Proper isometry

An even number of reflections in the same plane send each point from (x,y,z) back to itself, which is equivalent to an identity operation (a proper isometry), while an odd number of reflections in the same plane is equivalent to a single reflection (an improper isometry). The product of two reflections in separate planes is a proper isometry. We distinguish two cases (1) If the two planes are parallel and spaced by a distance d, the isometry is a translation by 2d in a direction perpendicular to the two planes (2) if the two planes intersect at an angle a, the isometry is a proper rotation (or, simply, rotation) through 2a about the line of intersection (the axis of rotation). Translations and rotations are continuous operations because motion can be controlled by continuous changes in the distance or angle between the two mirrors. 

Finally, the product of two reflections in intersecting planes combined with two reflections in parallel planes that are perpendicular to the line of intersection—that is, the product of altogether four reflections and therefore a proper isometry— is a rotation-translation or screw displacement. This also is a continuous transformation. 

In analogy to combinations of even and odd integers under addition, the product of n proper isometries is a proper isometry regardless of whether n is even or odd the product of n improper isometries is a proper isometry if n is even and an improper isometry if n is odd and the product of a proper isometry and an improper isometry is an improper isometry. 

A set of points on the sphere may be transformed by isometries or congruence mappings that preserve the distances between all pairs of points. All isometries, in turn, can be built up from three basic types of transformations. (i) rotations about an axis, (ii) mirror reflections in a plane, and (iii) parallel displacements of all points. If the mappings are restricted to a fixed sphere, parallel displacements play no role, and the congruence transformations reduce to proper isometries or (rigid body)... 

Translations and rotations are particular isometries, i.e. the distances between any two points in space are maintained. As translations do not really matter in our situation, we restrict our attention to linear isometries, i.e. to isometries that fix the origin of 3D space. The reason is that we can embed a molecule in 3D space so that the origin of the space coincides with the molecule s barycenter, which is fixed under every symmetry operation. It turns out that linear isometries are compositions of rotations and reflections, orthogonal linear mappings where the representing matrix has determinant 1. The determinant of a proper rotation is +1, that of an improper rotation -1. Reflections are improper rotations. 

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