Everything about Improper Rotation totally explained
In 3D
geometry, an
improper rotation, also called
rotoreflection or
rotary reflection is, depending on context, a
linear transformation or
affine transformation which is the combination of a
rotation about an axis and an inversion about the origin.
Equivalently it's the combination of a rotation and an
inversion in a point on the axis. Therefore it's also called a
rotoinversion or
rotary inversion.
In both cases the operations commute. Rotoreflection and rotoinversion are the same if they differ in angle of rotation by 180°, and the point of inversion is in the plane of reflection.
An improper rotation of an object thus produces a rotation of its
mirror image. The axis is called the
rotation-reflection axis. This is called an
n-fold improper rotation if the angle of rotation is 360°/
n. The notation
Sn (
S for
Spiegel, German for
mirror) denotes the symmetry group generated by an
n-fold improper rotation (not to be confused with the same notation for
symmetric groups). The notation
is used for
n-fold rotoinversion, for example rotation by an angle of rotation of 360°/
n with inversion.
In the wider sense, an improper rotation is an
indirect isometry, for example, an element of
E(3)
E+(3) (see
Euclidean group): it can also be a pure reflection in a plane, or have a
glide plane. An indirect isometry is an
affine transformation with an
orthogonal matrix that has a determinant of −1.
A
proper rotation is an ordinary rotation. In the wider sense, a proper rotation is a
direct isometry, for example, an element of
E+(3): it can also be the identity, a rotation with a translation along the axis, or a pure translation. A direct isometry is an affine transformation with an orthogonal matrix that has a determinant of 1.
In the wider senses, the composition of two improper rotations is a proper rotation, and the product of an improper and a proper rotation is an improper rotation.
When studying the symmetry of a physical system under an improper rotation (for example, if a system has a mirror symmetry plane), it's important to distinguish between
vectors and
pseudovectors (as well as
scalars and
pseudoscalars, and in general; between
tensors and
pseudotensors), since the latter transform differently under proper and improper rotations (pseudovectors are invariant under inversion).
Further Information
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