Returning to Arthur Christmas

As promised, since I’ve got the chance, I want to return to the question of the reindeer behavior as shown in the Aardman movie Arthur Christmas, and what would ultimately happen to them if the reindeer carry on as Grand-Santa claims they will. (Again, this does require spoiling a plot point of the film and so I tuck the rest behind a cut.)

After Arthur and Grand-Santa fall out of the sleigh, Grand-Santa claims the reindeer, lacking anyone to lead them, will continue flying in a straight line. The reindeer must end up somewhere; so, where?

I’m going to make explicit a couple of assumptions which take us away from realism but allow the problem to be handled better, in that if we were a little more realistic we’d come to the same conclusion but it would take more words and more complicated reasoning. Among the assumptions: the Earth is a perfect sphere, rather than the barely lumpy geoid it actually is; that the reindeer do keep to a straight line even though, realistically, they’d probably tend to go a little left or right of true straight and how far they’re off course would probably vary in time; that they aren’t going to ram into anything; and that whatever Grand-Santa means by a straight line has to be something a reasonable person would agree is a straight line.

If the reindeer keep a constant compass bearing, then they’ll either end at the North or South pole — either they’re heading right for it, or they’ll take the loxodrome curve which spirals into it — or they’re heading due East or due West and will loop around the line of latitude before coming back to the stranded Arthur and Grand-Santa. But there are other possible straight lines.

The constant compass bearing isn’t the only straight line you can have on a sphere, though. Actually, most people probably wouldn’t think it a very straight line except over a short distance or while looking at the Mercator projection map, since it curves so, especially when you get near enough a pole that the spiraling stands out. What people probably think of as a straight line is what they might get if you imagined a person just forever walking straight ahead while leaving a trail denoting where she’s been. This would be a Great Circle, a path that eventually loops back on itself and which cuts the rest of the world into two equal slices, things that were on the walker’s left side and things on the walker’s right.

That’s closer to what Rocket the Pony was speculating in this comment, that the reindeer would follow a satellite-like course. The Great Circle route traces out, yes, a circle in space; at the center of the circle is the center of the sphere on which it’s drawn; and it’s a closed figure. If you follow the path of the Great Circle you get back eventually to wherever you started. The reindeer and sleigh would return inevitably to Arthur and Grand-Santa, although the reindeer would be at altitude.

There are some nagging doubts to clear up, though. The first is, are we sure the reindeer could necessarily fly in a Great Circle? Is it possible they’d set off on a path that, yes, traces out a circle on the surface of the Earth, but that doesn’t have as it center the center of the Earth? (This is known as a Little Circle.) Put another way, can you draw a great circle from any arbitrarily selected point on the Earth with a starting point in any arbitrarily selected direction?

This is the sort of thing that bothers me when I try to shower. By the time I finished shampooing I was convinced that yes, you can draw just that and the reindeer are flying a Great Circle. But it’s good practice to take nagging doubts and consider them until you have an answer and an explanation. My answer came to “certainly” and my explanation was “rotational symmetry”, which satisfies me, although you might want to have it sketched out in more detail if the question bothers you.

Another nagging doubt is: we haven’t got any idea how fast the reindeer move, and we haven’t used any information about how big the Earth is. Shouldn’t that come into play somewhere? At least, how long it takes the reindeer to travel the circumference of the Earth seems like it ought to appear somewhere in this study. For now, though, it’s all right that it doesn’t, or at least so I tell myself. All we care about is whether they’ll ever see the reindeer again. Whether they have to wait ten minutes or ten million years we don’t care just yet. The reindeer speed and the size of the Earth tell us how long Arthur and Grand-Santa should expect to spend waiting, but right now we’re just wondering if their wait will ever end.

The biggest nagging doubt, and again, Rocket the Pony touched on this, is that the Earth rotates. The reindeer take some time to travel their Great Circle route, while Arthur and Grand-Santa are on the ground, moving east compared to where they had been. Are the reindeer tracking the surface of the Earth, automatically compensating for this rotation, or are they soaring like satellites so that when they do make it around the Earth they’re considerably west of the Clauses?

If we suppose the reindeer are tracking the surface of the Earth, the problem is solved apart from having to figure out what we mean by a straight line on the curved surface of the Earth that rotates with the planet, but we can turn that over to someone keeping a geography blog.

If we suppose that the reindeer are not tracking the surface of the Earth, that they’re moving above like satellites do and that the rotation of the Earth means that, from the ground’s perspective, they’re some distance west when they finish an orbit then we have a much more interesting question about whether they’ll ever be overhead again.