Regarding Quaternions
Author
I am trying (probably fruitlessly) to get as much conceptual understanding of Quaternions as I can. In trying to wrap my head around it, I thought of this:
If you consider an 'object', say a triangle, in 2D space, you cannot rotate that object around either of the existing axes (x and Y), as it would then rotate out of the space altogether. You can only rotate it around the 'Z' axis, which of course does not actually exist in your 2D space.
I'm just wondering if this is in any way, shape or form related to the usefulness of Quaternions in describing rotations in 3D space.
Quote:Original post by CaveDwellerNot really.
I'm just wondering if this is in any way, shape or form related to the usefulness of Quaternions in describing rotations in 3D space.
This is just MHO, but I think it's best to develop a solid understanding of axis-angle rotations in general (in particular the vector rotation equation and its matrix equivalent) before tackling quaternions. Quaternions simply provide another means of expressing axis-angle rotations and various operations on them (interpolation, concatenation, and so on). Once you have a good understanding of rotations and how to manipulate them via matrix transforms, quaternions (at least with respect to their use as a rotation representation) will seem much less mysterious.
quaternions are just an interesting abstract mathematical number system that has all the same properties as our 10 digit number system (i.e. the arithmetic axioms are all the same: commutivity of addition/multiplication, etc). It just so happens that one can use them to represent rotations nicely. They do not conceptually have anything to do with rotation, they can just do rotation nicely.
-me
-me
Quote:Original post by Palidine
quaternions are just an interesting abstract mathematical number system that has all the same properties as our 10 digit number system (i.e. the arithmetic axioms are all the same: commutivity of addition/multiplication, etc). It just so happens that one can use them to represent rotations nicely. They do not conceptually have anything to do with rotation, they can just do rotation nicely.
-me
Quaternions certainly aren't commutative. Just like rotations in 3d space, and matrices.
They form what is known as a "division ring" which is very nearly a field (same structure as the real numbers) except that they aren't commutative under multiplication. The key property that they share is that every non-zero quaternion has a unique multiplicative inverse (conjugate of the quaternion scaled by the quaternions modulus)... they are pretty similar to complex numbers though, except complex numbers do form a field since they commute under multiplication.
[EDIT: Also, they have a fair bit to do with rotation... they were originally developed by Hamilton when he was investigating 3d rotations... and the cross product came directly from quaternion algebra - which predates formalised linear algebra I believe]
The best way to understand the relationship between quaternions and rotations is to read some introductions to Geometric Algebra. There's an interactive OpenGL tutorial that's pretty good from the University of Amsterdam but the site seems to be down at the moment. If you can't get to that page, try David Hestene's GA page or Jaap Suter's GA Primer. Quaternions are mathematically the same as Rotors in GA but GA gives a much better framework for understanding their geometric basis.
Author
Thank you all for the responses, and thank you for the links - those look like very informative pages.
I'm fairly comfortable with the math of quaternions; I probably should have made my original question a little clearer: I just found it interesting that you can't do rotations at all in 2D space without reference to an axis that does not exist in that space. There are many references in the various quaternion papers and tutorials that refer to them as essentially a 4D entity. It just struck me that in both cases, completely accurate rotations seemed to require a reference to an axis or 'entity' from a higher dimension. i.e. in both cases, rotations in n space are better handled by reference to n+1 space. I wondered if there were some common principal behind that, but I suspect that I may be tilting at windmills on this one.
Thanks again.
I'm fairly comfortable with the math of quaternions; I probably should have made my original question a little clearer: I just found it interesting that you can't do rotations at all in 2D space without reference to an axis that does not exist in that space. There are many references in the various quaternion papers and tutorials that refer to them as essentially a 4D entity. It just struck me that in both cases, completely accurate rotations seemed to require a reference to an axis or 'entity' from a higher dimension. i.e. in both cases, rotations in n space are better handled by reference to n+1 space. I wondered if there were some common principal behind that, but I suspect that I may be tilting at windmills on this one.
Thanks again.
Quote:Original post by CaveDweller
I just found it interesting that you can't do rotations at all in 2D space without reference to an axis that does not exist in that space. There are many references in the various quaternion papers and tutorials that refer to them as essentially a 4D entity. It just struck me that in both cases, completely accurate rotations seemed to require a reference to an axis or 'entity' from a higher dimension.
In 2D, you rotate around a point, not an axis. You can extend that point into a third dimension, but it's not related to the rotation.
Author
Quote:Original post by JohnBolton
In 2D, you rotate around a point, not an axis. You can extend that point into a third dimension, but it's not related to the rotation.
Doh! Well, that's embarassing.
I guess I really need to get some solid fundamentals before I out-think myself again.
Thanks.
It's probably better to think about rotating in a plane rather than bringing axes into it. This then works in any dimension (>= 2). The tip about Geometric Algebra is a good one, nowadays a good google search will bring up quite a few nice intros to get your teeth into.
Author
I just wanted to say thanks for the suggestions of looking into GA. The links eventually get cross-referential; it appears to be a relatively small community with something of an evangelical attitude, but I'm finding the subject fascinating.
It was somewhere during about the 4th paper that some light finally started to go on (and I still haven't gotten to quaternions as such). Intriguing stuff.
It was somewhere during about the 4th paper that some light finally started to go on (and I still haven't gotten to quaternions as such). Intriguing stuff.
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