Sunday, July 25, 2010

Facing East or Facing West?

Updated below to add a flat projection of the Magic Ovals.
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It seems that a few months ago, Muslim clerics in Indonesia made a clerical error and told the country's faithful that they should face West when praying in order to be facing Mecca. They recovered, though, with the help of experts in cosmography and astronomy, who verified that by facing West from Indonesia the faithful would actually be facing Kenya and southern Somalia. Something like that. Google is my friend.

Naturally enough there was plenty of fun to be had ridiculing religious foolishness. I managed to honor my Grandfather's good advice along the lines of saying nothing if you can't think of something nice, and found myself playing with Google Earth. I don't know why these clerics needed expert cosmographers and astronomers to look into the question. Maybe it's against the religion to... Never mind. For some reason the question interested me.

Using Google Earth, I picked a few points roughly defining the extension of Indonesia. I plotted some lines that start out from those spots directly towards the West. Sure enough, they wind up in Kenya and southern Somalia. Then I made some green lines running from those same spots in Indonesia directly to Mecca. Here's the picture:



So, there is no place in Indonesia where one can face directly West to face Mecca. There must be such places, though, so I used the ruler tool to draw a line directly from Mecca to wherever I wanted to, and looked for places where the end point would be arriving directly from the West. I plotted a few such points and saw that they formed a line, like this (click the picture to make it bigger):



Wait a minute, it doesn't look like lines starting out from the magic arc are going West!

True, they just start out facing West. In so doing they define part of a "great circle" line directly to Mecca. Here's a closer view of the lines to Mecca as they start out from the magic arc. You can see that, in each case, the line starts out directly West.







If you try to continue the magic arc up through the Arctic and back down into the Atlantic Ocean (to form a magic circle, say), you find that it doesn't work. If you start a line to the West towards Mecca, before you get there it'll flip around and wind up starting out East instead. I guess that's because we're generally after the shortest distance, but trying to go West to Mecca from the Atlantic portion of a magic circle would lead to a path longer than the Easterly path. That's just a definitional thing, I think; just like the Equator, any great circle goes completely around the Earth.

(I'm starting to confuse a Great Circle with my magic circle. My magic circle is not a Great Circle because it does not have the center of the earth as its center. My magic circle just defines the points on the earth where any Magic Great Circle crossing it in an east-west direction will also pass through Mecca. I think.)

It turns out that there is a magic circle, though. It's actually more like a magic oval. I'll capitalize that. Magic Oval. It's just that, in the Atlantic you have to start out towards the East, not the West, like this:


Putting the two Magic Arcs together to form a Magic Oval, you get this:


As one might expect, there's a similar Magic Oval in the Southern Hemisphere. It looks something like this:


(I'm going to have to come back and put a nose and mustache in that picture.)

Since Google Earth works on a global projection (Is that the term a cartographer might use?), you can't see the whole thing all at once. If Google Earth has a way to present the whole thing into a flat projection, I have not learned how to do it yet. So, instead, I made a couple of videos showing the Earth with the Magic Ovals and Mecca Lines in polar and equatorial rotations. Let's see how that works.

First the Equatorial rotation:

video

Now the Polar rotation:

video

I don't know how to project the Magic Circles onto a flat projection in order to be able to see the whole thing at once, but someone made a similar thing that's available on Wikipedia. I think it's really quite beautiful. The Wikipedia image shows equidistant lines, whereas I am trying to show where one can face Mecca by facing either directly East or directly West, but what the hell. If I can figure out how to make Google Earth flatten out the Magic Ovals into one flat image, I'll post it later. Otherwise, hey, it's been interesting and fun. For me, anyway.

UPDATE:

It turns out that if you put all the relevant Google Earth items into one folder under My Places, you can select that folder and export it as a KMZ file. Then you can go to Google Maps MyMaps and import the KMZ file.

Projecting the Magic Ovals above on to a flat projection in Google Maps produces the image that follows:



Above is just a screen snag of the browser window showing the relevant part of the online version, which rolls the left and right edges (repeats the opposite edge to fill in what would otherwise be blank space).

The flat projection isn't quite what I'd expected, which was something more like the equidistant projection at the Wikipedia link. Clearly, I have a great deal to learn about map projections.

The flat projection above (a cylindrical projection, I think) looks to me sort of like an angel. Maybe a wire-model manga angel about to stomp on a poor heathen like me.

Hey! This could be a hidden sign from above, donchathink? Maybe I should copyright this image and sell it on eBay or something.

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