Laboratory in Pasadena, California, used
three different types of facets to make their quasi-parabolic dish.  They
made the facets using a material called "foamglas", which was easily
machined by simply rubbing the foamglass section over an aluminum master
mold covered with sandpaper.  The foam glass, once contoured, had to be
coated with LOTS of butyl rubber sealant, and then a thin glass facet was
bonded to this using vacuum bags.  It was labor intensive, but pretty darn
accurate.

I hope this math note is useful to y'all out there with--and
without--calculus training.  If you haven't had calculus and thought there'd
never be a use for it, well, here's your motivation to go buy that book or
take that night school course on it!