Great Circle routes..

majdrew

majdrew

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Hi Folks!

You know when you convert a rhumb into a gt.circle route, and you have a series of smaller rhumb lines..

Does this mean that you steer every rhumb line, and alter course, say, 10 times across the Atlantic? That's not right is it?

A curved gt. circle route on a chart is a straight line in reality, so surely it's just one course to steer? eg. the equator, if you were to sail around the equator you could steer 90°T and get back where you started, and not have to make any course alteration..

Or am I wrong?, definitely confused! Explain someone please!
 
A great circle is a straight line on a curved surface. The equator is the only great circle which crosses all the lines of longitude at the same angle - 90 degrees. Think of a great circle from London to Tokyo. It would pass close to the North Pole and as it did so the compass needle on the aircaft would swing from something like 30 degrees left of the nose (11 O'Clock) to 30 degrees left of the tail (7 O'Clock).

The extreme case is a great circle from the equator on the meridian to the equator at 180 degrees longitude. This passes directly over the pole and the heading is 000 degress for the first half of the trip and 180 degrees for the second half. The nearer you get to the poles the more pronounced the effect.
 
Majdrew - changing course often is exactly what you are doing. If you plot your GC co-ordinates on a Mercator chart you should get a wonderful curve. Pre GPS I seem to remember working them out for every 5 degree Long intervals on voyages from Capetown to Oz.....but it was a long time ago!
 
No one in a small boat sails great circle courses. You always sail according to wind and currents.

It is perfectly feasible to follow a great circle if you really want to. Just do a 'goto' on the gps and sail to keep the XTE = zero. The problem is that while it might be the shortest course, it will almost without exception be slower or more dangerous than a properly planned route. For example -

Canaries to Caribbean: Gt circle will mean you spend at least half the distance outside the trade winds.

UK to New England: Gt circle will take you into icebergs and fog.
 
Dont forget, your compass reads North & all your courses are taken from North (000/360*) clockwise.

As you progress west to east, the angle of your course line to any meridian (N/S line) you are crossing, is continuously changing (as is your local time).

You change your course accordingly, when convenient to do so, perhaps each Noon etc.
 
[ QUOTE ]

Does this mean that you steer every rhumb line, and alter course, say, 10 times across the Atlantic?


[/ QUOTE ]

Got it in one /forums/images/graemlins/cool.gif

But as the other forumites say, the GC routes are only useful for full powered vessels of more than 150 feet in length and they take you where you don't want to be.

Better a couple of rhumb lines that make the most of the trades for a sailing or low powered vessel.
 
A really good way of visualizing great circles is to take a globe and stretch a piece of string between your start and end points. The string naturally takes up the shortest route between the two points, which of course is the great circle route. For demonstration purposes, a cheap inflatable globe will do, but don't navigate by it!

If you are good at visualizing things in 3D, you can visualize it another way. The great circle is the intersection of the surface of the Earth with a plane passing through the centre of the Earth and also through the start and end points. Three points completely define a plane in three dimensions.

Both these visualizations will quickly convince you that indeed, your bearing does steadily change throughout a Great Circle route. In fact, the Rhumb line course (in a really pathological case such as to a point very near the Pole but 180 degrees in longitude away from you) will be a VERY long way to go! The case I described will follow a spiral with a half turn in it on a rhumb-line course.

Ain't navigating on a sphere fun!
 
The definition of a Rhumb Line is one that crosses all meridians at the same angle. Therefore if you set from anywhere (other than the equator) and maintained exactly the same course, you would eventually spiral into the Poles.

As a result, a GC track is a staright line on a curved surface, and when transferred to a flat surface (a Mercator chart) would be a curve. This is impossible to (a) draw accuractly and (b) results in a series of straight lines between waypoints.

In sum: If following a GC then YES - you would have to alter course every so often as you followed a Rhumb line from waypoint to waypoint. How far apart these waypoints are will entirely depend on speed (eg every 5 or 10 degrees of Long).

Following maths may help:

Great Circles may be drawn onto a Gnomonic Chart and then the waypoints transferred to a Mercator Chart for Navigational purposes. On the Mercator Chart, Rhumb Line tracks will then be drawn from Waypoint to Waypoint.

Navigational Great Circle calculations involve solution of the spherical triangle formed by the Meridians of Departure and destination and the Great Circle passing through them.

A derivation of Napier’s rules for finding distance is shown as follows:-

Cos Distance = Cos D’Long x Cos Lat A x Cos Lat B ± Sin Lat A x Sin Lat B

If the Latitudes are the Same – ADD
If the Latitudes are Different – SUBTRACT

The resultant distance from the above calculation will be expressed in degrees and minutes. To turn into miles, you must multiply by 60.

To find the Initial Course, ABC formulae may be used. The Course a ship will follow, changes continuously along a Great Circle, so the Initial Course is only good for the Departure position.

Therefore:

A = Tan Lat A ÷ Tan D’Long

If D’long < 90 - name opposite to Lat A
If D’Long > 90 - name same as Lat A

B = Tan Lat B ÷ Sin D’Long

Name same as Lat B

C = A ± B

Same names, +
Different Names, –

Tan Initial Course = (1 ÷ C ÷ Cos Lat A)

The result will be in Quadrantal format.
N/S named same as C
E/W named same as D’Long

The result must now be converted to 360 notation.

I teach this stuff professionally - so if I can help any further - please PM me.

Hope this helps.
 
