Tuesday, May 13, 2014

Parabola.

Using mathematics or what people have used mathematics for can be fun. One such item is a parabola. It can be used for everything from capturing radio waves to sound. There are a zillion formulas for building a parabola. One such formula is y=x^2/(4+b). But then, if you look closely on the net, there are examples you can use without all the math. In one article, I found a small picture of a parabola form that did not seem useful.




Then I thought to myself, what if I enlarged the picture, could the picture be more useful? Saving the picture to storage was the first task. Then we loaded the picture into a viewing program. There you could enlarge the picture and save it in the expanded format.


So far so good. Then I printed out the picture. Still do not have a parabola yet. Then it was time to get out the scissors and cut between the splines. Easy enough. Now just one last step in that we need to attach the splines together. Cellophane tape makes that easy.


If you add foil before you cut the parabola up, can make things interesting.



Then I thought, what about an even larger parabola. Poster size maybe. Extend out the spline edges with a straight edge (not like the crude extensions in the picture).



The we could make an even larger parabola all based on the original tiny picture. Let you do that yourself. Let's see solar cooker, sound umbrella, hat, or etc etc.

Update: Found another parabola to print out. The original was all black and emptied the ink cartridge.  Went into gimp and reversed the colors.



Time for an even larger one!

x     y
0     0.0000
3     0.1875
6     0.7500
9     1.6875
12     3.0000
15     4.6875
18     6.7500
21     9.1875
24     12.0000
27     15.1875
30     18.7500
33     22.6875
36     27.0000
39     31.6875
42     36.7500
45     42.1875
48     48.0000
The formula for a parabola is:

y = x² ÷ (4 × p)

where p is the distance from the bottom of the parabola to the focal point, and x and y are cartesian coordinates of points along the parabola.

For a parabola that is to be 96 inches across (that is, -48 inches to +48 inches relative to the focal point) and 48 inches deep, with a focal point 12 inches above the bottom of the parabola, the formula generates the numbers shown in the table on the right.

Sometimes it is useful to be able to locate the focal point after the fact. Rearranging the above formula

p = x² ÷ (4 × y)

where x is the width (from the focal point) of the parabola, y is the depth of the parabola, and f is the distance ahead of the bottom of the parabola of the focal point. For our above 96 inch wide and 48 inch deep parabola, f solves to 12 inches.

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