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The answer to What are quaternions and how are they used to represent spacecraft dynamics? gives us a thorough overview of the topic and is worth a good read.

I saw the HNQ Why are quaternions more popular than tessarines despite being non-commutative? and wondered:

Quesiton: Are bicomplex numbers including tessarines ever used in spaceflight (as an alternative to quaternions)?

See Bicomplex number for more.

uhoh
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    The most obvious use of quaternions in spacecraft is to represent rotation (and related concepts such as orientation), and the top answer to the "Why are quaternions more popular" question notes that they can't model 3D rotations. The only other potential use I can think of would be some signal processing operation, and it seems unlikely for such an obscure detail to be generally known...it could be buried deep inside some library. – Christopher James Huff Feb 16 '21 at 04:15
  • @ChristopherJamesHuff I see, thanks! Does "they" = bicomplex numbers as a whole, or just tessarines? If it's the former then maybe the "as an alternative to quaternions" doesn't work and the question needs to be modified. – uhoh Feb 16 '21 at 05:03
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    The answer seems to be general to bicomplex numbers. I'm not sure they're an alternative to quaternions at all, they're just similar in some respects, being another type of hypercomplex number. – Christopher James Huff Feb 16 '21 at 13:01
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    Also note the similarly named but quite different dual-complex numbers (https://en.wikipedia.org/wiki/Dual-complex_number). These are closer related to quaternions than to bicomplex numbers, and seem more applicable to spacecraft control than the latter. – Christopher James Huff Feb 16 '21 at 13:10
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    This is a bit tricky: Once math equations are written down, the implementation of , complex numbers, quaternions, etc. comes down to multidimensional arrays of numbers with some specified formulas for multiplying, combining, etc. these arrays of what are, after all, rational reals because that's all a computer deals with (other than naming them "classes" as an attribute). – Carl Witthoft Feb 16 '21 at 14:27
  • @ChristopherJamesHuff I've been trying to revive my awareness of complex numbers ever since this answer moved my problem into the complex plane (I wonder if dual complex numbers have a helpful application there?), so I'm always looking to learn about other ways that moving a problem to the complex plane provides utility. (see also How did Gaussian and Eisenstein integers get their names?) – uhoh Feb 17 '21 at 00:00
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    @CarlWitthoft Heh. SW engineers wish computers handled rational reals. Unfortunately, floating point numbers aren't quite the same. It's of course all reduced to simple arithmetic operations supported by the CPU's ALU/FPU, but I think it's reasonable to consider the software to "use" the abstractions used in its development. A problem is that what's an abstraction at one level is an implementation detail covered by another abstraction at another. And the abstraction may not even be in actual code, but in a comment giving the higher level math explaining an implementation. – Christopher James Huff Feb 17 '21 at 15:19
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    The linked question at HNQ is nonsense. Quaternions are useful precisely because they don't commute, and they don't commute in the same way rotations in 3D space don't commute. Using a commutative algebra to represent 3D rotations doesn't make sense. – David Hammen Feb 25 '21 at 15:46
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    @uhoh - Just because I don't know of any applications of bicomplex numbers in space exploration does not mean that they are not used anywhere in space exploration. It's hard to prove a negative. But as an alternative to quaternions? No. It doesn't make sense. – David Hammen Feb 26 '21 at 01:33

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Are bicomplex numbers including tessarines ever used in spaceflight (as an alternative to quaternions)?

Without the parenthetical remark, the answer is "I don't know". But with that parenthetical expression, the answer is all-caps "NO". Unit quaternions are used in space exploration precisely because they map nicely to rotations in three dimensional space. I wrote nicely, rather than perfectly as unit quaternions are a double cover to proper 3D rotations. A unit quaternion and its negative represent the same rotation. That problem is easily dealt with.

Tessarines don't have anywhere close to the right structure to represent rotations.

David Hammen
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Tessarines are quite like numbers, and quaternions are more like 3D vectors plus one more special dimension.

It is natural that vectors are useful in kinematics.

On the other hand, split-complex numbers are related to Minkowski space, so if space farers ever need extensive calculations in special relativity, split numbers may be of use.

Anixx
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