How to Use a Parabolic Curve to Collect Sun Energy

Suppose, for instance, that you desire to create a
parabolic mirror, or half-mirror, or extended half-curve of
rectangular shape (see Figure 1). You know you need to
calculate and draw the supports and surface for the mirror
you want . . . but that’s about all you know. So you begin
to look around and — hot dog! — you discover an article or an
advertisement or some other piece of literature which
glibly leads you to believe that almost any old curve will
do the job.

Do not be fooled by such misinformation! A parabolic curve
is the only curve that will collect the sun’s rays over a
broad surface and then-under conditions of ideal
efficiency-direct all those rays to a single given spot or
surface. No other curve or shape will do this. For maximum
efficiency and maximum focus, your curve must be parabolic.

As an example of the misstatements I refer to, I direct
your attention to George Salmon’s works on conics, higher
plane curves, analytic geometry, and higher algebra . . .
which are considered to be standard authorities in the
field by many knowledgeable experts. Yet, on page 199 of
Salmon’s Treatise on Conic Sections (6th Edition, Dover),
Article 209 states that ” . . . If we suppose one vertex
and focus of an ellipse given, while its axis major
increases without limit, the curve will ultimately become a
parabola.” THIS IS NOT TRUE! I also refer you to Article
214 on page 202 of the same book: ” . . . and we shall
show, in the present section, that a parabola may in every
respect be considered as an ellipse, having one of its foci
at this distance and the other at infinity.” Again, NOT
TRUE!

The following, however, IS true and is — I believe — described
in the simplest way possible, while retaining all the
accuracy of an exercise done by a licensed civil engineer.

Given: the focal length only. This can be any distance you
want to work with and is nothing but the distance from the
back of your planned curvature (see Figure 2 in the Image Gallery) — at the center — to the focus (the spot where the heat is to be
directed). Let’s say you’ve decided to use a focal length
of four feet. A completed parabolic curve, across the
focus, will have a diameter four times that focal length
or, in this case, a diameter of 16 feet (4 by 4). A
half-curve, then, will have a height of eight feet . . .
and here’s an easy way to seek that half-curve:

Draw the focal line out to its required length on a large
sheet of smooth paper (Kraft building paper is fine). Figure
3 shows this focal length — Of — on the sheet of paper. It also
shows a second line — fP — drawn at right angles to and twice as
long (eight feet) as Of.

We know, of course, that the parabolic curve we’re seeking
will run, in some fashion, between points 0 and P. And,
although we have a rough idea of the area in which that
curve will fall, we’re not yet sure of its exact course. So
we’re ready to get down to the finer definition of our
curve, and we’re going to begin that definition by drawing
in a number of lines that are parallel to fP and spaced one
inch apart (see Figure 4 in the Image Gallery). These lines need be put in only in the near vicinity of where our final curve must lie, but
they should be measured and drawn accurately. You will,
when finished with this step, have a total (counting iP) of
48 parallel lines drawn on your sheet of paper.

Now (see Figure 5 in the Image Gallery) find an accurate straightedge that is at least twice as long as the focal length Of (or, to put it another
way, at least as long as fP). Place the corner of one end
of the straightedge precisely on point f and-taking care to
keep that corner exactly on frotate the face of the
straight-edge from Of down to fP. As you touch each of the
48 parallel lines from the top down, add one inch to the
length (48 inches) of Of and make a dot. (The first dot
will be made on the first parallel line down and 49 inches
from f, the second dot will be on the second parallel line
down and 50 inches from f, etc.) Continue on until you
scribe your last dot on the bottom line and 96 inches from
f. The series of dots you’ve just made will define a
parabolic curve with a four-foot focal length.

Now connect the dots by very accurately placing a flexible
metal, plastic, or wooden strip across them and carefully
drawing a line from O-cutting through all the scribed
points in between — to P (see Figure 6 in the Image Gallery).

This completed line is the curve you seek. It is NOT part
of an ellipse. It is NOT part of a circle. It is NOT a
hyperbolic. It is a true parabolic curve and you can now
use it in the construction of your solar heating unit or
cooker. By doubling the drawn curve back over the focal
line (pick up point P, swing it over Of, and lay it down
again an fP distance on the other side of Of), you can
quickly and easily convert the eight-foot-long half-curve
you’ve just drawn into a 16-foot-long full curve. When the
focal line of this true parabolic curve is pointed
precisely at the sun, ALL the Incoming solar rays which
strike the curve and are reflected will focus at f . . .
and, believe me, it gets HOT.

OK. Now for some variations. Suppose, for instance, that
you want to construct a parabolic reflector which has the
same focal length (four feet) of the curve we’ve just drawn
. . . but a smaller diameter. Easy. Starting at the base
line fp, just measure up as far as you like, slice off the
bottom of the drawing, and keep the rest. (If you cut your
drawing as shown in Figure 7 (see Figure 7 in the Image Gallery), the remaining portion of the curve can be used to build a collector much like the one
shown in Figure 1a.)

