View Full Version : great circle

flyer_by_the_wire

1st Mar 2011, 14:57

hi!!

i dunno how but m a little confused regarding great circles and small circles.:ugh:

The shortest path between two points on a plane is a straight line. However, on the surface of a sphere there are no straight lines. Instead, the shortest distance between any two points on a sphere is a segment of a circle.Ok.:ok:

So between 2 points where a great circle ain't formed,we have a small circle.

Now the smaller arc of this small circle would be the shortest distance between these 2 points.

so y do we say that only the great circle distance is the shortest distance between 2 points,:eek:

Shouldn't it be that a circular distance(great circle or small circle) represents the shortest distance.

thnx in advance

Forget the circle word and use sphere instead.

Sit inside the SPHERE and draw lines on the outside wall.

The GREAT CIRCLE is not the literal meaning.

The circle does not exist.

on the surface of a sphere there are no straight lines. - only if you are talking in 3-D. In laymen's terms there are an infinite number of 'straight lines' on the surface of a sphere. See how much you need to turn left and right to go around the equator.

BOAC yes indeed when we use a paper map it is flat..uses two dimensions x and y..

I can remembr the formula for the volume of a sphere think that is 4/3PI R to the cube.

so we have 1,34 x 3.142 x the cube of the radius

but thats all I can remember. The surface equation I can't remember. Think one nautical mile is that distance subtended by one minute..or have I got that wrong?

HazelNuts39

1st Mar 2011, 15:38

So between 2 points where a great circle ain't formed,we have a small circle.The smallest circle you can draw on the sphere has the distance between your 2 points as its diameter, and the 2 points divide that circle into two equal arcs of 180 degrees. If you take either arc to get from A to B, you're not taking the shortest distance. To get that, you have to draw the greatest circle possible, which has the center of the earth as its center.

regards,

HN39

Upper Air

1st Mar 2011, 17:33

Great Circle TRACK - track.

The shortest line from a to b over the Earth.

Shorter than this one would need Thunderbird 4. (was it TB4 the one with the drill on it. . ?)

You can work out the Great Circle TRACK using one of many internet GCT thangs and they will do it for you.

There is formula for working out the Great Circle Track . .which is:

TheChitterneFlyer

1st Mar 2011, 17:54

A Great Circle is any line of Longditude, and, the Equator. A Small Circle is any line of Latitude (excluding the Equator).

Don't get confused by straight lines on a chart... dependant upon the Chart Projection a straight line could be anything... a Rhumb Line or a great Circle Track.

Good luck... it's a less than easy science to understand.

In simplistic terms, stretch a piece of string between any two points on a football... that's a Great Circle! If you were to have a piece of string (tied from end to end) that was any less (in length) than the circumference of the football and that you laid it onto the surface of the football... that's a Small Circle.

Cheers

TCF

rudderrudderrat

1st Mar 2011, 18:13

Hi Fly by the wire,

Stretch an elastic band between any two points on a Globe. It will be on the great circle between those points.

As HN39 mentioned, the centre of any great circle is at the centre of the sphere. An orbiting satellite moves around a great circle.

fellman

1st Mar 2011, 19:24

I'm not sure that I fully understand the OP's problem, but I think it might help to point out that the shortest distance between two points is only PART of a great circle. The full great circle goes all the way round the sphere, not just along the shortest line between the points.

Loose rivets

1st Mar 2011, 19:30

Chop the world in two equal halves - at any angle.

Travel over however much of the cut-line you need, and Voilà, you have it.

TheChitterneFlyer

1st Mar 2011, 20:10

So between 2 points where a great circle ain't formed,we have a small circle.

Now the smaller arc of this small circle would be the shortest distance between these 2 points.

NOOOOO... it's not the "smaller arc"... the shortest distance remains "part" of a Great Circle... the shortest distance between two globally defined points. If you were to continue "through" that defined point you'd eventually arrive back at the same position... a "Great Circle".

