How to Make a Parabolic Dish Out of Plywood! Or Anything Else! The neat thing about a parabola is that if you choose to make a square matrix of pieces that form the inner shape of a parabola, the pieces all have parallel top edge curvatures, thus can be cut out of one piece of, say, plywood. Then you can half-slot the pieces so that they fit together like a 3D jigsaw puzzle, or kid's construction set. Then you can set corner pieces of wood with screws at strategic spots to give it rigidity. A parabola is a "locus" (set) of points equidistant from a point, the focus, and a line on the convex side of it, the directrix, which is, from observing that the closest point on the parabola is the apex, or center, also at a same distance from the apex as is the focus, and is obviously also both perpendicular to a line drawn from the focus and through the apex to it, and symmetrical on either side of that drawn line. The formula for a given parabola of desired focal length comes simply from the fact that the distance between any point on the parabola and the focus point is equal to the distance from that same point on the parabola to a plane (or line in 2-D) in back of the reflector shape which is the same distance from the apex, or center, of the parabola as is the focus. This plane(line) is called the directrix, and it is perpencidular to a line drawn through the apex of the parabola to the focus. Note that the distance from the point on the parabola to the directrix is the closest distance, or the perpendicular distance to the plane(line). The reason I use the words plane(line) is that each point on a paraboloid, a 3-D parabola of rotation, is given by the formula which gives only a vertical slice through the focus and apex of the paraboloid and the directrix plane, the slice thus yielding a line directrix for each rotation "slice" of the "spun" parabola in 2-D. Now for this simple case it is good to use the origin of an x-y Cartesian, (named after DeCartes, don't freak), to graph these points so that you have a big paper pattern when you make one the size you want to mark the material with for cutting. And it is most useful to call the apex of the intended parabola the origin, or point (0,0), (x=0, y=0). Now, since the focus is "above" the apex, then its x-coordinate = 0, so the focus is just the point (0,f), where f is the focal length to the focus in the positive y direction. The directrix is clearly the line "below" the apex, so that it is the line on which all y = -f. Thus the apex is equidistant from the focus and the directrix and is on the parabola! The simple distance formulas come from old Pythagoras and algebra, and they are simply the square root of [the vertical distance squared plus the horizontal distance squared]. We need first the distance formula from the focus to a point on the parabola, and we'll call it (x,y), since that's what we're trying to find, (SQRT means square root and we'll use the ^2 symbol to mean squared) and so: Dfocus = SQRT [(x-0)^2 + (y-f)^2] We use the difference in coordinates to get rid of the coordinates themselves, since (y-f) would leave only their difference, which IS their vertical distance! Likewise for (x-0) or just x in this case. The focus is already AT the horizontal zero distance. Now, the distance from a point, which we do not know yet, to a line, sounds like a crazy thing to try to get, since we don't know what part of the parabola it lies on or how close that part is to the directrix line! But we DO! We know the vertical distance above the x axis is just its y-coordinate, y, and the distance from the x axis to all points on the directrix to be f, the focal length, but below the x-axis, and so in the negative direction, that is, all points on the directrix have y-coordinate = -f . Thus the distance from the directrix to any point on the parabola is simply: Ddirectrix = SQRT [(x-x)^2 + (y-(-f))^2] And so now, to satisfy the definition of a parabola, these two distances for all points (x,y) on the parabola must be equal, and so we simply set one equal to the other and solve for a simple equation!: Dfocus = Ddirectrix Or: SQRT [ (x-0)^2 + (y-f)^2 ] = SQRT [ (x-x)^2 + (y-(-f))^2 ] And now, since we have two things equal that are both square roots, we know that if we squared them they would still be equal, thus we can dispense with that and: (x-0)^2 + (y-f)^2 = (x-x)^2 + (y-(-f))^2 also: x^2 + (y^2 -yf + f^2) = 0 + (y + f)^2 and so: x^2 + y^2 -2yf + f^2 = y^2 +2yf + f^2 And now eliminating terms common to both sides: x^2 - 2yf = 2yf Or, adding 2yf to both sides: x^2 = 4yf And so finally dividing both sides by 4f, we get: y = (x^2)/4f And this is the equation for any parabola on a paper pattern using any measuring unit, mm's cm's inches, feet, miles, etc. As long as you use the same measure unit throughout and for the f or focal length you desire. Lay it out with shelf-paper or whatever, by graphing enough points to the accuracy you wish. If anyone wants proof that all pieces of a square matrix of slats that one wishes to use must be cut to the same parallel curvature, and so can be cut from the