Investigating the Use of the Steel Globetrotter to Draw Great Circle Tracks

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

[/FONT][/SIZE][/FONT]

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.

Well when I read this I immediately thought back to my engineering drawing training and the infamous “French curves”. Parenthetically, when that term first comes up in a room full of young men (now of a certain age) and when the term “French postcards” was in common usage and represented a furtive component of our ahem, education in certain matters. So the innuendo represented by “French curves” usually resulted in some titters.

But when I googled French curves there were the magic words “the shapes are segments of the Euler spiral”.

Now a days CAD programs simulate the use of the french curves with various items on the tool pallet. Although my physical set is rather small I’m sure there are some sets with much larger dimensions that might serve the purpose of directly laying out a great circle. Then again maybe not.

But speaking of innuendo, tell me more about the lesbian rules. How many are there? Are they carved in stone?

Before computer programs calculated variants as fast as you can feed your new design parameters, the design of the bow and stern parts of a ship’s hull was done with flexible splines on the drawing table.

Each transversal section was drawn, then all these sections were reunited in one drawing; the same for the longitudinal and horizontal sections. An experienced designer could then see, if the form was as intended and smooth in all three dimensions, or if corrections were necessary; he could “see” the finished hull.
The workshop used big splines to cut and bend the steel sheets.
Some splines became thicker to one end, to account for the changing theoretical radius of curvature of a parabola (or another curve) along its line.

The more fixed points are given along the line, where the spline must pass, the better.
At least, for a convex only curve, there must be 3 points along the track and the two end angles (a supplementary point at each end can force the spline to use these angles). Without that, the curve is undefined.

To show the difficulties with a Great Circle, some examples with virtual points in the North Atlantic;
Start at 60°N / 0° Prime Meridian; earth as a sphere, without ellipsoid complications:

Destination = 75°N / 60° W
StartBearing = 328° on GC, 304° Rhumb
Apex = 75.01°N / 62.2°W (just after the destination point, outside of track)

Destination = 60°N / 60° W
StartBearing = 297° on GC, 270° Rhumb
Apex = 63.4°N / 30°W (not surprisingly at half distance)

Destination = 45°N / 60°W
StartBearing = 274° on GC, 247° Rhumb
Apex = 60.1°N / 5.1°W (near Startpoint)

Destination = 30°N / 60°W
StartBearing = 261° on GC, 234° Rhumb
Apex = 60.5°N / 10.9°E (east of the Startpoint, outside of track)

…and a real one: Cape Race (46.7°N / 53.1°W) to Cape Hatteras (35.3°N / 75.5W)… sorry for the destruction of Sable Island
StartBearing = 244° on GC, 236° Rhumb
Apex = 51.9°N / 19.1°W (300NM off Ireland)

Without an exact apex-position along the track and the end-bearings, the best spline can do nothing usefull:

  • StartBearing depends on Sine and Cosine of StartLatLon and DestLatLon
  • Apex latitude depends on Sin(Startbearing) and Cos(StartLatitude)
  • Apex longitude depends on Sin(StartBearing and StartLatLon)

The maths are not complicated, but the results are not intuitive, at least for me.
Without the maths, it is doubtful whether a long experience may lead to significant results; with the maths, the spline is useless…

Urs - too late, I’ve already destroyed all the computers in the wheelhouse. Fittingly I used the same steel meter ruler for the task.

Yeah, French curves, I’m going to order three sets of giant French curves, one set each for the Indian, Atlantic and Pacific Oceans.

[QUOTE=Urs;192875]Before computer programs calculated variants as fast as you can feed your new design parameters, the design of the bow and stern parts of a ship’s hull was done with flexible splines on the drawing table.

Each transversal section was drawn, then all these sections were reunited in one drawing; the same for the longitudinal and horizontal sections. An experienced designer could then see, if the form was as intended and smooth in all three dimensions, or if corrections were necessary; he could “see” the finished hull.
The workshop used big splines to cut and bend the steel sheets.
Some splines became thicker to one end, to account for the changing theoretical radius of curvature of a parabola (or another curve) along its line.

