Bilbao Crystallographic Server Transformation matrix

The transformation matrix

• 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 (2x2) matrix



P =		P11 P12 P21 P22


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



(a', b') = (a, b)P = (a, b)		P11	P12		=	(aP11+bP21, aP12+bP22)
		P21	P22

• The origin shift p

The origin shift is described by the shift vector p=p₁a + p₂b. The coordinates of the new origin O^' with respect to the old coordinate sysem (a,b) are given by the (2x1) column



p=		p1
		p2




• Concise form

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

P₁₁a+P₂₁b, P₁₂a+P₂₂b; p₁,p₂

P = 1 1 -1 1

and p= 0 1/2