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.
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).
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
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.
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.
- Take care in marking out the design - use a sharp pencil and also make sure that perpendicular lines are at exact right angles. If you're sloppy then the edges of the segments won't meet properly.
- The design algorithm is based on an idea by Alex McEachern and Paul Boon published in the Amateur Scientist section of the magazine Scientific American sometime in the 1970s. The article was somewhat defective insofar as there were a number of errors (mathematical and editorial) and some unnecessary approximations which I have corrected.
- According to Gregory Kunkel, the original article was published in the December 1973 issue and corrections published the following February. See Gregory's method which is closely based on the Scientific American article. His page also has some pictures of a completed dish.
- I have come across another method - A Simple Technique of Fabrication of Paraboloidal Concentrators - this is a more mathematical treatment and, instead of using flat sections with straight boundaries, uses a continuous curve where the individual segments join.
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