How is this geometrical invariant called in English?

Question

I am studying affine and projective geometry and I have encountered some invariant: the cross ratio, which in Italia is called "birapporto" and another one which I do not know the name of in English. We referred to as "rapporto semplice"; if we are working on the real line as an affine space, and we have three points $x,y,z$ it is the ratio $\frac{x-z}{y-z}$. This quantity is invariant under affine trasformazionee of the line. Could you help me? I have tried to find out how it is called din English, but I didn't find anything... Thanks

Answer 1

In German, the cross ratio is called “Doppelverhältnis” which literally translates to “double ratio”. The origin of that is because it's a ratio of ratios:

$$(a,b;c,d)=\frac{(a-c)(b-d)}{(a-d)(b-c)}= \frac{\;\frac{a-c}{b-c}\;}{\;\frac{a-d}{b-d}\;}$$

The rightern expression is a ratio of two distance ratios: in the numerator the distance of $a$ and $b$ relative to $c$, in the denominator relative to $d$. Each of these fractions would be called a “Verhältnis”, a ratio. If you wanted to distinguish it more clearly from the double ratio, you might call it a simple ratio, or perhaps a single ratio.

I would guess that your Italian prefix “bi” denotes something “double”, that “rapporto” means “ratio”, and that “semplice” means “simple”. I would think “simple ratio” would be an appropriate term to use in this context. But I would not expect it to be understood without definition.

You could go for “ratio of distances”, “ratio of oriented lengths” or something like that to provide a bit more detail of what you mean. But $\frac{a-b}{c-d}$ would be a ratio of distances, too. And invariant under affine transformations as long as all four are collinear. If you want your exact definition, for three points with $z$ occurring in both numerator and denominator, then I know no universally understood term for it, so I'd pick any of those suggested above and combine it with a short definition at first use.