Name for ratio of three co-linear points in affine geometry

Question

Given three points $a_1$, $a_2$, $a_3$ in an affine space such that $\dim \langle a_1, a_2, a_3\rangle = 1$ (i.e. they are co-linear) and $a_1\ne a_2$, my professor defined the "simple ratio" (making a literal translation) as the unique scalar $$ (a_1a_2a_3) := \lambda \in\mathbb{K} \qquad\text{such that}\qquad \overrightarrow{a_1a_3}=\lambda\, \overrightarrow{a_1a_2}\,. $$

I haven't found any equivalent term in English sources. Is such a concept defined in other places, or does it happen to be "popular" where I live?

For example, this notation is used in "Álgebra Lineal i Geometria" by M. Castellet and I. Llerena.

Answer 1

The Wikipedia Projective harmonic conjugate article uses the term "division ratio" which is rare in English unfortunately. More precisely, it states "C divides AB externally" and gives the ratio $AC:BC$ which corresponds to your $\lambda$ and where $ABC$ are three points in order on a line. A third point can divide a line segment "internally" where the ratio $\lambda$ is negative. Further it also states:

In terms of a double ratio, given points a and b on an affine line, the division ratio of a point x is $$ t(x)=\frac {x-a}{x-b}.$$

It seems to me that the term "division ratio" is what you want, although "simple ratio" is simpler.

Answer 2

I don't think I've ever seen this affine version named before, but I have seen the projective geometry concept that it's a case of: the cross-ratio $$(a_4,a_1;a_2,a_3) = \frac{a_1a_3\cdot a_4a_2}{a_1a_2\cdot a_4a_3}$$ If we choose $a_4$ to be the point at $\infty$ on the line, then $\frac{a_4a_3}{a_4a_2}=1$ and this becomes $\frac{a_1a_3}{a_1a_2}=\lambda$.

Answer 3

Ratios between segments are preserved in Affine Geometry. The simple ratio is not just popular, it has been widely used. Sometimes they define it a bit differently (instead of $A_1A_3/A_1A_2$ you take, for example, $A_1A_3/A_2A_3$. All of those are related, and what is more important, all these are preserved under affine transformations. BTW, Castellet and Llerena is a great book!