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I understand that an orthodrome (line on a great circle) is the line with the shortest distance between two points on a sphere.

If an aircraft flies straight ahead, i.e. there is no wind, the rudder is not actuated and the aircraft is parallel w.r.t. the surface of the earth underneath it, will it follow an orthodrome/great circle?

I also understand that a loxodrome is the line that one follows when keeping a constant heading. This is the point where I'm confused, because wouldn't that mean that when flying straight ahead/along an orthodrome, the heading/bearing would constantly change? In other words, does this mean that one needs to turn in order to keep a constant heading?

EDIT: My question is not "which line does one follow when keeping a constant heading" (a loxodrome, I know that) but rather "what line does one follow when flying straight ahead (in the hypothetical case of no wind and no control inputs)?"

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    $\begingroup$ Closely related, maybe a dupe? Also this question. $\endgroup$ – Pondlife Nov 8 '17 at 21:32
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    $\begingroup$ Possible duplicate of What trajectory do I fly if bearing angle is constant? $\endgroup$ – mins Nov 8 '17 at 22:23
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    $\begingroup$ What effect does the Earth’s rotation (I.e. Coriolis force) have? $\endgroup$ – Jim Garrison Nov 8 '17 at 23:51
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    $\begingroup$ @JimGarrison, none because the discussion is based upon ground track and does not address winds in any way. The problem is actually a geometric problem but if we assume the Earth to be spheriod it maps to navigation on the surface of the sphere. $\endgroup$ – mongo Nov 9 '17 at 3:34
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    $\begingroup$ @Daniel the second thing you said: it is not meaningful to speak of great circles on ellipsoids. When you cut an ellipsoid with a plane going through the center, what you obtain looks like a great circle, but it is not. $\endgroup$ – Federico Nov 9 '17 at 12:20
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When flying an orthodrome, with few exceptions, the heading will be constantly changing. In theory the change is continuous, but in practice most aircraft are quantatized to the nearest degree.

An exception are missiles, where because of a higher speed, having a more precise heading is more critical. Therefore most longer range missiles will internally use a higher granularity in the heading (like 0.01 degree or less).

To be clear, a loxodrome is a course which crosses all meridians of longitude at the same angle, and has a constant bearing measured to true or magnetic north. Loxodrome are also called rhumb lines. All loxodromes spiral from one pole to the other pole, except longitudinal loxodromes.

An orthodrome is also called a great circle route, and is characterized with heading changes (for most headings) to allow the vessel/aircraft to fly the shortest path along the surface of the earth to get to another point on the earth. Assuming that the earth is a sphere, an orthodrome is defined by a plane which goes through the center of the sphere, and the curved lines formed by the outer portion of the sphere intersecting the plane form what is known as a great circle route. The equator and meridians of longitude and their inverse lines on the other side of the sphere, form orthodromes. In those examples, travel on the equator is a constant heading of 090 or 270. On longitude lines, the heading is either N or S, until polar passage. An equatorial orthodrome cannot be a loxodrome. A longitudinal orthodrome is a loxodrome, although a rather uninteresting one.

It all seems rather straight forward now, right? There are wikis on both rhumb lines / loxodromes and also great circle routes or orthodromes. The graphics in them may help understand things.

https://upload.wikimedia.org/wikipedia/commons/thumb/0/04/Great_circle_hemispheres.png/220px-Great_circle_hemispheres.png

https://upload.wikimedia.org/wikipedia/commons/thumb/d/d6/Loxodrome.png/220px-Loxodrome.png

Addendum to readdress question from OP:

If an aircraft flies straight ahead, i.e. there is no wind, the rudder is not actuated and the aircraft is parallel w.r.t. the surface of the earth underneath it, will it follow an orthodrome/great circle?

The answer is always yes.

What the OP appears to be asking is whether the airplane, flying with only an inertial reference (NOT an inertial guidance system) and a fixed distance above the surface of the earth, can be accomplished without a change in direction, except for the circling of the earth.

So to explain this, let's call the earth a sphere for this discussion. If a great circle route is extended, it will scribe a line which wraps around the sphere, and divides the sphere into two identical half-spheres. Those half-spheres can be made with a single directional cut of the sphere. If you will, the sphere is split in half by a plane (geometric type, not aeronautical type) and the circle formed by that plane is the great circle route wrapping around the sphere.

