Linear isometry

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. 

Having a distance at hand, we can introduce linear isometries as mappings y) IR IR that keep the zero vector and distances hxed, i.e. we require that, for allx,y IR ... 

Here, the left side is fixed under linear isometry, and the squares x x = x and y y = lyp on the right hand side are fixed as well, so that x y is also retained. Moreover, the law of cosines gives the identity... 

Thus, the scalar product as well as lengths are retained, along with the cosine and therefore also the angle. In the next step we shall see that lineru isometries are in fact linear mappings. 

Thus, we have shown that mappings cp IR - IR which fix the origin and retain distances, are linear. This is sufficient justification to call them linear isometries. It remains to discuss briefly why these linear mappings are represented by orthogonal matrices. 

Hence, the matrix d> representing the linear isometry satisfies... 

Conversely, orthogonal matrices represent linear isometries since their columns contain an ON-basis. 

These mappings of 3D space are the mappings represented by orthogonal matrices, i.e. they are elements of the group of matrices representing linear isometries, also denoted by O3, for sake of simplicity of notation ... 

The point group is generated by two linear isometries, a reflection at the plane spanned by 6(1 and 62, and a 180° rotation with axis Cj. Although both these isometries have the same effect on the skeleton, both must be considered since a reflection changes chiral substituents, while a rotation does not. These generators are represented by... 

The point group P was shown to be the set of four linear isometries p,-, i e 4, linear mappings of 3D space, represented by the following matrices (with respect to the chosen basis) ... 

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