Some time ago some old-timer told me if I ever found myself sailing second mate and I couldn’t figure out how to plot great circle courses a long (a meter or yardstick) steel ruler (aka the globetrotter) could be laid on the chart edge-wise and bent to the shape of a great circle.
I always managed to get by without using that technique but I have wondered from time to time if it would work. On a few occasions I’ve laid the ruler on the ship track and surprisingly it seems to be a perfect match.
I know that a great circle plotted on a Mercator chart is a sine curve so I wondered if a bent ruler, or some part of a ruler was a sine curve or something close to it.
It turns out that it’s an interesting question, so interesting in fact that Laplace, Bernoulli, Euler and others have been there and back.
The curve formed by bending a ruler crosses paths with font design, turns in railroad tracks, pendulums and capillary action just to start. You can also come across lesbian rule.
Here is from a paper: [FONT=CMR17][SIZE=5]The Euler spiral: a mathematical history by [FONT=CMR12]Raph Levien
The beautiful Euler spiral, defined by the linear relationship between curvature and arclength, was first proposed as a problem of elasticity by James Bernoulli, then solved accurately by Leonhard Euler. Since then, it has been independently reinvented twice, first by Augustin Fresnel to compute diffraction of light through a slit, and again by Arthur Talbot to produce an ideal shape for a railway transition curve connecting a straight section with a section of given curvature. Though it has gathered many names throughout its history, the curve retains its aesthetic and mathematical beauty as Euler had
clearly visualized. Its equation is related to the Gamma function, the Gauss error function (erf), and is a special case of the confluent hypergeometric function.
These curves are used boat and ship building as well, anyone who had built a boat likely bent splines to lay out the lines. Splines are basicly the same as bending rulers.
As to the practicality of using the globetrotter for great circle tracks, I gave up on the math and had to resort to empirical methods. The method works best a low latitudes, seems to be right on the money, but errors increase at higher latitudes. I almost snapped my little plastic ruler in half tying to make it match up to the great circle tracks that pass into the Bering Sea displayed on a computer screen with a nav program but did OK closer to the equator.
For accuracy you really need the vertex as well as point of arrival and departure. Otherwise you have to estimate how much to bend the ruler. Maybe with a little practice you could develop a feel for the needed amount of bending required and get good results without the vertex.