A matrix will preserve or reverse orientation according to whether the determinant of the matrix is positive or negative.

Geometrically, this consists of all the elements of GL(n) that preserve both orientation and volume of the various geometric solids in Euclidean space.

Rotations are direct isometries, i.e., isometries preserving orientation.

Each of these groups has a subgroup of transformations which preserve orientation: those where the determinant of A is positive.

Unlike in the three-dimensional case, we can equivalently restrict the affine transformations to those that preserve orientation.

Those which preserve orientation are called proper, and as linear transformations they have determinant +1.

In his book Principles of Mathematics (1903), Russell considered a motion to be a Euclidean isometry that preserves orientation.

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

Even though it appears to be a well-behaved function, it is not holomorphic; it reverses orientation whereas holomorphic functions locally preserve orientation.

There is a subgroup E(n) of the direct isometries, i.e., isometries preserving orientation, also called rigid motions; they are the rigid body moves.