Just a word of warning; there are cases where the Cosine rule will give answers that are dominated by errors in the least significant digits of the computation, where the sine and cosine of the angles are close together. By the way, most mathematicians and map projection people call the relationship you called Napiers rules simply the Cosine Rule, and its counterpart the for sides and angles the Sine Rule. I've been working in this field for nearly thirty years and not heard the usage Napiers Rules in common usage anywhere.

Oh, and the ambiguity in the sign goes away if you use the correct sign convention in the equations from the start of the computation. It's only ambiguous if you restrict the angles to positive angles. If you use the convention west is negative, south is negative, and apply it consistently with the correct signs for the trigonometric functions, the equations automatically end up with the correct sign.
 
Agreed - there can be and will be errors, depending on the degree of precision used - but then this is the same for Mercator Sailing and /or any other calculation made. My understanding was that the Cosine and Sine formulae was an ultimate DERIVATION of Napiers, but am happy to be corrected if I'm wrong. The one thing I am not is a mathematician!!! Also agreed, if mathematical signs are entered into the calculator, you don't need to worry about resultant signs, but I personnaly don't like this method - it's all a matter of choice. Not saying it's wrong - just don't like it.

Regarding Napiers in general - I know they can only be used in a spherical triangle if there is 90 degrees in it (which is usually at the vertex - but not necessarily so). Interesting that you don't use them. I find them a really easy and quick way of getting the required result and SQA expect them to be used in Mates/Masters Nav exams - but if you are used to other ways/methods then I can understand your preference (please don't read this as a dig even though it might read as one - I am always more than happy to be corrected if I'm wrong, but like the opportunity to be able to try and show people where I'm coming from!)

Regards
 
[ QUOTE ]
The definition of a Rhumb Line is one that crosses all meridians at the same angle. Therefore if you set from anywhere (other than the equator) and maintained exactly the same course, you would eventually spiral into the Poles.

[/ QUOTE ]
Sorry to be picky but not true. A rhumb line course due East or West from any point on the earth's surface will return to the same point and never get any closer to either pole.
 
A man sets off from his camp and travels due North for 10 miles. He sees the tracks of a bear and follows it due West for 10 miles until he catches up with it and shoots it. He then drags it 10 miles due South until he arrives back at his camp.

What colour was the bear?

When you think you know the answer, highlight this post /forums/images/graemlins/wink.gif

<span style="color:white">There was no bear. Bears do not live at the South pole. </span>
 
No worries; no dig perceived!

I'm coming from a different mathematical background; I work with map projections and spherical trigonometry in that context. Of course, I stick to the conventions I am used to; to do otherwise is to invite confusion!

I'm used to having to use the sign convention; if you don't when writing projection software, your code will only work in about 1/4 of the world! And, if you use it from the beginning of the calculation, it is a very safe way to proceed.

People may be interested in a command line utility called geod. It's a bit of a computer geek's tool, but it does extremely accurate great circle computations - it is reputed to be good to millimetres! I didn't write it; it's been around for a long time, but it's a very powerful little utility, and very stable.
 
[Sorry to be picky but not true. A rhumb line course due East or West from any point on the earth's surface will return to the same point and never get any closer to either pole.

[/ QUOTE ]

Unfortunately you are wrong. This would only be true if you started at the Equator. From anywhere else, you would evenetually spiral into the poles. Remember, the definition of a Rhumb line is that you cut each meridian at exactly the same angle. If you look at a globe, you would see that if you cut or crossed each meridian at exactly the same angle you would spiral in to the poles. Good image of what I'm trying to say at following link:
www.navworld.com/navcerebrations/flight3.gif
 
I don't know whether it's in the current chart catalogue, but I have in front of me an old copy of Chart 5095B. It's a gnomonic chart of much of the North Atlantic, showing at any point the course to steer to follow a great circle route to the Mona Passage. The Mona Passage is between Haiti and Puerto Rico, and is the route usually followed to the Panama Canal.

For a big ship following the GC route, it's easy. Fix your position, look at the chart and read off the course that you should be steering. At Bishop Rock you should be steering 260 deg T, but the course gradually changes until you reach the Passage steering about 215 deg. The rhumb line bearing is about 232 deg.

The problem is that the chart only works if you are aiming for the Mona Passage (or, by taking reciprocal courses, starting there and aiming for somewhere in the eastern Atlantic. I suspect that the only reason that this chart was produced was the large number of ships wanting to go through the Panama Canal. There may well be other similar charts for other routes, and I would expect one for the Pacific entrance to the Canal

Incidentally, the edition date is 1937, with corrections in 1939 and 1955.
 
A line that intersects each meridian at 90° is called a Line of Latitude.

I've not noticed any lines of latitude that form spirals!
 
Not playing anymore - if you're going to uintroduce logic to the argument (and at the same time teach me to read you posts properly) it's not fair (lol) /forums/images/graemlins/smirk.gif
 
[ QUOTE ]
A man sets off from his camp and travels due North for 10 miles. He sees the tracks of a bear and follows it due West for 10 miles until he catches up with it and shoots it. He then drags it 10 miles due South until he arrives back at his camp.

What colour was the bear?

When you think you know the answer, highlight this post /forums/images/graemlins/wink.gif

<span style="color:white">There was no bear. Bears do not live at the South pole. </span>

[/ QUOTE ]

So what if he shot a snowleopard instead? /forums/images/graemlins/wink.gif
 
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