GarageYears

1st Mar 2011, 20:35

A great circle, also known as a Riemannian circle, of a sphere is the intersection of the sphere and a plane which passes through the center point of the sphere, as distinct from a small circle. Any diameter of any great circle coincides with a diameter of the sphere, and therefore all great circles have the same circumference as each other, and have the same center as the sphere. A great circle is the largest circle that can be drawn on any given sphere. Every circle in Euclidean space is a great circle of exactly one sphere.

For any two points on the surface of a sphere there is a great circle through the two points. The minor arc of a great circle between two points is the shortest surface-path between them. In this sense the minor arc is analogous to “straight lines” in spherical geometry. The length of the minor arc of great circle is taken as the distance of two points on a surface of a sphere, namely great-circle distance. The great circles are the geodesics of the sphere.

When long distance aviation or nautical routes are drawn on a flat map (for instance, the Mercator projection), they often look curved. This is because they lie on great circles. A route that would appear as a straight line on the map would actually be longer.

No credit to me - thank Wikipedia - but I think the point is the bit in bold. This is the confusion to most folk.

I think the point to understand is that it is the "minor arc" of the great circle that presents the shortest route between two points. Other circles can be drawn, but will result in a longer minor arc (least that's what I understand).

Dimitris

1st Mar 2011, 20:36

Great circle - Wikipedia, the free encyclopedia (http://en.wikipedia.org/wiki/Great_circle)

A great circle, also known as a Riemannian circle (http://en.wikipedia.org/wiki/Riemannian_circle), of a sphere (http://en.wikipedia.org/wiki/Sphere) is the intersection of the sphere and a plane (http://en.wikipedia.org/wiki/Fundamental_plane) which passes through the center point of the sphere, as distinct from a small circle (http://en.wikipedia.org/wiki/Small_circle).

For any two points on the surface of a sphere there is a great circle through the two points. The minor arc of a great circle between two points is the shortest surface-path between them.

Strictly speaking the Earth is not a perfect sphere (http://en.wikipedia.org/wiki/Reference_ellipsoid) (it is an oblate spheroid or ellipsoid – i.e., slightly compressed at the poles), which means that the shortest distance between two points (a geodesic) is not quite a great circle. Nevertheless, the sphere model can be considered a first approximation.

Great circle routes are used by ships and aircraft where currents and winds are not a significant factor. Flight lengths (http://en.wikipedia.org/wiki/Flight_length) can therefore often be approximated to the great-circle distance (http://en.wikipedia.org/wiki/Great-circle_distance) between two airports. For aircraft travelling west between continents in the northern hemisphere these paths will extend northward near or into the Arctic (http://en.wikipedia.org/wiki/Arctic) region, however easterly flights will often fly a more southerly track to take advantage of the jet stream (http://en.wikipedia.org/wiki/Jet_stream).

GarageYears

1st Mar 2011, 20:37

Gotcha by a whole minute!

keith williams

1st Mar 2011, 22:46

From the original post

So between 2 points where a great circle ain't formed,we have a small circle.

There are no two points on the earth which cannot be joined by a great circle.

Now the smaller arc of this small circle would be the shortest distance between these 2 points.

Try testing this argument on a football or other suitably small sphere.

Draw a circle with a diameter that is less than that of the sphere. Draw dots at two points on this circle.

Draw a straight line across the circle joining the two points.

You now have three paths between the two points. One is the straight line and the other two are arcs of the circle.

If you measure these paths you will find that the shortest is the straight line.

Now extend the straight line until the two ends meet. You will find that you have drawn a circle that has its centre at the centre of the sphere. This is the great circle on which the two points lie.

So the shortest distance between the two points is part of the great circle on which the two point lie.

flyer_by_the_wire

2nd Mar 2011, 11:50

@ all above

thank u so much for your valuable input.

@Keith

But i really don't get the idea as to how you can draw a great circle between any two points.Consider two places placed @ the same lattitude(eg.15 degrees N) but different longitudes(say 18 degree east and 20 degree east)..

Now these two places are joined by the same parallel of lattitude(which is a small circle).

So how can a great circle be drawn through these 2 points?A small circle joins these 2 places.And hence the shortest distance b/w these 2 should be the small circle!:ugh:

rudderrudderrat

2nd Mar 2011, 12:11

Hi flyer by the wire.

