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.


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.

  • $\begingroup$ Please see my reference to Coxeter's book. I must admit that I am more familiar with Russian terminology, where the adjective "simple" is used instead. $\endgroup$ – GReyes Feb 17 at 9:14

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 $\frac{a_1a_3}{a_1a_2}=\lambda$.

  • $\begingroup$ The cross ratio is an invariant in Projective Geometry (simple ratio is not invariant there, and you need a ratio of ratios). Observe that the cross ratio is a ratio of simple ratios. $\endgroup$ – GReyes Feb 16 at 19:37

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!

  • $\begingroup$ For example, the midpoint divides the segment in a simple ratio =2, according with your definition. The midpoint is an affine invariant. $\endgroup$ – GReyes Feb 16 at 19:40
  • $\begingroup$ Do you have a reference for it being named in English? $\endgroup$ – jmerry Feb 16 at 20:10
  • $\begingroup$ I was a bit surprised not to find the term "simple ratio" as often as expected. But you can check, for example, Coxeter's book (a classic) books.google.com/… $\endgroup$ – GReyes Feb 17 at 6:21

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