The heading of an airplane flying a great circle route will with few exceptions, be constantly changing. The exceptions are when the plane is the equator or is a meridian of longitude and the corresponding reciprocal meridian.

So once again, when the path of the aircraft is constrained to the surface or some fixed distance above the surface, the inertial direction relative to the surface of the sphere, will remain a constant direction. Since only great circles which pass through the poles or travel the equator have one axis fixed relative to lat/long of the earth, they will have constant headings. The ones which are polar will have heading flips at the poles. All other orthodromes will have continuously changing headings.

There is just one more aspect of navigation that I would be remiss to not mention, and that is Transport Wander, which may be observed on a heading indicator on an aircraft, and is the function of the sin(track angle) * delta longitude/flight hours * tan(latitude)/60. The polarity changes with east vs west and northern vs southern hemispheres.

To the OP, I am sorry that I misinterpreted your question, and for the resultant confusion.

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  • $\begingroup$ The line of constant heading isn't a straight line by any definition. $\endgroup$ – DJClayworth Nov 9 '17 at 0:22
  • $\begingroup$ "When flying an orthodrome, with few exceptions, the heading will be constantly changing." Alright, that makes sense. But when flying without control inputs and without wind, will the aircraft follow this orthodrome? $\endgroup$ – Daniel Nov 9 '17 at 10:53
  • $\begingroup$ @mongo What do you mean with "it will not fly a path on it's own"? It will have to follow some path right? Like a straight line along the ellipse (geodesic), if I got it right. $\endgroup$ – Daniel Nov 9 '17 at 21:08
  • $\begingroup$ Alright, thanks to everyone who answered, now it appears more clear! With the combination of the answer I seem to finally understand it. $\endgroup$ – Daniel Nov 11 '17 at 11:09
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There isn't a perfect answer to this, but effectively yes.

Anything that moved in a completely straight line would of course leave the curved surface of the Earth and go off in to space. And realistically no vehicle of any kind could perform the (lack of) manouvers described above with no control changes.

To perform the best approximation of 'straight flight' while staying on the surface of the Earth would mean deviating from the true straight line only in a direction perpendicular to the Earth's surface. This in turn means staying in the plane defined by containing the vector of direction of initial motion and also the centre of the Earth. The intersection of the Earth's surface with this plane - i.e. the path the craft would actually follow- would be the great circle. And in answer to the last part, yes, when flying a great circle the heading constantly changes.

EDIT: This does, of course, assume a reference frame relative to the Earth, as pointed out in comments..

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    $\begingroup$ Upvoted, however, since your reference frame rotates, the great circle would be disturbed by the coriolis force (unless you are moving in the plane of rotation, i.e. heading E/W along the equator). $\endgroup$ – Waked Nov 9 '17 at 1:01
  • $\begingroup$ Your explanation with the "plane defined by containing the vector of direction [...] and also the centre of the Earth" actually helped me quite a lot! So just to recap: without control inputs or wind, one will stay on the great circle, with a constantly changing heading? $\endgroup$ – Daniel Nov 9 '17 at 10:50
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    $\begingroup$ @Daniel assuming no wind and a trimmed aircraft then yes. $\endgroup$ – Notts90 Nov 10 '17 at 21:45
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If you go straight north-south, you follow longitude lines which are straight.

If you go east-west at the equator, that latitude line is straight. All others hit the earth's surface at a bit of an angle, and so they are curved.

Imagine you follow the 89 deg 59 minute parallel, which is 1 minute (1/60 degree) south of the North Pole. Like every parallel, it's a circle. It's close enough that at altitude, you could see the north pole. You could also see past the north pole to the 89deg59' parallel ... on the other side of the North Pole. The parallel you are on! You'd be over there in a few minutes, heading 090 the whole time. And obviously you'd have to turn to get there, since you are flying a circle.

At lower parallels, the effect isn't nearly as extreme, but it's still a turn.

If you fly diagonally, at a heading not a multiple of 90, then east-west travel is part of your path, so a slight turn is part of the heading.

How can the path be curved if you are flying straight? It's not... The location grid system is curved because Earth is curved.

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  • $\begingroup$ Alright, that's a nice explanation for why maintaining a constant heading requires turning! So I guess from this one can deduce that flying completely straight means a constant change in heading? $\endgroup$ – Daniel Nov 10 '17 at 16:50
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    $\begingroup$ @daniel precisely. $\endgroup$ – Harper - Reinstate Monica Nov 10 '17 at 17:14

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