You can join two points with an infinite number of curved lines (including a Line of Latitude). However, only one line is the shortest distance, and it will lay on a Great Circle. (Lines of Longitude & the Equator are NOT the ONLY Great Circles.)

I suggest you buy yourself a globe of the world, and an elastic band. It will only take you a couple of minutes to see the difference.

keith williams

2nd Mar 2011, 13:10

@Keith

But i really don't get the idea as to how you can draw a great circle between any two points.Consider two places placed @ the same lattitude(eg.15 degrees N) but different longitudes(say 18 degree east and 20 degree east).. Now these two places are joined by the same parallel of lattitude(which is a small circle). So how can a great circle be drawn through these 2 points?A small circle joins these 2 places.And hence the shortest distance b/w these 2 should be the small circle!:ugh:

Don't worry, lots of people have problems visualising this type of thing. But it is quite easy to demonstrate it to yourself.

Take an apple or some similar sperical(ish) fruit. The moreaccuarte will be the results.

Draw your line of latitude on it and draw two dots on that line to represent your two positions.

Place the apple on a table and roll it across the surface until one of the dots is at the top of the apple. Place a knife on that dot and while keeping the knife on the top dot, move the knife until it is also directly above the other dot. Now cut straight down through the apple to the table.

You will find that you have cut through both dots and also through the centre of the apple. The edges of the cut surfaces are the great circle on which the two dots lie.

Mr Optimistic

2nd Mar 2011, 13:10

If you 'chop' the sphere perpendicular to the N-S axis you will get the small cicle line of latitude route. If you angle the cut downwards so it passes through the centre of the earth you will get the great circle track. In the northern hemisphere the line where the surface is sliced for the great circle will lie north of the constant latitude line. You can make any number of 'cuts' you want taking in both points by angling the blade.

Upper Air

2nd Mar 2011, 13:20

A semi great circle.

LOL:oh:?

TheChitterneFlyer

2nd Mar 2011, 13:33

If you 'chop' the sphere perpendicular to the N-S axis you will get the small cicle line of latitude route.

Except at the equator...

dan1165

2nd Mar 2011, 13:35

The shortest distance between 2 points on the sphere is "orthodromy" and is a part of a great circle . Distance given between 2 navaids :8 .

The longest distance is "loxodromy" . It's a path crossing the meridians at a constant angle .

Dimitris

2nd Mar 2011, 15:47

Dude... forget drawing lines, imagine a plane that passes from the 2 points and the center of the earth. There is only one such plane (3 points define a plane). The section of that plane with the surface of the earth is the 'great circle' between those two points.

Check google earth (not google maps) and click 'measure'. The distances and routes it gives you are great circles. Even at same latitude it doesn't follow a line of latitude (unless both points are on equator)

Alternatively:

Take an orange. Mark 2 points. Then imagine cutting with a knife that passes from the two points and the centre of the orange. The cut is a 'great circle'.

Geometry...

flyer_by_the_wire

2nd Mar 2011, 17:24

@Keith

Amazing way of explaination..sliced 7 tomatoes..got it..thnx:D

Another basic question which has been troubling me is the fact that meridians of longitude are rhumb lines.By definition rhumb lines are lines which cut all the meridians at equal angle.So how come a meridian becomes a rhumb line.

Looking forward to another great explaination.Thnx.

Green Guard

2nd Mar 2011, 19:31

You do not draw any track or great circle on any globe .

Either you draw it on a Jepp Chart or any other chart with similar globe projection.

Or Calculate

Dist= asin ( sin Lat1 x sin Lat 2 + cos Lat 1 x cos Lat 2 x cos ( Long 2- Long 1))

Track Heading will constantly change with the distance....

even if both Lat1 and Lat 2 are on the same paralel....unless they are on equator.

(That is why it was not possible to use it while flying constant heading, before the time of INS etc)

P.S.

Or just use your RNAV computer

biscuit74

2nd Mar 2011, 19:55

I'm not 100% sure what projection a Jeppesen chart uses- sounds as if it is a Lambert projection.

