A post which started with a waking reverie on Wednesday morning just past – but which turned into something more educational.
Let us suppose we have two places on the earth, on the globe, a fair distance apart, far enough apart that the curvature of the earth matters. Let us call them A and B and what we want to do is find a route from A to B. We do not allow A and B to coincide, because then the whole problem collapses.
The idea is that when we pray or when we are buried at A we want to be facing B. The problem is to know where B is, in what direction do we face. For Christians, B is often Rome or Jerusalem; while for Moslems, it is often Mecca.
Direction proper is defined by lines of longitude and latitude. At any point on the earth this gives us north, south, east and west. Intermediate directions are given by making the appropriate angle on the plane which is tangent to the earth at the point in question, in this case A. (1)
We simplify things a bit by saying that the earth is a perfect sphere. There are no hills, no mountains, no valleys and no rivers. At any given point, neglecting curvature, it is all perfectly flat.
We note that according to reference 1, the diameter of the earth is 12,756 km and that the circumference is 40,070 km. With the ratio of the two being close to that important number we call π, approximately 3.1416. We note also that the earth, leaving aside the humps and bumps already mentioned, is not actually a sphere, although it is reasonably close to so being.
A route from A to B generally defined would be any line running along the surface of the globe which started at A and ended at B, a line which was smooth, with no sudden changes of direction, and which did not intersect itself on its way from A to B. Such a route is sketched in the figure above. We might then say that if we look along the start of the route from A, we are facing B. This looking and facing would have a direction, as defined at (1) above.
The trouble with this is that such a route could start off in any direction we chose and still get there in the end. So we could look in any direction we fancied and still claim to be facing B. Nevertheless, if we were looking along the only road which led from A to B, there would be some sense in it. After all, one can’t go tramping over the hills.
Nevertheless, let’s try restricting our routes to something more reasonable. Let’s say that we only allow routes which are the result of intersecting a plane with the sphere that is the earth. Leaving aside the limiting case of tangency, the intersection of a plane with a sphere is always a circle, so our routes are always parts of circles, the radius of which must be less than or equal to the radius of the earth.
Any such circle which passes through A and B will define two routes, with one usually longer than the other. We are usually interested in that other, the shorter of the two routes. We can visualise those circles by rotating our defining plane on the axis AB and we can see that there will always be exactly one plane which defines a circle on which A and B are diametrically opposite, which means that the two routes will have the same length.
Note that, as we move along such a route, our direction in the sense of (1) above will usually be continuously and smoothly changing, although not as drastically as it might under the first scenario, illustrated above. Unlike a straight line on a plane, such a route does not have a well defined direction.
However, it is well known that the shortest such route will be found by taking a plane which passes through A, B and the centre of the earth. A sketch of a proof of this is offered below.
There will be exactly one such plane, except in the case that these three points are collinear and A is diametrically opposite to B with respect to the earth as a whole, when there will be an infinite number of planes and an infinite number of routes of length equal to half the circumference of the globe. We really can take our pick.
Otherwise we might say that we are facing B from A when we are looking along that shortest route. Which depending on circumstances, might be north, south, east or west – or any direction in-between. And the shortest route from London to Vancouver involves flying over the Canadian Arctic – and is a good deal shorter than the route suggested by the Mercator maps which we will come to shortly.
Which all goes to show that one should not make directional rules about praying and burying, unless the two points concerned, say A and B, are reasonably but not too close together. Say more than ten miles but less than a hundred miles.
Geometrical digression
In what follows we suggest lines on which one might prove that the shortest distance between two points on the earth lies on the (nearly always) unique great circle which connects them.
A sphere intersects a plane in a circle and part of that circle is our route from A to B. The centre of any such circle is going to be on the perpendicular bisector of the line AB. Two such circles are shown above.
It can be shown the upper arc of the small circle is entirely outside the large circle and the lower arc is entirely inside, as shown here. We want to show that the upper arc of the small circle is longer than the upper arc of the large circle, from which it follows that a great circle, that is to say a circle the centre of which coincides with that of the earth, provides the shortest route between two points on the surface of that earth.
