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How do you set correct heading while cruising across latitudes?

On a flat map, point A might be to point B at, say, heading 060 (intentionally no cardinal headings taken for example, as they don't create this situation). But on a curved surface, like spherical earth, the heading doesn't remain constant- it keeps changing as you move away from or toward the equator, sending aircraft miles off-target over long distances, as a result of pole-bound spiraling of the path.

Is there a formula to determine the optimum heading at the outset (even though it may not 'seem' correct), thereby, reaching the desired point B on the globe at the end of the cruise?

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  • $\begingroup$ Related (but not talking about navigation automatisms): KLAX >> HDG 090° >> Cross the Atlantic >> Where are we now?. Spherical navigation is even more complex with magnetic declination. $\endgroup$ – mins Jun 11 '15 at 20:45
  • $\begingroup$ Flying long distances, almost always your heading will have to change. This is more pronounced at higher lattitudes as the meridians come together at the poles. Suppose you fly from Tokyo to New York . The shortest route takes you almost over the North Pole. You'd be heading northerly at first and past the top of the world you have a southerly heading. $\endgroup$ – Chris V Jun 11 '15 at 21:36
  • $\begingroup$ At the risk of being pedantic: when an autopilot holds a heading, it does just that: it maintains that heading and doesn't adjust it for great circle effects or magnetic variation or wind drift or anything else -- the autopilot holds that heading. When the autopilot is coupled to a navigation solution (such as "NAV" or "LNAV" coming from an FMC or a GPS), then the A/P will fly whatever route the nav solution commands, which typically (though not necessarily) does compensate for all such things... by changing your heading as required to maintain the desired course. $\endgroup$ – Ralph J Jun 14 '15 at 0:50
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Yes, almost all modern computer navigation systems take this into account.

The track between two points along the "spherical" earth is called the great-circle track. Except for a N/S heading (or a E/W heading at the equator), the heading will vary along the track. Ed Williams has compiled a formulary for great circle navigation here, mostly derived from the Haversine formula. The formula you are interested in is "Course between points" for calculating the initial heading to fly from point A to point B.

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You wrote:

On a flat map, point A might be to point B at, say, heading 060. [...] Is there a formula to determine the optimum heading at the outset (even though it may not 'seem' correct), thereby, reaching the desired point B on the globe at the end of the cruise?

As written in the other answer, this is not how large distanced are flown, as aircrafts follow great circles for a minimal distance. But if you want to fly a constant heading from your outset, this can be done:

Imagine, you have one of these illuminated globes and put it into a paper cylinder, such that the cylinder touches the equator everywhere. All contours of the earth are projected against the cylinder, and you can retrace them with a pen. Unfold the cylinder, and you get a mercator projection: enter image description here

Mathematically, this can be calculated by this conversion of earth coordinates:

$x=\text{<longitude>}$

$y=\mathop{\rm arsinh}(\tan(\text{<latitude>}))$

and drawing this into a coordinate system where unit length is equal for x and y. (the labels on the y-axis are not y-coordinates, they denote the latitude.)

As you see, the face of the earth is vertically warped, and the size of continents is not preserved. For example, Greenland seems to be of the same size as Africa, though it's actually only 7%. Also, distances are not preserved.

BUT: Angles are preserved!

Draw a straight line from your location to any destination, and read the angle between this line and the vertical lines. Follow this heading, and you will reach your destination. Mathematically, you can determine the course by the following formula, but keep in mind that the arctan function does not always give you the correct formula. You need to know if you have to fly NE, SE, SW or NW and apply this do the value. (Can be done automatically with more math, too.)

$$\alpha=\arctan\frac{x_{dest}-x_{depart}}{y_{dest}-y_{depart}}$$


As said above, distances are not preserved. This straight lines are by far not the shortest routes and your passengers will complain about the long flight, as well as your airline about all the burned fuel...


EDIT:

To answer the question in the comment:

In a standard mercator projection as above, only pure north-south tracks as well as tracks on the equator lie on great circles and so are the shortest route.

Of course, you can put that paper cylinder around your globe in an other orientation, for example touching the 0-meridian instead of the equator. This way, you get the transversal mercator projection, which looks like this:

enter image description here
(by Lars H. Rohwedder, http://commons.wikimedia.org/wiki/File:Transversal_Mercator_0.jpg)

In this map, straight horizontal lines will lie on great circles, but the heading (angle to meridians) will change during flight. You can also use other arbitrary orientations for the cylinder, like this 45° projection, looking even more weired.

However, if you are searching for a constant heading and don't care about distance, the standard mercator projection gives the answer.

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  • $\begingroup$ So what this is saying is that there is no straight line path is the shortest distance (follows the great circle) or at least close to it? Or can you do a mercator projection that centers around the poles, for instance to approximate the shortest distance between New York and Tokyo? $\endgroup$ – Michael Jun 12 '15 at 4:38
  • $\begingroup$ @Michael along the meridians of longitude (i.e. directly northerly and southerly tracks, for example Stockholm - Cape Town), great circles equals rhumb lines (with "rhumb line" being another word for a straight line on a mercator projection) $\endgroup$ – Waked Jun 12 '15 at 6:03
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Yes. They fully take into account that the world is flat.

Just kidding. Long distance route planning is a complicated process. Usually commercial flights follow fixed, well-established routes. For example, when flying from North America to Europe, aircraft generally use North Atlantic Tracks. These tracks are designed in a way that is fully cognizant of the oblate sphericity of the Earth.

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    $\begingroup$ if only it were that easy... i could just fly my plane into one of those tracks and be at my destination, no problem. sorry, this question is asking how the autopilot handles intermediate heading changes required to stay in that track. $\endgroup$ – Erich Jun 12 '15 at 5:08
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    $\begingroup$ "when flying to Europe aircraft generally use North Atlantic Tracks." You seem to have forgotten that planes fly to Europe from places other than North America. $\endgroup$ – David Richerby Jun 12 '15 at 8:33

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