The more fixed points are given along the line, where the spline must pass, the better.
At least, for a convex only curve, there must be 3 points along the track and the two end angles (a supplementary point at each end can force the spline to use these angles). Without that, the curve is undefined.

[Snipped the part about using splines to plot great circle tracks]

[/QUOTE]

Speaking as someone who has done a fair amount of lofting (which is what is being described in the first part of the quoted post) while working with vintage model sailing yachts, permit me to add a few observations:

If you are trying to achieve a “fair” hull shape (informally, one without bumps or “wiggles”) then the principle tool is indeed a set of splines and “ducks” to hold the splines in place. However, the general rule is to never use more than five ducks for a given shape. In contrast to a navigation plot, a fair curve does not have to be a mathematically defined curve. The main exercise is to achieve fair curves on the diagonals, the angular “slices” of the hull that track, e.g., the bilge curve; it is quite possible to have a good looking set of curves in three dimensions and end up with a hull that is not fair. Curves that are too tight to be done with splines are drawn with ship’s curves or “Copenhagen Curves,” which look like french curves but aren’t. My set is one of my most treasured possessions. At times I will be scaling up a drawing by a long-dead designer and will find that one of my ship’s curves fits his curve exactly, which makes it likely I selected the same one he did when making the original drawing. Kind of a spooky feeling.

I remember my high school drafting instructor taking the class to a boatyard in Oakland CA where we saw the full-size splines and Copenhagen Curves the loftsmen used to lay out the parts of the wooden fishing boats that still sailed from San Francisco. One of my relatives, who built wooden fishing boats in Washington state, did not work from a drawing but lofted a hull directly on the materials using the basic dimensions given by the customer.

Final fairing on a wooden model yacht uses a “long board,” a long, thin board about 1" wide with sandpaper glued to it. This acts as kind of a spline to take down the last of the “bumps and wiggles.”

And when working from published plans, it’s wise to remember the final advice provided by a lofting textbook: “A fair line overrides any given dimension.”

Cheers,

Earl

Thank you, Earl, I fully agree. I am not as expert; only basic knowledge remains, and some souvenirs.
I worked with ship designers and builders (rivers and lakes) for about a year - 50 years ago.

Cheers,
Urs

[QUOTE=Urs;192893]Thank you, Earl, I fully agree. I am not as expert; only basic knowledge remains, and some souvenirs.
I worked with ship designers and builders (rivers and lakes) for about a year - 50 years ago.

Cheers,
Urs[/QUOTE]

Oh, yes, I was just trying to fill in some of the blanks for those under the traditional “threescore and ten.” :slight_smile:

I forgot to add this link to a description of traditional sailing craft design, as applied to models: http://usvmyg.org/design/design.htm

Cheers,

Earl

Yes, Earl, your link shows exactly what I meant.

With sailboats, every frame along the boat is important and unique.
Freighters, tankers or passenger ships normally have a long section amidships with identical frames; that simplifies the diagram.

And yes, our memory becomes a bit selective…

Cheers,
Urs

[QUOTE=Urs;192875]

The maths are not complicated, but the results are not intuitive, at least for me.
Without the maths, it is doubtful whether a long experience may lead to significant results; with the maths, the spline is useless…[/QUOTE]

The math I’m looking for is the nature of the spline on a Mercator chart, not looking for a GC formula for a sphere. In general subject of the OP is Euler’s Sprial.

I don’t understand your data either, why are you comparing GC initial courses to Rhumb lines? Need to compare a track laid out with a ruler to one using GC calculations.

[QUOTE=Kennebec Captain;192932]The math I’m looking for is the nature of the spline on a Mercator chart, not looking for a GC formula for a sphere. In general subject of the OP is Euler’s Sprial.

I don’t understand your data either, why are you comparing GC initial courses to Rhumb lines? Need to compare a track laid out with a ruler to one using GC calculations.[/QUOTE]

My goal was to show the very different places where the apex could be, relative to the start and end points; somewhere inside the track, or far away outside.
The indication of GC and rhumb bearings was just to help imagine that non-graphic explanation.

