3 ms·
For rotations in 3D space, another nice representation is based on antisymmetric matrices [1]. All rotations can be represented as exp(A), where A has the form:
by guyomes 2y ago
For rotations in 3D space, another nice representation is based on antisymmetric matrices [1]. All rotations can be represented as exp(A), where A has the form:
0 -z y
z 0 -x
-y x 0
Moreover, this generalizes to other dimensions. This allows to see that in 2D, rotations have one degree of freedom, in 3D they have three, and in 4D they have six.
[1]: https://en.wikipedia.org/wiki/Rotation_matrix#Exponential_map https://en.wikipedia.org/wiki/Rotation_matrix#Exponential_ma...
- jasomill 2y agoThe two dimensional case is my favorite: let I be the identity matrix and J the "imaginary unit matrix" 0 -1 1 0, then JJ = -I = exp(Jπ) and exp(Jθ) = cos(θ) -sin(θ) sin(θ) cos(θ).