Special Orthogonal Group

The special orthogonal group, denoted as $SO(n)$, is a group of $n\times n$ matrices that have the property of being both orthogonal and having a determinant of 1.

The Rotation Matrix is a special orthogonal group.

Why is det(R) = 1 needed?

The first property, $R{R}^T=1$, is the definition of an Orthogonal Matrix, which guarantees to preserve volume. However, it doesn’t guarantee preservation of orientation.

$det(R)=1$ guarantees that orientation is preserved.

Both are required, if you only ensure that $det(R)=1$, there might be stretching, but volume and orientation is maintained.

• Orthogonality ensures that the transformation preserves the lengths of vectors and the angles between them

Related

• Special Euclidean Group