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

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.

## 3 Answers

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.

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

• 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. – 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!

• For example, the midpoint divides the segment in a simple ratio =2, according with your definition. The midpoint is an affine invariant. – GReyes Feb 16 at 19:40
• Do you have a reference for it being named in English? – jmerry Feb 16 at 20:10
• 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/… – GReyes Feb 17 at 6:21