Navigating on a Sphere

I had to think about it for a bit but that’s right, the rate of change of the compass bearing is generally less then other steering errors.

I started thinking about how I navigate in the woods. I use a hand-held magnetic compass (a Silva Type 15T). The standard practice is to set the compass on the heading I want to travel and then find a landmark in the distance (a tall tree for example) on that bearing and travel till I reach the landmark, and then repeat the process.

So as long as I travel toward the landmark it’s a great circle track as unlike the ships compass which updates constantly I only update when I stop and pull the compass out for the next leg.

However each time a get a new bearing I’m starting off on new GC. So overall the track would be a rhumb line.

Likewise if I am steering on an E/W range the compass heading will constantly change (except on the equator).

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It was interesting to look at how the Polynesians found their way around the Pacific. The Maori first explored New Zealand and then settled here in about 800.
Navigators throughout the pacific had the status of high chief and were selected for the task as early as 6 years old. The ability to undertake voyages between islands was lost by the Maori race well before contact with Europeans because they had everything they needed in New Zealand .
As far as I am aware the last people to retain the ability were the people of Palau.
The voyages were always undertaken at a certain time of the year. They were aware of stars on the prime vertical and had memorised the positions of islands and the direction to sail as indicated by stars such as Altair and sailed towards them much as you did with your compass changing over to another star when the primary star got too far above the horizon. They had their own names for all the stars they used to navigate.
During daylight the course was maintained by reference to the wind and swell. They had an extraordinary ability to read swell patterns and their changes around reefs and islands. According to Tongan sources it took about 16 years of training and they had an arrangement not unlike a small Stonehenge where the pupils lay on their backs in the centre to align themselves with the constellations that they used.

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I would think that this is more the definition of following a rhumb line, since you have defined your course by a steady bearing to your waypoint. In order for it to be a great circle, your chosen waypoint (tree in your example), would have to be the exact point of final destination. Consider how the bearing was initially chosen. If you plotted it on a Mercator projection map, you would have had to first plot a great circle and then project where the intermediate waypoints were and the proper bearings to keep you on that circle. If instead, you gathered the bearing from a handheld GPS, you would again need to find intermediate bearings depending on how may waypoints (trees) you used. Correct?

Edit: I rethought my answer. Still rhumb line if you took the initial bearing from a map. But if you took the bearing from a GPS, this would be closer to approximating a GC, as long as you continued to get a new GPS bearing after each tree you stop at.

I can see my waypoint (big pine tree) so I’m not going to follow a steady bearing on the compass (rl). I’m going to walk straight towards it. Visual bearings are GC same as RDFs.

The waypoint will be on a rhumb line between me and my destination but I’m not going to use a rhumb line to reach it.


If you assume that you can’t see the destination, then you need either a map with a known starting point, or a GPS to figure out the bearing that leads you to the first tree. If you use a map, although you’ll follow a GC between each tree, the overall trek will be a rhumb line. If you use GPS, you’ll have to recalculate a new bearing after each tree in order to follow a GC.

Your waypoint should not be too distant. Humans are not made to walk straight from one point to a visible other one; the path will be a curve.

On a large rectangular and free place, people marching from one corner to the opposite one will not take the diagonal. They follow the joints in the concrete or other marks on the soil, until the angular difference between their heading and the destination becomes obviously too large.
With fresh snow covering all marks, the first walker will make a curved line and all followers will take his steps.

I was intrigued and tested myself:
I could only walk straight if I had a front and a back mark on the destination as reference, say a street sign and behind a corner or a door of a house.

Why is this?
Maybe for our ancestors, going straight from A to B, was not essential for their survival…

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Well, in practice you can only travel short distances before you have to use the compass again, I would say in dense woods being able to see 30 meters would be good going. Of course it varies.

With this compass you can see the needle in the mirror and then sight your next waypoint in the sight,

I use it to find property corners in the woods, can come out close if you’re careful.

Need to rotate the compass until the needle and the arrow are aligned. Arrow is set to desired bearing.

Yes, this.

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Wow. I never knew a walk in the woods could get so bloody complicated.

