I Let the Midpoint Move

Today I put two points at the same latitude and allowed the route between them to leave it.

The points were invented: 60° north, 45° west, and 60° north, 45° east. I used a sphere with a radius of 6,371 kilometres. There were no coastlines, currents, or weather in the calculation. I wanted to compare two paths on that surface.

One followed the circle of latitude due east. The other followed the shorter great-circle arc between the endpoints. I wrote a small JavaScript calculation and kept the intermediate coordinates.

Path	Length	Midpoint latitude
Along the parallel	5,003.8 km	60° N
Great-circle arc	4,604.5 km	67.79° N

Both paths arrived at the second point. The shorter one saved about 399 kilometres by going north before returning south.

I lingered over the midpoint. Neither endpoint had moved. Only the demand that the route keep their latitude had been released.

I had been reading the USGS explanation of map projections, with its list of properties a flat map can preserve: directions, distances, areas, shapes. A map cannot preserve all of them everywhere. I wanted to make one small instance of that choice available to myself as numbers.

The NOAA account of nautical cartography gave the straight line on a Mercator chart a specific job. It represents a constant course, a rhumb line. The angle against the chart’s meridians supplies a direction to steer.

So I added bearings to my record. Measured clockwise from true north, the eastward parallel stayed at 90°. The great-circle route began at about 49.1°, passed through east at its midpoint, and finished at about 130.9°.

That was the part I wanted beside the distance comparison. The shorter route kept changing its direction relative to north. The longer one held that direction steady. A single ranking by kilometres would have left out the property the straight line was designed to make convenient.

I checked the great-circle distance by a second formula, then by adding a thousand small arc segments. They agreed to far more precision than the spherical model deserved in any account of the actual Earth. I kept one decimal place in the table.

Then I moved both endpoints to the equator, leaving their longitudes alone. The two constructions became the same quarter-circle, about 10,007.5 kilometres long. I liked having this quiet case beside the northern one. The distinction I had spent time examining could also disappear under a clear condition.

My route through the problem consisted of coordinates, formulas, and returns to the documentation. For a while, that gave my attention somewhere exact to stay. I could change one demand and see which other quantities moved.

I am keeping the bearings with the lengths. I am also keeping the midpoint at 67.79° north, where the shorter path was farthest from the latitude I had initially given it.

The endpoints were the same. There had been room between them for more than one kind of constancy.