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Isometries

An isometry W, see also Section 2.1,

  1. maps each point P of the point space onto exactly one image point $\tilde{{\it P}}$: $\tilde{{\it P}}=\textsf{W}\,{\it P}$;
  2. is a mapping of the point space onto itself which leaves all distances and thus all angles invariant.
There are different types of isometries which will be characterized in this section. For this characterization the notion of fixed points is essential.

Definition (D 3.1.1) Let W be an isometry and P a point of space. Then P is called a fixed point of the isometry W if it is mapped onto itself (another term: is left invariant) by W, i.e. if the image point $\tilde{{\it P}}$ is equal to the original point $P$: $\tilde{P}=\textsf{W}\,P = P$.

The isometries are classified by their fixed points, and the fixed points are often used to characterize the isometries in visual geometric terms, see the following types of isometries. Besides the `proper' fixed points there are further objects which are not fixed or left invariant pointwise but only as a whole. Lines and planes of this kind are of great interest in crystallography, see the following examples.

Kinds of isometries.

The kinds 1. to 4. of isometries in the following list preserve the so-called `handedness' of the objects: if a right (left) glove is mapped by one of these isometries, then the image is also a right (left) glove of equal size and shape. Such isometries are also called isometries of the first kind or proper isometries. The kinds 5. to 8. change the `handedness': the image of a right glove is a left one, of a left glove is a right one. These kinds of isometries are often called isometries of the second kind or improper isometries.

  1. Identity I. The identity mapping maps each point onto itself, each point of space is a fixed point. All lines and planes of the space are left invariant as well.
  2. Translation T. By a translation each point of the point space is shifted in the same direction by the same amount, such that the translation vector r from each original point P to its image point $\tilde{{\it P}}$ is independent of the point P. There is no proper fixed point. Nevertheless, each line L parallel to r is mapped onto itself as a whole, as is each plane which contains L. [The identity mapping may be considered as a special translation with r = o, where o is the zero vector of length zero, see Section 1.3. Except if it is mentioned explicitly, the term `translation' is used for proper translations only, i.e. for translations with $\mathbf{r} \ne \mathbf{o}$.]
  3. Rotation. Each rotation maps a line of points onto itself pointwise. This line is called the rotation axis. The whole space is rotated around this axis by an angle $\Phi$, the rotation angle. The unit vector $\mathbf{u}_{\circ}$ parallel to the rotation axis is called the direction of the rotation axis. Each plane perpendicular to the rotation axis is mapped onto itself as a whole: it is rotated about the intersection point of the plane with the rotation axis. For a 2-fold rotation also each plane containing the rotation axis is left invariant as a whole. [The identity operation may be considered as a special rotation with the rotation angle $\Phi=0^{\circ}$. Except if it is mentioned explicitly, the term `rotation' is used for proper rotations only, i.e. for rotations with $\Phi \ne 0^{\circ}$.]
  4. Screw rotation. A screw rotation is a combination of a rotation ( $\mathbf{u}_{\circ}$ is the direction of the rotation axis) and a translation with its translation vector parallel to u$_{\circ}$. A screw rotation leaves no point fixed, the rotation axis of the involved rotation is called the screw axis, and the vector of the involved translation is the screw vector. The screw axis is not left fixed pointwise but as a whole only (it is shifted parallel to itself by the involved translation). In general the result of the combination of 2 isometries depends on the sequence in which the isometries are performed. The screw rotation, however, is independent of the sequence of its 2 components.
  5. Inversion. An inversion is the reflection of the whole space in a point $P$, which is called the center of inversion. The point $P$ is the only fixed point. Each line or plane through $P$ is mapped onto itself as a whole because it is reflected in $P$. The inversion is an isometry of the second kind: any right glove is mapped onto a left one and vice versa.
  6. Rotoinversion. A rotoinversion can be understood as a combination of a rotation with $\Phi\neq0^{\circ}$ and $\Phi\neq180^{\circ}$ and an inversion, where the center of inversion is placed on the rotation axis of the rotation. A rotoinversion is an isometry of the second kind. The inversion point (which is no longer a center of inversion !) is the only fixed point; the axis of the rotation, now called rotoinversion axis, is the only line mapped onto itself as a whole, and the plane through the inversion point and perpendicular to the rotoinversion axis is the only plane mapped onto itself as a whole. Again, a rotoinversion does not depend on the sequence in which its components are performed.
  7. Reflection. A reflection is another isometry of the second kind. Each point of space is reflected in a plane, the reflection plane or mirror plane, such that all points of this plane, and only these points, are fixed points. In addition, each line and each plane perpendicular to the mirror plane is left invariant as a whole.
  8. Glide reflection. A glide reflection is an isometry of the second kind as well. It can be conceived as a combination of a reflection in a plane and a translation parallel to this plane. The mirror plane of the reflection is now called a glide plane. The translation vector of the translation involved is called the glide vector g. There is no fixed point of a glide reflection. Left invariant as a whole are the glide plane and those planes which are perpendicular to the glide plane and parallel to g as well as those lines of the glide plane which are parallel to g.
Crystallographic symmetry operations may belong to any of these kinds of isometries. They are designated in text and formulae by the so-called Hermann-Mauguin symbols and in drawings by specific symbols which are all listed in IT A, Section 1 as well as in the Brief Teaching Edition of Vol. A. Although each kind of isometries is represented among the crystallographic symmetry operations, there are restrictions which will be dealt with in the next 2 sections.


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