As promised, I have more to say about the grid above my head.
#1 and #2 can be found here.
3) A Fourier Series for the Grid
So I came to think, if I can fit a sine wave to a grid, can I superpose sine waves in a Fourier Series fashion to form the grid itself? I figure it might be possible, but a better idea would be to consider it from a differential calculus perspective.
Consider a single pulse of the wave from x=0 to x=L. For the equivalent square, the derivative is +1 from 0 < x < L/2, -1 from L/2 < x < L. For the sine wave the range from -1 to +1 remains the same, but the derivative is continuous and it smoothly decreases from +1 to -1 instead of suddenly jumping. So we consider the Fourier Series for the derivative. It would have to start off with a single sinusoidal wave. But as terms are added, the derivative from 0 < x < L/2 would approach a constant 1, while the function from L/2 < x < L would approach a constant -1. Meanwhile, the limit would approach a nonexistence at x=L/2, and the derivative would cease to be continuous at that point.
This function that I am describing above – the derivative of the grid – is well-known. It’s called the square wave: Link. Its Fourier Series is given as follows:
Taking the integral of the Series yields the [top half of] the grid, more or less minus a few details here and there.
4) Integration; Storm Prediction Center Watch Boxes
So speaking of integration, the illustration provided in the previous post of the bed schematic lends itself to the area problem. How can I add up the squares represented by the grid itself to best approximate the area of the region of the bed NOT covered by the bed sheet? There may be an analytical method to this, but for now I am reduced down to guessing, which I do quite frequently when I’m lying down in my bunk all by myself. Of course, if I had more squares, an appropriate sum of the area of the squares approaches the area of the “naked” region itself.
This may be a rudimentary method, but the national Storm Prediction Center (SPC) does kinda the same thing in its issuing of weather watch boxes. These “boxes” are parallelograms delineated by a set of coordinates. Here’s an old prototypical example of an SPC watch box:
Now note that, along the boundaries of the watch, it’s difficult to say whether you are in the watch itself or not – unless you use an approximation method. Instead of using squares and saying “are you in this square or not”, the SPC uses current county outlines to “integrate” the watch box region. [Okay, they’re not interested in integrating, but I just like using that analogy.] It’s a little more precise than using a linear extrapolation between coordinate points on a spherical topology, and a lot more straightforward to the public than using arbitrary squares.
The problem is, the approximation’s going to be off if you’re using awkwardly shaped “differential elements” where counties just don’t jive very well with the parallelogram. Or if your threat area doesn’t fit well with the parallelogram. So recently, SPC has shifted philosophy by placing the county outline first, and keeping the parallelogram as a simple proxy used to make Weather Channel graphics more spiffy. But in reality, the parallelogram has been largely discarded in favor of county outlines.
Still, in day to day rests on my bed, I imagine the oval of the bed not covered by the sheet as a watch box. The grid represents my counties. It’s fun to think about. Terrible approximations, just like SPC watches used to be.
5) Drawing Tornadoes
Find two arbitrary points on the grid. From each point, follow their gridlines down towards where they converge. Stop following the lines anywhere before that convergence point, but do not cross or go over that convergence point. Draw a horizontal line across the two “top” points and across the two “bottom” points where you stopped.
Congrats, you drew a tornado!
[Insert illustration here, later.]