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Only if you don’t want to get lost :slight_smile:



At the risk of being a contrarian, I’m not sure anyone shooting an azimuth on a trek through the woods would or should be doing any kind of great circle. In the military we learned that since you naturally move to the right or to the left, shoot an azimuth to your tree or bush and alternate which way you circle around it. Maybe I’m misunderstanding, but if you are shooting an azimuth and moving in some sort of arc, depending on your distance, you could end up missing your mark. If the distance is great. You could miss by quite a bit. In a leaders course, that would fail you.

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The point that was being made is that on a sphere, the shortest distance between two points is not a constant bearing rhumb line (unless you are following a meridian or the equator). It is always going to be a great circle path from A to B. If you follow a constant bearing rhumb line and keep going, you will always end up at a pole. The trek through the woods was just to illustrate that when you move from A to B by line of sight alone, you are technically not keeping a constant compass bearing.

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Okay. Thanks. I realize a great circle can be shorter than a rhumb line over distances. Although technically you might not be keeping a constant bearing over land we sure tried to!

A great circle is the shortest distance between two points on a sphere, over any distance, mile, meter or millimeter, doesn’t matter.

By definition, find the shortest path it’s going to be a GC.

Thank you for making that point. I understand the concept. I think attempting it in the woods is a mistake.

Some years ago, I wrote a program to compare distances and bearings on a spherical and an ellipsoid earth. In addition, it calculates some nice values with no direct utility. The formulae being not always self-explaining, I added remarks to remember what I did, as this one:

Geodesics vs. Great Circles >>>

The shortest path between two points on a curved surface is a Geodesic:
on a sphere, all Geodesics are part of a Great Circle,
on an oblate ellipsoid (the earth), only some Geodesics are part of a Great Circle, all meridians and the equator (the latter not for near antipodal distances).

A Great Circle always closes on itself without an intervening self-intersection, the sphere (and exceptionally the ellipsoid) is “cut” into two equal halves;
an oblique Geodesic on an ellipsoid self-intersects near its apex and nadir; it does not close on itself and therefore cannot contain a “cutting” surface.

A non-GC Geodesic around the ellipsoid earth returns to the original longitude higher or lower than the original latitude:
North or South difference depends on the initial bearing and on the geolatitude of the initial point.

Some sample values, calculated as ‘direct solution’ (given LatStart + InitBearing + Distance), by Vincenty and Karney (identical results) >>>

>   LatStart  InitBearing   Distance         Δ Latitude on Start-Longitude
>    +47.57°      15°       40066.6 km       0.5270°N = 58.6 km N of Start
>    +47.57°      30°       40061.0 km       0.4729°N   52.6 km N
>    +47.57°      45°       40053.4 km       0.3857°N   42.9 km N
>    +47.57°      60°       40045.9 km       0.2724°N   30.3 km N
>    +47.57°      75°       40040.4 km       0.1398°N   15.5 km N
>    +47.57°      90°       40038.5 km       0.0029°S    0.3 km S      
>    +47.57°     120°       40046.5 km       0.2796°S   31.1 km S
>    +47.57°     150°       40062.2 km       0.4837°S   53.8 km S
>    -47.57°      90°       40038.5 km       0.0029°N    0.3 km N
>    +47.57°      89.69905° 40038.53628 km   <1 millimeter off StartPos  >>>

The last path returns quasi to its startpoint, its initial bearing is exactly at the limit between N and S deflection (analogously with all other start latitudes).
However, it is not a true Great Circle, the half-distance point is 81 meters South of the true geometric antipodal point and,
with the same initial bearing, but double distance (twice around the earth), the path returns to the initial longitude 645 meters South of the start-point.

Nothing I’ve posted here was intended to be used as practical advice, just making observations.

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But, if the inconviences of using GC with a hand-held compass in the woods was ignored…

The difference in bearing up to 40 miles, give or take is going to be 0.1 degree or less. Hand-held compass are considered to be within only about 2 degrees in practice. Even with the tripod mounted surveyors compass the estimated error is about 1/4 degree.


No I understand. I appreciate it. I was just making an observation from the personal experience of tromping around the woods with a ruck, a weapon, and a compass.

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