Here we have two very different curves:

  • Transform a straight line on sphere into a curve on a flat, forcefully distorted, Mercator chart. I do not know if the resulting curve has a name, or if someone deemed it necessary to describe it with a formula.
    All GC curves I have seen were calculated intermediate points on the sphere, reported on a paper chart or transposed into a flat diagram by a program. Thus, the ever changing distortion of the chart, with the latitude, is automatically respected.

  • A spline works similarly to a single flat spring or a steel beam, with working lengths from start to apex, and from apex to destination, with inflexible start, middle and end points. It works “inversed”, the force is not applied along the beam, but at the fixed ends; the flexion or deflection must be the same.
    This curve is probably named, the construction analysis must have formulae to calculate its profile. However, this curve cannot take care of the chart distortion.

Can you combine or interchange these two different curves? I do not really know, but I remain skeptical…

[QUOTE=Urs;192941]
Here we have two very different curves:

  • Transform a straight line on sphere into a curve on a flat, forcefully distorted, Mercator chart. I do not know if the resulting curve has a name, or if someone deemed it necessary to describe it with a formula.
    All GC curves I have seen were calculated intermediate points on the sphere, reported on a paper chart or transposed into a flat diagram by a program. Thus, the ever changing distortion of the chart, with the latitude, is automatically respected.
    [/QUOTE]

That curve does have a name. It was mentioned in the OP. It’s called a sine curve.

This is from Bowditch, chapter 24:

On a Mercator chart, a great circle appears as a sine curve extending equal distances each side of the equator.

As for the bending ruler, it’s a elastica curve. This post Kangaroo Curvature Analysis - shows a computer generated graph of an elastica curve with a sine curve overlay.

Google searchs that include “Euler” turns up too much stuff. Better luck with a search for elastica curves.

A post A script for elastic bending (aka the elastica curve) turns up on the first page. At the bottom of that post is a link to: Bending of a long, thin elastic rod or wire (finding shape & height) at a physics forum. One of the post links to The Geometry of Bending which has overlay of the sine curve and elastica curve.

A book shows up in the search - Structural Geology, on page 229 [I]“a low-amplitude undulating elastica is hardly distinguishable from a sine curve.”[/I] The same book mentions river meanders and ox-bow lakes as taking the shape of an elastica curve.

That would explain why the bending ruler works best at low latitudes and smaller DLo.

As a practical matter less and less plotting is done on paper charts now. Pre-ECDIS we used to plot GC tracks using rhumb line segments but the officers on watch would follow the GC on the GPS. As a result the hourly fixes would fall in an curve above (in the Notheren hemisphere) each segment, rejoining the drawm track at the plotted GC points. In this case a steel ruler would given more accurate results as it would have been closer to a sine curve then the rhumb line segments.

Using old charts with poorly erased lines or by picking a point or two out of the postion log from previous voyages a navigational-challenged mate might have been able to get away with using the globetrotter to lay out passable GC tracks in some cases. The ruler could be used to mark off end points of each rhumb line segment so the smooth elastica curve didn’t give the game away.

[QUOTE=KPChief;192864]Well when I read this I immediately thought back to my engineering drawing training and the infamous “French curves”. Parenthetically, when that term first comes up in a room full of young men (now of a certain age) and when the term “French postcards” was in common usage and represented a furtive component of our ahem, education in certain matters. So the innuendo represented by “French curves” usually resulted in some titters.

But when I googled French curves there were the magic words “the shapes are segments of the Euler spiral”.

[/QUOTE]

Allow me to assure you that the engine cadets are still tittering about french curves. I was required to buy a set for my drawing class, but I need more practice using them: our instruction wasn’t top-notch for drawing. Shame actually: lots of important, tangential (excuse the pun) deep thinking has been done by people studying curves and technical drawing.

I wonder if the titterers aren’t on to something, though. Could it be that all beautiful forms are Euler curves? Ships, maps, orange peels, women, spur gears. Maybe I should start carrying my french curves around in my purse to test this.