When One Basis Vector Changes

Changing from one basis set to another is easy — multiply by a matrix. When the basis sets differ by just one vector I get confused, if we are to exploit the faster arithmetic. Here is some simple algebra that unconfused me.

In 3D we have two basis sets: {X, Y, A} and {X, Y, B}. Note the symmetry. We can express A in the 2nd set and B in the 1st set thus:
A = a_xX + a_yY + a_bB
B = b_xX + b_yY + b_aA
Substituting our expression for B into the expression for A yields:
A = a_xX + a_yY + a_b(b_xX + b_yY + b_aA)
A = (a_x + a_bb_x)X + (a_y + a_bb_y)Y + (a_bb_a)A
0 = (a_x + a_bb_x)X + (a_y + a_bb_y)Y + (a_bb_a − 1)A

Since {X, Y, A} is a basis the coefficients in parens in this last equation must all be 0. We thus learn:
a_b = 1/b_a
a_x = − a_bb_x
a_y = − a_bb_y

In routine cvz2f we have the natural normal to the facet already expressed in zone bcc coordinates as ufn.(k). ufn.(k) plays the role of B above—we have expressed B and the natural normal in the other basis. To express zone basis element k in facet coordinates, is like expressing A in basis {X, Y, B}.
a = 1. /. w.(k) acts as “a_b = 1/b_a”.

Now cvz2f now flips the sign of the vector component for the natural normal. All the users of the routine needed it!