Geometrically distinguishing between relative positioning of 3 points on a line

Question

Scenario: we have a line in a 2D plane, and we place 3 points on it each having a different color. For example:

we see in this case that the blue point is the first point, the green is the second and the red the last one.

Here's a second scenario where the red point is behind the green point, i.e. green comes last:

I am trying to find out if by a simple geometric criterion one can figure out if the red point is past the green point (as in the 1st image) or if the red point is behind the green one and past the blue (as in the 2nd image).

The drawn vectors are my attempt to solve the mentioned problem: with the vectors defined as: $\vec{x}$ connecting blue to green, $\vec{y}$ connecting green to red, and $\vec{z}$ connecting blue to red, we can form the shown triangles, which relates them as $\vec{x} + \vec{y} = \vec{z}.$ And the triangle inequality tells us the norm of $z$ is smaller or equal to the sum of the two other norms. But I am getting stuck in how to use this triangular relation in order to distinguish between the two cases.

Question:

• Is there a simple geometric criterion, possibly based on this vectorial formulation, that for a given set of 3 points, allows us to distinguish between the 1st image and the 2nd? For simplicity we assume the blue point to be always the first point, but the relative positioning of the next two points can change.

Answer 1

There is no way to distinguish order strictly by the vector representations. You have to know something more about the curve to determine the order. For example, in your second example, you would get the exact same three vectors from a curve that headed more directly towards $x$, then turned tightly to head back towards $y$. But in that case the order of the points on the curve are "origin, $x$, $y$", instead of the "origin, $y$, $x$" in your example.

How else you handle this is going to depend on what you are trying to do, and whether there is anything more that you know about the curve. If you just have a collection of points, and are trying to guess the order in which they are traversed, then an "always go to the closest" rule (with some tie-breaker clause) is the best you can hope for. Either that or "minimize the sum of the angles over the whole path".