On the Parabolic Curve of Primary Mirrors

Abstract: In order for a parabolic mirror to work, light has to reflect off of
every point on it and be directed in a straight line to the focus. With this in
mind, we know a light ray traveling parallel to the y-axis and reflecting off of
the point (x, f) has to be directed to the focus in a straight line parallel to
the x-axis. In order for a vertical light ray to reflect in this way, the point
(x, f) on the mirror has to have a slope of 45 degrees or, in other words, a
slope of one. This means that the derivative of the parabolic function at the
point (x, f) has to equal 1.[Nature and Science. 2006;4(2):23-24].

Keywords: parabolic curve; parabolic mirror


In order to make a reflecting telescope such as a Newtonian Reflector, the
primary mirror is often ground into the shape of a parabola. The reason for this
is that when light enters the telescope tube, it is reflected off of the
parabolic mirror and, due to the unique parabolic shape, is focused at a single
point (the focus) which is a certain distance away from the center of the mirror
(the focal length). The purpose of this paper is to try to describe which
parabolic curve is suitable to fit a known focal length. The general solution
has been known for a long time, but I was able to use a kind of guess and check
method to find a similar solution on my own. This is one way in which the
solution can be derived. We will assume the parabola is centered at the origin,
so we know the general form of the parabola will be similar to y = ax^2 where a
is some particular coefficient depending on the given focal length.

In order for a parabolic mirror to work, light has to reflect off of every point
on it and be directed in a straight line to the focus. With this in mind, we
know a light ray traveling parallel to the y-axis and reflecting off of the
point (x, f) has to be directed to the focus in a straight line parallel to the
x-axis. In order for a vertical light ray to reflect in this way, the point (x,
f) on the mirror has to have a slope of 45 degrees or, in other words, a slope
of one. This means that the derivative of the parabolic function at the point
(x, f) has to equal 1.

So, knowing that the parabolic function will take the form y = ax^2, and that
the derivative of this function evaluated at the point (x, f) must equal 1, we
can figure out the general equation for any parabolic mirror with a particular
focal length.

Take the case where f, the focal length, equals 4. In order to find the
x-coordinate for (x, f) we need to solve y = ax^2 for x.

4 = ax^2
(4/a) = x^2
(2/sqrt(a)) = x

We know that at the point ((2/sqrt(a)), 4) the slope of the parabolic curve must
equal 1. To find an equation for the slope of the tangent line to the parabolic
curve at any point we take the derivative of y = ax^2.

y = ax^2
y’ = 2ax

We know x = 2/sqrt(a) and that the slope at this point has to be 1. Plugging in
y’ = 1 and x = 2/sqrt(a) we get:

1 = (4a)/sqrt(a)

Rearranging we get:

Sqrt(a) = 4a
a = 16a^2
1 = 16a
1/16 = a

Plugging a back into our original parabolic equation we get:

y = (1/16)x^2

which will produce a focal length of 4.

We have just found the parabolic equation for a mirror that produces a focal
length of 4. However, we can generalize this equation for any focal length. Take
f to be the focal length, but this time, instead of assigning a value to f, we
will leave it like it is.

f = ax^2
f/a = x^2
sqrt(f)/sqrt(a) = x

f’ = 2ax
1 = (2a*sqrt(f))/sqrt(a)
sqrt(a) = 2a*sqrt(f)
a = 4fa^2
1 = 4fa
1/(4f) = a

Therefore, by plugging a back into the parabolic equation we get the general
solution:

y = 1/(4f)x^2

where y is the curve that a parabolic mirror takes with any focal length f.

Make Big Paraboloid Reflectors Using Plane Segments
This page describes a simple algorithm (downloadable as an Excel spreadsheet)
that calculates the dimensions of cardboard sections that when assembled will
form a parabolic dish (paraboloid). The design allows free choice of focal
length, aperture and overall size. The dish can be used for concentrating energy
in the form of sound to make a highly senstive and directional microphone, or
(when covered with a metallic reflector or made from metal sheeting) a solar
furnace or a collector for radio waves.
Introduction
Parabolic reflectors (or paraboloids) and mirrors are used in astronomical
telescopes, car headlights and satellite dishes. The paraboloid has the unique
property that an on-axis parallel beam of radiation will be reflected by the
surface and concentrated at its focus (or conversely, a point source located at
the focus will produce a parallel beam on reflection). This feature is
illustrated in the diagram below - parallel rays enter from the left and are
brought to a focus at a single point.

