"You've got to make sure each one has its proper rotation on the pole, and that they all keep spinning."

Two rigid bodies are said to be different (not copies) if there is no proper rotation from one to the other.

Both are physical quantities which assume a single value which is invariant under proper rotations.

But by insisting on determinant +1, we can restrict the matrices to proper rotations.

Therefore, it is impossible to pick the proper rotation using factor analysis alone.

He made sure he wore T-shirts and underwear in proper rotation.

Thus, the symmetry group of the sphere with proper rotations is .

In addition to preserving length, proper rotations must also preserve orientation.

An improper rotation can be understood as an inversion followed by a proper rotation.

Let us only consider the case, then, of matrices R that are proper rotations (the third eigenvalue is just 1).