One of the features generally known about me is that I'm not a mathematical sort of cove. In fact, pretty much the opposite.
Nevertheless, I was pretty pleased with myself for starting to read through Donald E. Sands' Introduction to Crystallography. It's a subject that I've never properly understood, and I thought that it might be worthwhile to spend a few hours trying to become acquainted with the subject in greater detail.
I made it as far as nine pages before I encountered this:
Calculations involving oblique coordinate systems are certainly more tedious than they would be if the axes were at right angles to each other, but compensation is provided by features such as the identity of the fractional coordinates of equivalent points in different unit cells. The following formulas will be useful.The volume V of a unit cell is given by
V = abc(1- cos2 α - cos2 β - cos2 γ + 2cos α cos β cos γ)1/2
The distance l between the points x1, y1, z1, and x2, y2, z2 is
l = [(x1 - x2)2a2 + (y1 - y2)2b2 + (z1 - z2)2c2
+ 2(x1 - x2)(y1 - y2)ab cos γ + 2(y1 - y2)(z1 - z2)bc cos α
+ 2(z1 - z2)(x1 - x2)ca cos β]1/2You should verify these formulas for the familiar case where α = β = γ = 90 degrees. Derivation of these formulas is accomplished easily by means of vector algebra.
Clearly, this isn't going to be quite as easy as I had hoped.