Rotors: A practical introduction for 3D graphics (2023)(jacquesheunis.com) |
Rotors: A practical introduction for 3D graphics (2023)(jacquesheunis.com) |
[1] https://en.wikipedia.org/wiki/Orthogonal_group#Canonical_for...
The 3-vector is not a bijective representation (starts repeating after length == 2*pi) but otherwise is the most elegant of them all, IMO. No need for rotors or quaternions. Plus you can simply use Rodrigues to get a rotation matrix back.
[1] https://en.wikipedia.org/wiki/Rodrigues%27_rotation_formula
quaterions:
0*1 + b*i + c*j + d*k
rotors:
0*1 + b*xy + c*yz + d*zx
Now... rotors do have some unique powers in that they're incredibly general. You don't need to hop from complex numbers to quaternions when you move between spaces and beyond, you can just use rotors for everything:
2d:
complex numbers
rotors
3d:
quaternions
rotors
4d:
octonions
rotors
Minkowski spacetime:
???
rotorsWhat's the difference between doing:
rotors: 01 + bxy + cyz + dzx
and
quaterions: 01 + bjk + cki + d*ij
?
edit: https://api.lib.kyushu-u.ac.jp/opac_download_md/410895/178c.... this seems to explain the difference
quaterions:
a*1 + b*i + c*j + d*k
rotors:
a*1 + b*xy + c*yz + d*zx
The exceptions are 0 degrees and 180 degree rotations (and 360, 540, etc...), which will have one and zero as the real components, respectively.
Quaternions are a concept specific to the 3-dimensional (Euclidean) space, in the same way as "complex" numbers (for whom "binions" would be a more appropriate name) are a concept specific to the 2-dimensional (Euclidean) space.
Neither quaternions nor "complex" numbers have anything to do with a 4-dimensional space of vectors.
Quaternions are a field that is a subset of the 2^3 = 8-dimensional geometric algebra associated with a 3-dimensional space of vectors, while the "complex" numbers are a field that is a subset of the 2^2 = 4-dimensional geometric algebra associated with a 2-dimensional space of vectors.
While vectors are associated to transformations of the corresponding affine space that are translations, quaternions/complex numbers are associated to transformations of the space that are rotations or similarities.
How could one even subject a statement like that to proof? If you insist that you thought about quaternions without thinking in 4D, and the author insists that you're just so used to thinking in 4D that you didn't even notice it, then who's to arbitrate that dispute?
(I'm sensitive to these issues because I'm a mathematician of the "visualizing 4D is just visualizing n dimensions and setting n = 4" variety, so I have no idea when I'm particularly thinking in 4, or any other specific number, of dimensions ….)
So the dispute is just using words differently. For quaternions, I think its very important to understand how they apply to everyday space / reality / imagination / intuition / Geometry. I wish it was something everyone understood.
To prove that statement to a normal person in the normal dimension way you'd need to define your 4 dimensions like space, time, or scale and explain examples of quaternions.
The geometric product works in any dimensions. They have a clear geometric intepretation. Rotations and translations can done using the same algebraic operations.
There are vectors in the wide sense, i.e. elements of a linear space. Linear spaces are a.k.a. vector spaces, where "vector" is used in the wide sense.
Then there are vectors in the strict sense, which is the sense corresponding to the etymology of the word "vector", which have additional properties over the axioms of a linear space.
Vectors in the strict sense are elements of some particular linear spaces that are associated with the translations of affine spaces, and which are also associated with geometric algebras, where the dimensions of the geometric algebras as linear spaces over the real numbers are 2^N, where N is the dimension of the set of vectors as a linear space over the real numbers.
Quaternions as a linear space over the real numbers happen to be 4-dimensional, but this 4-dimensional space has no relationship whatsoever with a 4-dimensional space that would be an extension of the familiar 3-dimensional space of the Euclidean geometry, which models the space in which we live.
Since the quaternions are means for describing transformations of the 3-dimensional space of Euclidean geometry, all applications of the quaternions include the 3D geometry in a more or less disguised form, in the same way as any application of complex numbers includes the geometry of the Euclidean plane, even if that is not obvious because the applications are described in an abstract way, using only the axioms of the field of quaternions or of the field of complex numbers.
Many applications of complex numbers in electronics or digital signal processing become far more easier to understand when one thinks about the geometric transformations of a plane that correspond to complex numbers, instead of thinking only about the axioms of the field of complex numbers. The same happens for quaternions.
The physical space in which we live and that we can imagine, is modeled mathematically as an affine space, i.e. as a space of points. We can also imagine affine spaces with more dimensions than 3.
Some linear spaces are vector spaces in the strict sense, being sets of the translations of an affine space. Other linear spaces, like the set of quaternions, are not vector spaces in the strict sense. In order to help our perception of such abstract linear spaces, we may use tools like graphs or drawings that map some part of the abstract linear space to an affine space that we can visualize, e.g. on a computer display, but we must keep in mind that this is just a mapping and that the nature of that abstract linear space is different from the spaces that we can see.