Every method of representing rotations using just 3 coordinates is bound to have unavoidable singularities. Same principle as gimbal lock. The fix is to add at least one coordinate. Using quaternions (4 components) is a common fix. You can also go all the way to a matrix representation, but that’s overkill unless you need the other properties (scaling, shearing, etc.).
For rotations in three dimensions, that’s true, but it’s not true in general that for an n-dimensional system with coordinate singularities, you need to add a coordinate to avoid the singularities. The standard six orbital parameters have coordinate singularities, but there are other sets of six parameters that give the same information that don’t. (Of course, the standard six also have other advantages, that are convenient enough that they usually outweigh the disadvantage of the singularities)
Here’s a map of the ETOPS ranges over Antarctica for an Airbus 330 with one engine out. The outer shaded region is more than 240 minutes from an airport, and the inner is more than 370 minutes.
Whoa, whoa, wait a minute – practical quaternions? Please say more! I think this is nearly as intriguing as adding antigravity modules to the aircraft to improve fuel economy…
Absolutely. I have written a couple of bits of code that plotted stuff all over the globe, and produced maps. Quaternions are the basic building blocks of the transforms you need. The same is true for 3D modelling of motions and even 3D graphics and games. My first real encounter with quaterions was with a motion tracker used for immersive vis sim.
The Universal Transverse Mercator coordinate system is augmented by the Universal Polar Stereographic to cover the entire globe in a sensible manner. UPS covers North of 84 and South of 80. Maybe the aircraft nav software gets unhappy crossing into UPS. (I doubt that is the reason.) UTM has the evil property that you can have multiple map references for the same actual location. It is fine if you are navigating entirely inside one of the transverse plane. Once you cross planes, it all goes to hell.
The MGRS (military grid reference system) is essentially the same thing, but with labelling changed.
The GEOREF (World Geographic Reference System) was used for aviation, but is a basically a way of labelling a position with increasing precision as you add length to the position designation. Not useful for navigation, and not seen much these days. (I had to make map displays for that too. It was the easy one.)
If you work in Cartesian coordinates and use quaternions for all your transformations it is trivial to make most things work out. You need to jump in and out of lat-long from time to time. However I can well understand why actual aircraft nav programs might prefer to do something else. Like stay in lat long. Doing proper navigation, stuff that involves weather, and other complicating factors, is going to require a lot more care than just slicing and dicing the geoid.
You can represent the orientation of a 3-dimensional object using three Euler angles, which might feel most natural because you have 3 parameters worth of information. But if you want to do things like move the object smoothly from one orientation to another, it becomes problematic. You can just find the angles for the initial position and for the final position, and move each of the angles linearly from the starting value to the ending value, but even if you’re not very close to the coordinate singularity, this will usually end up with a rather convoluted movement. If, on the other hand, you represent the orientation with a quaternion (four parameters) with the restriction that the magnitude is 1 (to reduce it by one parameter), then a simple transition of each of the parameters will, in fact, give you a simple motion, and you won’t have any singularities.
Aliens? Just asking cuz best (and scariest) motion tracker ever! (I get it is not that kind of motion tracker but had to ask)
60 nanometers is pretty close anyway.
And just so this post isn’t entirely snark, there is a similar trending thing about a “straight line” voyage from India to Alaska. It misses Antarctica, though
Rotations in 3-dimensional Euclidean space are (naturally…) represented by 3 × 3 orthogonal matrices with determinant 1. It is a 3-dimensional Lie group (i.e., it is smooth…), with the sphere as a quotient.
A 2-dimensional sphere, of course, is just \{\,(x,y,z) \mid x^2+y^2+z^2=1\,\} . What you can’t do is pick a two-parameter system of coordinates on the sphere that will not degenerate somewhere. One would hope aircraft navigational software would take this into account, but who knows? I am not convinced that a good solution is to never fly near a pole, never cross the Equator or the International Date Line, and so on just to avoid weird bugs; in fact you would assume that kind of stuff is the first thing they would test since it’s so obvious.
Hopefully you got the answers you needed already–but yes, quaternions have widespread use in computer graphics for representing rotations while avoiding the weird issues with yaw/pitch/roll representations or otherwise. As DPRK notes, 3x3 matrices are also commonplace and it’s easy to translate from one to the other. Typically one will use quaternions in the “early” parts of the pipeline and then switch to matrices when it comes time to render things on the GPU, since matrices are more comfortable for them.
One neat thing about them is that you can compute the log and exponent of a quaternion. From that, you can compute any real power of one: say: q^{0.5} or q^{0.01}. You can then use this for interpolation: q itself is the full rotation, while any power less than that is a partial rotation (since, for example, applying q^{0.01} a hundred times gives the full amount). You get nice smooth interpolation out of this, assuming that you are careful to go in the short direction rather than the long way (i.e., rotate by 10 degrees rather than 350 in the other direction).
Ascension Technology Flock of Birds.
Back then they were pretty much it for motion tracking.