We construct an even fan of lines from the centre of the large circle C, chopping the upper segment of the large circle in a large number of equal portions, and chopping the space between the two circles into an equal number of quadrilaterals, leaving aside the triangles at the two ends which are easily dealt with. The bottom two angles of the quadrilaterals are all the same obtuse angle, slightly larger than a right angle.
Such a quadrilateral, ABDC is shown above. Without loss of generality we extend a parallel from C to E on BD, between B and D. CE is then longer than AB. The angle θ is obtuse, so CD is the longest side of the triangle CDE and so CD is greater than CE is greater than AB. Summing the trapeziums, we show that the outer arc is indeed longer than the inner arc.
In the course of which I learned once again how to differentiate the arcsine function – a bit of trigonometry which I had long forgotten.
But this is not the end of the story
[Mercator projection of the world between 85°S and 85°N. Note the size comparison of Greenland and Africa. White for snow and ice, green probably for a fresh water supply]
In the foregoing, we restricted our attention to routes which lay on circles on the surface of the earth and came to the conclusion that, over long distances, direction to take to get from one place to another was not well defined. We now turn our attention to the Mercator projection, a type of projection which continues to be popular, if only because of its simplicity.
And in the days when sailors did their own navigation, it had the big advantage that you could set your course by the straight line between two points on the map – known to sailors as a rhumb line. Keep the ship on a north westerly heading – or whatever – and you will get there in the end. It was not necessary that you could see you destination, the compass was enough. No longer necessary to hug the coast, apt to be a long way round and quite possibly dangerous if the weather turned nasty. For all of which see references 2, 3 and 4, but perhaps 6 rather than 5. Reference 6, from the University of North Carolina, on the French Broad River, gives an accessible explanation of how the Mercator projection works, why it preserves angles and gives us rhumb lines. It also points up Mercator’s achievement in coming up with it in the mid sixteenth century, well before the tools of calculus were available.
With the bonus that I now know that what I had always thought was the Mercator projection was actually Lambert’s equal area projection, the results of which might look similar, but which are significantly different. Mercator’s projection was more complicated than I had realised, more than a straightforward projection from a line or a point inside a sphere onto the surrounding cylinder – although it still suffers from the same defect as all cylindrical projections in that it does bad things in high latitudes where lines of longitude come together. Perhaps I had been misled by thinking in terms of projective geometry.
One way of thinking about this is that if A and B are on the same line of latitude, one can get from A to B by heading east or west, depending on their longitude. One can go round and round in circles. If A and B are nearly on the same line of latitude, this trajectory becomes a helix, spiralling up or down, converging either to the north or the south pole, depending. Rather like the bath water going down the plug hole. But a helix is not a circle, so excluded from the earlier discussion.
Conclusions
There is clearly a choice to be made if one is at A and wants to face B, some hundreds of miles away and well out of the line of sight. Perhaps the dignitaries of the churches involved need to convene a conference to thrash the matter out. Perhaps on the site of Chalcedon, locale for an important conference at the time of the Monophysite heresies? For which, see reference 7.
PS 1: I suspect that I have posted about all this before, probably in fewer words, although search has failed to reveal it. But the exercise has served to reactivate my schoolboy geometry and filled a gap in my knowledge. And served as a reminder that one does not always know as much as one thinks one does!
PS 2: asking Google Images about the Wikipedia Mercator map of the world included above resulted in a lot of stuff about magic numbers. Still wondering why.
References
Reference 1: https://imagine.gsfc.nasa.gov/features/cosmic/earth_info.html.
Reference 2: https://en.wikipedia.org/wiki/Mercator_projection.
Reference 3: https://en.wikipedia.org/wiki/Conformal_map_projection.
Reference 4: https://en.wikipedia.org/wiki/Rhumb_line.
Reference 5: https://en.wikipedia.org/wiki/Geodesic_curvature. Where it gets rather too mathematical for me.
Reference 6: https://www.marksmath.org/classes/common/MapProjection.pdf.
Reference 7: https://en.wikipedia.org/wiki/Monophysitism.
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