On a Lambert, the great circle track between any two points on the chart is straight line on that chart. If you check such a line, you will see the angle at which it cuts each successive line of longitude changes progressively, showing that mapped onto the surface of the earth the track will be curved.

Does that help?

You can spot a Lambert projection because all the lines of longitude are straight lines pointing to (& meeting if projected to) the apex at the Pole. Lines of latitude will be arcs of a circle.

boguing

2nd Mar 2011, 20:39

The confusion occurs as a result of the way navigation, as an art and science developed.

Drawing lines of Longitude around the globe from the two poles was obvious - and easy to see that it simplified the definition of how far West or East a point is. Segment lines that make the chocolate orange possible.

The lines of latitude are not so obvious, it's one way of doing it, but it could be argued that adding another pair of poles on the equator could generate a second set of segment lines that would intersect at points to give relative coordinate.

The former method looks good on a map, but what it doesn't show is that it is not a straight line in any sense.

We're brought up on maps that have nice straight lines on them. Nobody tells us until we're older that it's all an enormous fudge. Not only are lines of latitude not straight, but the type of projection means that not much actually looks like a map from a satellite. The only viewpoint that would see a line of latitude as straight is from a satellite sitting in the same plane as that line. From there, every other line of latitude would not be straight.

All lines of longitude are great circles.

The only line of latitude that is a great circle is the equator.

A Rhumb line is a quick and dirty way to get a heading from A to B. It needs no finessing if you're going North or South, but the closer to East West you head, and the farther up or down the globe you go the more a Great Circle will save you distance. You'll also need to factor in crosswinds/tides (aircraft/ship), but that's obvious.

FBTW - you will need to stretch your mind a little! All meridians of longitude are both Rhumb and great circle. Brace yourself - they do cut meridians at the same angle = 0.

Thinks - that might be 180 :D

keith williams

2nd Mar 2011, 21:01

Many references define a Rhumb Line as a line that crosses all of the meridians at the same angle. This definition leads to a second characteristic, which is that a Rhumb Line is a line of constant heading.

Historically the radial lines on a compass rose were called Rhumb Lines. So sailing a rhumb line course simply meant sailing a constant heading course.

The meridians are clearly lines of constant heading, (True North or True South), so they satisfy this condition for being rhumb lines.

Now let’s look again at the statement that “A Rhumb Line is a line that crosses all of the meridians at the same angle.”

The first point to note is that a line cannot cross any meridian that is not in its path. So we could refine our definition a little bit to become “A Rhumb Line is a line that crosses all of the meridians in its path at the same angle.”

The meridians of longitude are not circles, but are semi-circles. They start at one pole and end at the other. Where one meridian ends, its anti-meridian begins. And where the anti-meridian ends, its meridian begins. This means that the meridians and anti-meridians meet at the poles, but never actually cross each other. So between the poles the meridians of longitude are rhumb lines, which never cross any other meridians.

For a track which goes directly over one of the poles there will be a constant heading followed by an abrupt reversal of heading at that pole, after which the heading will again be constant. So we could argue that this track is made up of two rhumb lines.

HazelNuts39

2nd Mar 2011, 21:20

From Wikipedia:

In navigation, a rhumb line (or loxodrome) is a line crossing all meridians of longitude at the same angle, i.e. a path derived from a defined initial bearing. That is, upon taking an initial bearing, one proceeds along the same bearing, without changing the direction as measured relative to true north.I suppose one can reverse that statement, so that any line that maintains a constant heading to true north is a rhumb line?

So the equator and the parallels are also rhumb lines?

regards,

HN39

Loose rivets

2nd Mar 2011, 22:20

I claim the prize for the shortest answer in post 10. However, I omitted to say one has to hold the two halves of the world together before walking along the cut line. Oh, and I also omitted to say the world is an oblate spheroid, so all the rules get twisted a tad.

Not much though. 12,756 / 12,714 km

Keith Williams;

So between the poles the meridians of latitude are rhumb lines, which never cross any other meridians.I assume you didn't intend what you wrote?