In addition to the aforementioned use of quarterions for transformations in computer graphics and geodesy, it also has real applications for simulating and controlling physical systems. Quaternions are widely used for guidance and control applications in simulating high performance aircraft and rockets, as well as in 3D multiaxis mechanism synthesis and control because it is easier to check the math and doesn’t suffer from virtual gimbal lock (the singularities mentioned upthread) where it is not possible to determine a unique path. Having done kinematic synthesis just using Euler angles and then doing the same synthesis with quaternions will reveal why it is so much easier to use the latter.
Stranger
That’s true, though it’s no different than saying that given a system with no singularities, I can usually come up with an alternative that does have them with the same number of coordinates. For instance, translations in 3D Euclidean space, represented as (X, Y, Z), has no singularities. If for some reason I want to represent it as (\frac{1}{X}, \frac{1}{Y}, \frac{1}{Z}), then singularities appear out of nowhere. But it’s kinda pointless.
Note that if you look at, say, a torus, you will normally need three charts to cover it, but (without increasing any dimension) if you are willing to cover it by a plane, now you can easily represent any point on the torus by two “angle” coordinates, just not uniquely.
Note, incidentally, that according to the Whitney embedding theorem you can smoothly embed any (not too crazy) smooth manifold in Euclidean space; at most you have to double the dimension.
I flew almost directly over the North Pole in January this year, on a Finnair flight from Osaka to Helsinki.
I don’t just mean the Arctic region, the flight path passed within at most a few km of 90°N. In order to avoid Russian airspace, the plane routed up through the Bering Strait and then directly over the pole to Helsinki.
My wife’s flight a week later took the same route and she was given a certificate by Finnair to say she had flown over the pole. I was jealous that I didn’t get one.
I’m not sure what you’re saying, here. A single chart seems to me to be perfectly adequate for a torus, and the nonuniqueness of representation is only due to adding a multiple of 360º to an angle giving an equivalent, coterminal angle, which is also an issue with longitude.
I suppose I am saying something like: if you wish to cover the torus by charts based on contractible open sets, then already, for topological reasons, you will need at least three. You can even map each set homeomorphically onto the entire Euclidean plane to define some coordinates.
A single chart is not enough to define the differentiable structure on a torus (you can do it with two), but you can construct the torus as a quotient \mathbb{R}^2/\mathbb{Z}^2 and get it that way. Exactly the same deal as a circle vs. the real line.
Just for fun, IIRC temperatures at the Pole of Cold in Antarctica dip below the freezing temperature of carbon dioxide. And that’s at ground level. Since it’s cold because the upper atmosphere is falling to the surface, I’m sure the temperature at 30,000 feet (or whatever) is even colder.
Does anything especially interesting happen to jets in such cold conditions? I imagine building up a layer of dry ice on the wings is neither fun nor a familiar problem to solve…
Fuel temperature can be a problem in polar or even high latitude non-polar operations. Fuel cooled below about -40C thickens and becomes a waxy petroleum jelly. Which neither pumps nor burns for shit. But first some of the constituents of the fuel start solidifying into microscopic (and eventually macroscopic) gooey sticky chunks. That’s not good either.
Heat is generated by air friction and ram compression as the airplane moves through the air. So there’s a type of temperature, called “total air temperature” (TAT) which takes that ram heating into account. At cruise altitudes and speeds, TAT is normally about 20C warmer than the ambient or “static air temperature” (SAT). So the aircraft structure is “feeling” TAT despite the much colder SAT nearby. Even over the US Midwest in summer the SAT at cruise altitudes is -40C to -50C. With TAT at typical cruise speeds being -20C to -30C.
When the fuel is loaded on the ground it’s at more or less +20C being as much of it is stored or piped underground. So there’s a large thermal mass there at the start of the flight that takes time to cool from +20C to -40C. As well, the fuel tanks typically contain heat exchangers with the hydraulic system, where hot hydraulic fluid is cooled by the fuel resting in the tanks and that fuel is incrementally warmed by the hydraulics.
But as a general matter, the fuel is as warm as it’s going to get when you take off and only cools asymptotically towards the TAT as you cruise. Once you start descent it begins to warm again, but usually not by much; there just isn’t enough time spent at the lower altitudes for it to rewarm much before you land.
For especially long flights through especially cold airmasses the atmospheric fuel cooling can eventually get to be a problem where speeding up, descending, or diverting are the only options to prevent thickening fuel and then a crisis. Nowadays with high quality polar weather forecasting and data these situations don’t tend to be surprises. And the flight is routed around the extreme cold or below it enough to not cause a midflight problem needing ad hoc resolution. But monitoring air and fuel temp as you drone along is just one more routine task for the crew and the computers.
The rest of the mechanics of the airplane are designed with an awareness of the fuel limitations, so they’re more robust to cold than the fuel is.
I assume (but feel free to fight my ignorance) that this is due to economic restraints of the GPS system (few people living in those latitudes, hence few satelites there - and its “good enough”) …
IOW there is nothing intrinsically in the GPS system that would make it less precise above the poles should those become really important? … right?