3 ms·
The same is true (with lower performance, I suspect) with the 3x3 matrix representation.
by BenFrantzDale 8y ago
The same is true (with lower performance, I suspect) with the 3x3 matrix representation.
- jacobolus 8y agoMatrices for rotation have the problem that you are using a 9-dimensional representation for a 3-dimensional quantity, so it is easy to make matrices that are not rotations (indeed the vast majority of the possible space of matrices consists of transformations that are not anywhere close to a pure rotation). Composing matrices leads to rounding errors. It is more complicated to invert matrices. As you say the matrix logarithm is more complicated. Etc. If you use a scalar + bivector “rotor” representation, that is only a 4-dimensional representation, which is easy to normalize to unit magnitude.
- jcagalawan 8y agoIsn’t the inversion a trivial transpose for rotations since they’re orthogonal?
- gugagore 8y agoIt's beautiful to see that the logarithm of a rotation matrix is a skew-symmetric matrix. The skew-symmetric matrices correspond to angular velocity, which behave like vectors. You can add and interpolate as you expect. Rotation matrices multiply. Skew-symmetric matrices add. The logarithmic and exponential map moves between the two spaces. Just letting you know that I think it's valuable to bring up matrices into this discussion, even if they do have problems in practice with rounding errors and efficiency.
- marctenbosch 8y agoSkew symmetric matrices are isomorphic to bivectors :)
- senderista 8y agoThat’s because skew-symmetric matrices are the Lie algebra of the Lie group of rotation matrices (and the exponential map takes a Lie algebra to a Lie group). In physics we call them the “infinitesimal generators” of the rotation group.
- gugagore 8y agoI love knowing about Lie algebras and groups, but I think I disagree about the "because". The matrix exponential and logarithm just are. You can define and compute them without invoking anything about Lie groups. I didn't know the term "infinitesimal generator". Thanks!
- BenFrantzDale 8y agoIndeed. And certainly since rotation matrices and quaternions are both representations of rotation, they should have a log that is isomorphic. Wilder still is the log of a 4x4 transformation matrix has the same tangent-vector properties, giving a coordinate-system-invariant rotation and translation.