EDIT :: OK folks, Keith has now fixed the original.

keith williams

2nd Mar 2011, 23:47

Absolutely mm43. Well spotted.

Dimitris

3rd Mar 2011, 12:34

Green Guard (http://www.pprune.org/members/19828-green-guard)

Join Date: Aug 2000

Location: RedIndia

Age: 96

Posts: 174

You do not draw any track or great circle on any globe .

Either you draw it on a Jepp Chart or any other chart with similar globe projection.

Or Calculate

Dist= asin ( sin Lat1 x sin Lat 2 + cos Lat 1 x cos Lat 2 x cos ( Long 2- Long 1))

Track Heading will constantly change with the distance....

even if both Lat1 and Lat 2 are on the same paralel....unless they are on equator.

(That is why it was not possible to use it while flying constant heading, before the time of INS etc)

P.S.

Or just use your RNAV computer

The problem is that the guy that started the post doesn't understand what a great circle is, he can 'use' the 'RNAV' sure, in the same way I (not a pilot) can 'use' the autopilot to 'fly' a plane.

Keep going like that and soon there will be Unmanned Airliners flying cause if its ok to use the RNAV cause you don't understand what a 'sphere' and a 'plane' are (geometrical plane, not the flying one), then it will be ok not to understand what 'lift' and 'glideslope' is cuase you can use the 'green dot'... I'd trust as a passenger more a computer than a 'RNAV user' or a 'green dot follower'. At least computers are deterministic.

delete if offending

cheers

(SLF)

Rivet gun

4th Mar 2011, 12:01

If you have i tunes installed, you could follow this link.

Connecting to the iTunes Store. (http://itunes.apple.com/institution/the-open-university/id380206132)

Then under categories on the right hand side, click mathematics. Then click on the the picture with the hot air baloon, then start with the the video at line 5 titled "spherical geometry", followed by "looking for theta" and "great circle distance"

Type1106

4th Mar 2011, 12:23

Try this site to visualise the problem- Great Circle Mapper. Great Circle Mapper (http://www.gcmap.com/)

1106

Upper Air

5th Apr 2011, 11:09

No, the Equator strangely enough is in fact a Great Circle. One of the reasons being is that is does not `concave` to the nearer pole - like rhumb lines do. As the most lear-ned chaps and chapessess have said - if you put an elestic band round a large orange around the middle, with the pip bit at the top . . . now, if you move the elastic band up to say . . half way up the orange towards the top . . and it stays on . . then, then the track the elastic band is forming over the surface of the orange is now, is now, concave to the nearer pole - the North Pole of the orange.

In other words your EASTerly track will follow the curve of the Earth - you will always be heading EAST but you will not be TRACKING to your EASTerly DESTINATION you will be following the curvature of the Earth.

So, Inertial Navigation Systems can calculate a DIRECT GREAT CIRCLE TRACK - = A STRAIGHT LINE to the destination WITHOUT GOING AROUND THE CURVATURE OF THE EARTH. In other words it cuts the corner or in this case the bend.

So, say you are in Western Europe France, Paris say and you want to go to Eastern China. If you go due East South East you may eventually arrive there but it would take a long time and a lot of fuel.

So, you may wish to head North East from Paris ! Yep, North East and go East over the North Siberian coast all the way round through Mogolia or wherever your "DIRECT" Great Circle Track Takes you to your destination.

So, your initial heading is North East! and your subsequesnt headings will be East and South East respectively.

There are formulae for working all this out, which you will get on your ATPL course and the INS/IRS systems do it all using accellerometers and different types of gyro systems and flux things and well it gets very involved but enough to say it maintains the Heading of the aircraft to co-incide with the Track required instead of basing the heading (as you correctly said) on Magnetic North.

Just to completely freak you out - it does in fact calculate from Mag North in order to ascertain which way it is pointing, how far it is from the Pole as a waypoint etc., if only to inform the pilot, so it does "have a reference" to Mag North but will or will not use it to calc this TRACK - as it will effectively just look at the destination co-ordinates and draw a stright line to it (Great Circle Track) and maintain THAT line to Destination.

Hope that helps.