same piece of material, I can supply the proof, although the diagram accompanying the equations would be hard to render here! But I DO guarantee it!! Try it empirically with cardboard! I did sit down and prove it for a school project once and I have looked at it recently. I would not have guessed it worked out so easily and practically stood up and proved itself! You know, those great proofs that look horrific, and then everything falls out leaving a simple elegant term? It is like that! This surface will act as a perfect concentrator, whether for sound, light, radio waves, or even tennis balls if it's strong enough. Enjoy! -Steve Walz rstevew@armory.com ----------------------------------------------------------- The Parabolic Dish: How? Have you EVER GRAPHED anything, on graph paper? If not simply obtain: 1) the most simple algebra book. 2) read it through where it talks about graphing functions in x and y. 3) a parabola facing upwards is y = x*x/4f, where f is the focal distance you will need to place your receiver in feet, cm's, inches, or whatever as long as you use the same unit for everything related to creating the shape you need. YOU decide the focus you need, and IT affects how steeply curved your parabola is!!! Longer focus, shallower mirror. 4) Graph the function from the negative diameter/2 you need, to the positive diameter/2 you need, in values of x. 5) Obtain the resulting values for y. They are your heights at every radius, (which is diameter/2 ), above being absolutely flat! 6) Do it on shelf paper so you can get the scale right full size. 7) Place on a piece of plywood and make a bunch of curves that shape in parallel to each other and about 4 inches wide or more apart. 8) If you make them the full diameter, then you can make an interslotted deck of symmetrically curved slats which will intermesh like pieces of divider carboard on edge in a corrugated box of bottles or glassware! 9) Cut off the ends vertically as needed to fit the diameter of your dish at each edge and make the slots for intermeshing downward for the "North-South group" of slats, (say, as you have it laid out in your back yard), and upward slots for the other "East-West" group of slats, so that the slots go only halfway through the width vertically! 10)Assemble the slats in a rectangluar intermesh by sliding the "N-S group" down into the upward facing slots of the "E-W group" which lay at right angles to the other group standing on edge, like rockers from a big rocking chair. You may wish to assemble this first and then you'll see what I mean about cutting off the ends vertically inside the diameter! If you DON'T cut off the corners it will still work fine, but it will look like you bent the world's biggest square plywood spice rack into a parabolic dish!!!:-) 11)Rigidify this assembly with wood blocks at intersections of the plywood held with screws through the plywood into the blocks, and you will have a parabolic dish! The metal mesh, or even plain paper and aluminum foil work beautifully to concentrate either sunlight to do some light welding, or else to bring in satellite signals, OR you have your very own radio-telescope! You can cook weenies on it as well, but they burn fast with a 6 foot dish, so don't put them right IN the focal point, but a bit inward or outward! A twelve foot dish CAN weld with aluminum foil!!! And you'll have to use maybe six or eight inch wide slats of plywood for that size! The denser the mesh of slats the stronger and more conforming parabolically the better, but the heavier it will be. I have seen bright folks make them out of steel rod welded into a frame member like an aircraft strut fuselage assembly, with triangle trussing. Should I try to draw a circle with a rectangular (square) intermesh of cells from above in ASCII or shouldn't I? Nawh, if you would really and truly want me to design all of it for you, tell me your diameter and focal length and send me $50 to make the patterns!! Oh, you COULD also use a radial spoke arrangement, but the one I use here is stronger. Liberal use of glue with the wood blocks, (hardwood), and screws, is recommended! Mount it on an alt-azimuthal mount with sturdy metal or plumbing pipe assembly (the kind like on little cheap telescopes, you know, up and down and right and left around?) and you have got it. So there! And remember to put a long pipe on it and counterweight its weight with slidable concrete weights in coffee cans, or such. The counterweight pipe is to pass right through the axis of symmetry of the parabola, (the line you could rotate it around like an umbrella, and have it still look the same from any angle). P.S., if anyone doubts that one can be so cavalier with using the same curve throughout the design, I can supply them with a proof that it will work. It is elegant, and I first derived it myself and then have seen it in a couple solar energy books! One simply uses less of the curve, and in ANY direction at any radius across the dish, as long as it is a symmetrical piece of the same curve about the apex, or point of the parabolic shape. -Steve Walz rstevew@armory.com