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.

[the focusing action of a parabola]

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)



[parabola]
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.

[the dish]

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:

[length of centreline of a general segment]
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:

1. Number the rows at the left.
2. In the next column put the value of the x coordinate (each row
increases by the value of the x increment you've chosen).
3. Calculate the corresponding value of y and put it in the next
column; y = a x x².
4. In the column labelled y1: copy the value for y from the next row.
5. 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.
6. 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.
7. 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.

[Dish design table]

I have set up an Excel spreadsheet to do all the calculations - download
<http://graffiti.virgin.net/ljmayes.mal/var/Paraboloid4.xls> 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.

[Dimensions of a segment]
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.


By yours Dr.BHUDIA-Science Group Of INDIA.
http://uk.groups.yahoo.com/group/venustransit_2004/ <../../../>
President:"Kutch Science Foundation".
Founder :"Kutch Amateurs Astronomers Club - Bhuj - Kutch".
Life Member:"kutch Itihaas Parishad".
kutchscience@...
<http://in.groups.yahoo.com/group/kutchitihasparishad/post?postID=y-45gB\
N_vSZJVeec3ZN6y7XZ3wn8A42UKA3Sbf1z_wbnNzeLg4ojrgKFkw5EKL4FbewvLPv5UWcPLp\
X_2yTyUA> , kutchscience@...
<http://in.groups.yahoo.com/group/kutchitihasparishad/post?postID=xcg0RB\
zdxEE8lrpzSYLeUKjemnQLFYDF6v7xKuf7ryZu8X6v8CvuLJMQqTPj7245dd50Z3aJ9_3PtD\
vldB4> ,
http://uk.geocities.com/wildlifeofkutch/
<http://uk.geocities.com/wildlifeofkutch/>
http://www.geocities.com/kutchscience
<http://www.geocities.com/kutchscience>
http://profiles.yahoo.com/kutchscience2000
<http://profiles.yahoo.com/kutchscience2000>
http://in.groups.yahoo.com/group/scienceclubofindia
<http://in.groups.yahoo.com/group/scienceclubofindia>
http://in.groups.yahoo.com/group/kutchscience
<http://in.groups.yahoo.com/group/kutchscience>
http://in.groups.yahoo.com/group/kachchh
<http://in.groups.yahoo.com/group/kachchh>
http://in.groups.yahoo.com/group/bhuj
<http://in.groups.yahoo.com/group/bhuj> Do visit our ABOVE
Clubs/Groups of Science club of India, Science Group of India.


[Non-text portions of this message have been removed]