Figure 1: The focusing action of a parabola
The above examples of parabolic reflectors all use a smooth surface as the
reflector; but a parabolic surface can be approximated using an array of flat
surfaces (small plane mirrors). Provided that the size of each reflector is kept
small then the errors will not be significant for several applications - such as
a solar concentrator (or solar furnace), a sound mirror or a radio receiving or
transmitting dish. The size of each individual mirror needs to be smaller than
the target (a microphone, saucepan or radio antenna). In this design (apart from
those at the centre) the individuals mirrors are quadrilaterals (or more
precisely, trapezia, since they have two parallel sides).
The material to make the dish is somewhat a matter of personal choice -
cardboard is fine for a microphone reflector and when covered with aluminium
foil, will make a solar concentrator. A cardboard paraboloid a metre or a metre
and a half in diameter can easily gather enough infrared rays from the sun to
cook a sausage (or your hand - be careful). A big cardboard paraboloid is easy
to make with very small focal ratios: f/0·25 or less. Light plywood can also be
used for a more durable dish at the expense of increased effort in construction
and additional weight. Sheet metal (or a metal mesh for a radio reflector) could
also be used. A major challenge with heavy structures is to support and steer
them and also prevent them from sagging and distorting (which will affect their
ability to focus properly).
The Principle
If you want to understand how the algorithm works, we'll need to have a look at
the maths (if you don't like the maths then skip this bit and go on to the
design section - you'll just have to take the design on trust).
We start by considering the parabola; this is a one dimensional curve and is a
section through a paraboloid - a paraboloid is formed by rotating a parabola
about its axis. The equation of a parabola is:
y = a.x²
where a is a constant.
For a parabola with a focal length of f:
a = 1/(4f)

Figure 2: Parabola - focal length = f
The axis of the parabola is coincident with the y-axis and the focus is located
at (0, f).
If the reflector depth is equal to the focal length then the edge of the mirror
and the focus both lie in the same plane - it makes locating the focus easy and
any supporting structure for the detector can be made flat (like the spokes of a
wheel). It follows that at the focal point the radius of the aperture is 2f and
that the focal ratio for this arrangement is f/0·25.
Now consider the actual dish (shown below in partial plan and section). The
section resembles the smooth curve shown above in figure 2 except that it is
made up from short straight lines. There are three features to note: firstly the
points that are joined by the lines lie on the parabolic curve; secondly, the
points are equally spaced along the x axis (which means the lengths of the
parallel sides of the trapezia are simple to calculate) and thirdly, the
distances between the points (measured along the parabola) increase with
distance from the centre.

Figure 3: Plan of the dish and section
When viewed from above, each segment comprises a simple triangle whose apex
angle is equal to 360° divided by the total number of segments (figure 4).
Multiplying the x distance by the tangent of half the apex angle gives the half
width of the triangle at x from the centre of the dish. This simple calculation
allows us to find the lengths of the parallel sides of the quadrilaterals.

Figure 4: Top View of a Single Section
When flattened out, the shape of the segment is not a simple triangle but a more
complicated shape; we need to calculate the distance between the parallel sides
and this allows us to then draw a complete segment. To get the linear distance
measured along the surface of the mirror we consider two adjacent points on the
parabola:
Figure 5: Calculating the length of a segment
The distance between the two points is found using the formula:
zn = (( xn+1 - xn )² + ( yn+1 - yn )²)½
Design
First decide how many sections you want to use - the plan above shows twelve -
having more sections means greater accuracy but also more work. Divide this
figure into 360° - this gives the angle at the vertex of each section. Now get
the tangent of half this angle (in this example the angle is 30° so we need to
find tan(15°) which is 0·268). Secondly choose the size of the increment in x -
this should be no larger than the detector placed at the focus - say 2 inches
for a microphone or 4 inches for a hamburger. Now choose a focal length - that's
the distance from the bottom of the dish to the focal point. Calculate the value
of a by multiplying f by four and taking the reciprocal of the result. For
example, if f is 8 then a will be 1/(4 x 8) = 1/32 = 0·03125
Then set up the table as follows:

Number the rows at the left.
In the next column put the value of the x coordinate (each row increases by the
value of the x increment you've chosen).
Calculate the corresponding value of y and put it in the next column; y = a x
x².
In the column labelled y1: copy the value for y from the next row.
For each row: calculate the square of the difference between y1 and y, add it to
the square of the value of the x increment. z is found by taking the square root
of this sum.
In each row calculate Vd which is equal to the value of z for that row plus all
the values of z in the preceding rows.
The 'from centre' distance is the half width of the section at the distance Vd
from the dish centre - it is calculated by multiplying the value of x in the
next row by the tangent already found.
The process can be repeated for as many rows as desired to increase the size of
the aperture for a given focal length.

I have set up an Excel spreadsheet to do all the calculations - download here.
If you use this you only need to choose the number of sections, the focal length
and the x increment to get the design.
Construction
Use the last two columns of the table - mark out a line on the card with the
distances given by Vd marked. Now measure perpendicular lines whose lengths are
given in the last column. Cut out the segment (and then repeat 11 more times -
phew!). Score along the perpendicular lines. Now, when the edges are joined with
adhesive tape or an equivalent material, the segments automatically bend into
the desired paraboloid.
Figure 6: Marking out the segment
To stiffen the dish, I add a cardboard ring which I attach to the edge of the
dish using hot melt glue. Good luck. I will appreciate feeback from anyone who
has a go at constructing one of these.
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