Bilbao Crystallographic Server Transformation matrix

## The transformation matrix

A general change of the coordinate system involves both an origin shift and a change of the basis and is described by the matrix-column pair (P, p).
• The matrix P

• This matrix is often referred to as the linear part of the coordinate transformation and it describes a change of direction and/or length of the basis vectors.

The matrix P is a (3x3) matrix

P =  P11 P12 P13 P21 P22 P23 P31 P32 P33

and relates the new basis (a', b', c') to the old basis (a, b, c) according to

 P11 P12 P13 (a', b', c') = (a, b, c)P = (a, b, c) P21 P22 P23 = (aP11+bP21+cP31, aP12+bP22+cP32, aP13+bP23+cP33) P31 P32 P33

• The origin shift p

• The origin shift is described by the shift vector p=p1a + p2b + p3c. The coordinates of the new origin O' with respect to the old coordinate sysem (a,b,c) are given by the (3x1) column

 p1 p= p2 p3

• Concise form

• The data on the matrix-column pair (P, p) are oftenn written in the following concise form:

P11a+P21b+P31c, P12a+P22b+P32c, P13a+P23b+P33c; p1,p2,p3

For example:

The expression (P, p) = (a-b,a+b,2c; 0,0,1/2) stands for

P =
 1 1 0 -1 1 0 0 0 2
 0 and p= 0 1/2

### Euler angles

According to Euler's rotation theorem, any rotation can be discribed using three angles (θ, φ, Ψ).

A general rotation can be written as:

R =
 Cosθ -Sinθ 0 Sinθ Cosθ 0 0 0 1
 1 0 0 0 Cosφ -Sinφ 0 Sinφ Cosφ
 CosΨ 0 SinΨ 0 1 0 -SinΨ 0 CosΨ

[*] For more information: International Tables for Crystallography. Vol. A, Space Group Symmetry. Ed. Theo Hahn (3rd ed.), Dordrecht, Kluwer Academic Publishers, Section "Transformations in crystallography", 1995.

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