Find a unit vector exactly halfway between two connected line segments

Question

Seems like a quick and easy question but I haven't been able to find exactly what I'm looking for so:

How to calculate a unit length vector that is pointing along the line which is exactly 50% of the angle of two connected line segments?

A picture speaks a thousands words (that are also better than my explanation!)
So basically I want to calculate the blue unit vector given the two red line segments (which are actually 3 points and are therefore guaranteed to be connected)

The red segments are of arbitrary length, and the result doesn't have to be a unit it would just be easier for me.
It would also be handy to have a way to force the resultant vector to point in a certain direction (relative to the input segments), this isn't essential as I think I can work this one out - as the input line segments ultimately form an n-gon.

Any examples would be ideal in C++ please, but other languages welcome.

Many thanks for any pointers.

Answer 1

Create and normalize two vectors from your red segment, starting from their comon vertex, then add the results (component by component). Then you can normalize the output if you want to get a unit vector.

The problem is that you'll always end up is the second case, because the angle between 2 vectors will always be less than 180°. But then of course you can just create the opposite vector and see which one suits better your purpose.

Also there's a special case when both vectors are aligned and the average will be 0 (but you can easily detect it).

Answer 2

I think you can get the direction to be consistent by treating this as if you're generating a 2D vertex normal. That is:

1. Take each of the red vectors, swap the x and y components and negate one of them to create the normals.

2. Normalize them.

3. Sum those two vectors, and renormalize.

You'll probably also want to test for the case where the two red lines are overlapping each other - the final renormalization will try to divide by zero there.

Answer 3

Let A and B be your vectors:

The summing vectors do not need to be unit vectors but simply equal in length so, if |A| >= |B|, you can:

which is more numerically stable since you have only a fraction and the bigger denominator

The same result can be obtained by substraction, once again the vectors has to be equal in length

This works only for non-convex angles; you can simply test if your angle is convex and multiply H by -1