So, to first order, an infinitesimal rotation matrix is an orthogonal matrix.Īgain discarding second order effects, note that the angle simply doubles. The product isĭiffering from an identity matrix by second order infinitesimals, discarded here.
(In 3 dimensions the trace of any rotation matrix must equal 1 + 2 cos(Angle) therefore the angle of rotation must be infinitesimal)įirst, test the orthogonality condition, Q TQ = I. To understand what this means, one considers These matrices do not satisfy all the same properties as ordinary finite rotation matrices under the usual treatment of infinitesimals. Where dθ is vanishingly small and A ∈ so(3). Īn actual "differential rotation", or infinitesimal rotation matrix has the form Informally, an element of is the difference between the matrix of an infinitesimal rotation and the identity matrix, but "scaled up by a factor of infinity". The matrices in the Lie algebra are not themselves rotations the skew-symmetric matrices are derivatives. (The vector cross product can be expressed as the product of a skew-symmetric matrix and a vector). It is also a semi-simple group, in fact a simple group with the exception SO(4).Īnd consists of all skew-symmetric 3 × 3 matrices. It is compact and connected, but not simply connected.
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