Rotors: A practical introduction for 3D graphics (2023) (opens in new tab)

(jacquesheunis.com)

77 pointsbladeee1y ago20 comments

20 comments

Geometric Algebra supporters keep advertising that rotors are great since they work in any dimension, which makes me wonder: would an arbitrary n-dimensional SVD-like decomposition benefit from using rotors instead of rotation matrices, and if so how? And if not, why?

thesz1y ago

There are 2^n coefficients in the general GA transform (versor). One should be very, very careful dealing with GA versors of high dimensions.

chombier1y ago

Yes, but from the canonical form of rotation matrices [1] I would expect such matrices to be represented as a sum of bi-vectors/rotors, which should take the same amount of data?

[1] https://en.wikipedia.org/wiki/Orthogonal_group#Canonical_for...

rsp19841y ago

If you take the Matrix logarithm of an SO(3) (3x3 rotation matrix) you get a 3-vector that represents the axis of rotation, scaled by the rotation amount (in radians). This is also a cheap operation using the inverse Rodrigues formula [1].

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

mecsred1y ago

Thanks for the link to the Rodrigues form, that's quite interesting. Slightly confused by your comment though, shouldn't the matrix logarithm produce another matrix?

rsp19841y ago

You're correct. The logarithm produces what's essentially a cross product matrix, 0 on the diagonal and symmetric off-diagonal. The off-diag elements are the 3-vector I was talking about. Thanks for pointing that out.

nyrikki1y ago

SO(3) is nonabelian, and isn't simply connected, which is why the surjective homomorphsin to SU(2) is valuable, particularly in 3D graphics.

itishappy1y ago

Composing axis-angle representations gets real weird real fast. You can convert them into 9 element rotation matrices, but then you lose the benefits of storing them using only 3 elements in the first place.

koolala1y ago

Saying quaternions require thinking in 4 dimensions seems like a lie with no proof. The geometric product is just the quaternion product broken up into scaler and vector parts.

itishappy1y ago

It's no lie, quaternions do actually have 4 dimensions. The part I take issue with is that rotors also require 4 dimensions to represent 3d rotations, they're just labeled slightly more intuitively.

    quaterions:
    0*1 + b*i + c*j + d*k

    rotors:
    0*1 + b*xy + c*yz + d*zx

I've included real components, but when representing rotations they'll always be zero. (They'll be non-zero during intermediate calculations though, so you need to consider them!)

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:
    ???
    rotors

koolala1y ago

I don't consider variables to equal dimensions so I'm glad you put them in 3D too. People take the term 'dimension' very literally when thinking about things.

What'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

itishappy1y ago

I'm wrong. Too late to edit, correction below:

    quaterions:
    a*1 + b*i + c*j + d*k

    rotors:
    a*1 + b*xy + c*yz + d*zx

The representations I shared previously with zero real component are for the points under rotation, not the rotors themselves which have real components in the general form. Apologies for misinformation!

The exceptions are 0 degrees and 180 degree rotations (and 360, 540, etc...), which will have one and zero as the real components, respectively.

adrian_b1y ago

You are right.

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.

HelloNurse1y ago

Applications of quaternions to 3D geometry do not matter: as a field or vector space over real numbers quaternions are four dimensional because 1, i, j, k are linearly independent. Over complex numbers they are a two dimensional vector space instead.

3 more replies

JadeNB1y ago

> Saying quaternions require thinking in 4 dimensions seems like a lie with no proof.

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 ….)

koolala1y ago

I think of dimensions like 3D space. Normal people dimentions. Math calls independent variables 'dimensions' but it's a overloaded naming convention that causes people to think it has something to do with normal everyday dimensions like space. Math may as well use a different term than 'dimension' similar to Rank and there would be no confusion. Using the everyday dimention term makes a harmful misconception about quaternions for normal people because they are associated with understanding everyday geometry.

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.

ColinHayhurst1y ago

> The geometric product is just the quaternion product broken up into scaler and vector parts.

The geometric product works in any dimensions. They have a clear geometric intepretation. Rotations and translations can done using the same algebraic operations.

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20 comments

chombier1y ago

thesz1y ago

There are 2^n coefficients in the general GA transform (versor). One should be very, very careful dealing with GA versors of high dimensions.

chombier1y ago

Yes, but from the canonical form of rotation matrices [1] I would expect such matrices to be represented as a sum of bi-vectors/rotors, which should take the same amount of data?

[1] https://en.wikipedia.org/wiki/Orthogonal_group#Canonical_for...

rsp19841y ago

[1] https://en.wikipedia.org/wiki/Rodrigues%27_rotation_formula

mecsred1y ago

Thanks for the link to the Rodrigues form, that's quite interesting. Slightly confused by your comment though, shouldn't the matrix logarithm produce another matrix?

rsp19841y ago

nyrikki1y ago

SO(3) is nonabelian, and isn't simply connected, which is why the surjective homomorphsin to SU(2) is valuable, particularly in 3D graphics.

itishappy1y ago

koolala1y ago

Saying quaternions require thinking in 4 dimensions seems like a lie with no proof. The geometric product is just the quaternion product broken up into scaler and vector parts.

itishappy1y ago

It's no lie, quaternions do actually have 4 dimensions. The part I take issue with is that rotors also require 4 dimensions to represent 3d rotations, they're just labeled slightly more intuitively.

    quaterions:
    0*1 + b*i + c*j + d*k

    rotors:
    0*1 + b*xy + c*yz + d*zx

I've included real components, but when representing rotations they'll always be zero. (They'll be non-zero during intermediate calculations though, so you need to consider them!)

    2d:
    complex numbers
    rotors

    3d:
    quaternions
    rotors

    4d:
    octonions
    rotors

    Minkowski spacetime:
    ???
    rotors

koolala1y ago

I don't consider variables to equal dimensions so I'm glad you put them in 3D too. People take the term 'dimension' very literally when thinking about things.

What'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

itishappy1y ago

I'm wrong. Too late to edit, correction below:

    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.

adrian_b1y ago

You are right.

Neither quaternions nor "complex" numbers have anything to do with a 4-dimensional space of vectors.

HelloNurse1y ago

3 more replies

JadeNB1y ago

> Saying quaternions require thinking in 4 dimensions seems like a lie with no proof.

koolala1y ago

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.

ColinHayhurst1y ago

> The geometric product is just the quaternion product broken up into scaler and vector parts.

The geometric product works in any dimensions. They have a clear geometric intepretation. Rotations and translations can done using the same algebraic operations.

j / k navigate · click thread line to collapse