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It'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 beha
by gugagore 8y ago
It'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.