It’s a little more complicated — but not really much more
difficult — to extend a parabolic “dish” out to some endless
dimension . . . while still retaining a specific focal
length.

Using the focal length Of as one (1) unit, lay off on the
extended line, OX, ANY number of unit lengths that you
want. (See Figure 8 in which, as you’ll note, we’ve tilted
our original — Figures 3 through 7 — parabolic drawing 90 degrees to
the left to give us plenty of drawing space.)

If you’ll look at Figures 8 and 9 you’ll note that at the end
of each consecutive odd number on line OX, the next
vertical line will always be the next EVEN number above the
odd . . . while all the in-between numbers will be
fractional. (Example: The first odd number — 1 — on the base
line is followed by a vertical line labeled 2. The second
odd number — 3 — on the base line is followed by a vertical
line labeled 4. And so on. And the vertical lines in
between 2 and 4 — 2.82842 and 3.4641 — are fractional.)

All right. Note also that the evenly numbered vertical
lines (2, 4, 6, etc.) — the ones which separate the
odd-numbered unit measurements (1, 3, 5, etc.) — are each
longer than the preceding evenly numbered vertical line by
the square root of 4 (the square root of 4 is 2, and
vertical line 4 is 2 longer than vertical line 2, while
vertical line 6 is 2 longer than vertical line 4, etc.)

If you’ll study Figure 9 (see Figure 9 in the Image Gallery), you’ll further note that the squares of each vertical line (both evenly numbered and
fractional) form a very neat sequence, in which each square
advances by exactly 4. (That is: The square of vertical
line 2 is 4, the square of vertical line 2.82842 is 8, the
square of vertical line 3.4641 is 12, etc.)

Furthermore, if you’re familiar with right triangles,
you’ve probably already noticed that the hypotenuses beyond
f also form a series, each one greater than the last by 1.
(That Is: The hypotenuse — shown as a dotted line — of the
triangle running from f along OX to vertical line 2.82842,
up that vertical line and back down to f . . . is 3. The
hypotenuse-again shown as a dotted line-of the triangle
running from 1 along OX to vertical line 3.4641, up that
vertical line and back down to f . . . is 4. And so on.)

By computing and plotting out the squares for the vertical
lines involved, then, you will have the precise distances
from the base line to the curvature for a parabolic curve
extending as many units out as you care to take it.

It’s also very easy to double check any of these
computations, since every right triangle within the curve
will have a hypotenuse two units greater than its base.
(That Is: Point A in Figure 9 is a total of 7 units from f
and is located at the base of a vertical line measuring
5.65685 units. Seven times seven equals 49 and 5.65685
squared is 32. Forty-nine plus 32 equals 81 and 9 — which is
two units greater than 7 — is the root of 81.)

This order of numbers is a constant factor to any parabola.
Or, to put it another way, should you want a parabola of
ANY measurement, you have only to multiply or divide by the
necessary number to obtain the new dimensions. (Example: I
want a parabola with a focus of seven feet and I know that
every number above was calculated with the focal length Of
given as one (1) unit. Therefore, all I have to do to
figure those same numbers for a focal length of seven feet
is multiply by seven. And if I don’t want feet, I can just
as easily convert those numbers to meters, miles, or
anything else I do want. The ratio will always be the same.)

OK. You can relax. The hard part is over. Now that you know
HOW to calculate a parabolic curve of any size and/or
width, I’m going to save you the trouble. The chart
accompanying this article (see the diagrams in the Image Gallery) contains a list of measurements based on 10 verticals between each unit (this is, in short,
a list of coordinates for perfectionists who want to lay
out an absolutely accurate curve).

Please note that all of the fractional numbers on the chart
are based on an Of of 4. To use them for ANY focal length,
just divide by four and then multiply by the desired focal
length. (Example: I want a focal length of 11. So I divide
each number by 4 and then multiply the result by 11.)
Remember, too, that the charted numbers represent points on
the curvature measured up from the base in one-tenth
intervals. That’s cutting everything pretty fine, and you
can skip a few of the points if you wish.

And here’s one final tip about the accompanying chart: SAVE
IT FOR FUTURE REFERENCE. To my knowledge, the figures you
see here have never been printed before and they can save
hours of math work for anyone who might ever want to lay
out a true parabolic curve.

Of course, if you really dig math, here’s a bit of far-out
figuring that you can use to keep yourself occupied some
rainy afternoon: In any right triangle formed in any half
of a parabola and touching both the focus and the curve
(see Figure 11 in the Image Gallery), h plus a is always twice the focal length
and — naturally — h2 minus a2 always equals b2. You can use
those facts as further proof of exactness when scribing
your next parabolic curve.

EDITOR’S NOTE: George Donald Graham, the author of this
article, is also the author of Capture the Sun, a very
interesting little paperback book (Enterprises Unlimited,
Star Route, Ferndale, California 95536) that delves more
deeply into the calculation and construction of parabolic
solar energy devices. If you liked this article, you’ll
want to read the book. It’s available for $2.50 plus 75!
postage and handling from either the publisher or from
MOTHER’s Bookshelf.