calculating the weights for direction vectors to a target

Question

I have an entity that is allowed to move in a fixed amount of directions. Let's call a the angle between two directions. r is the length of each direction vector. Now the entity wants to head from its current position (x1,y1) to a target (x2,y2) in one of the fixed directions. Instead of getting the direction that the entity should head into, I want to get a weight for all directions that is based on how close the entity gets to the target with that direction, so that I can combine those weights with other things like avoiding some direction for example.

I thought about calculating the distance from the new positions to the target and then use the closest distance (the one in which the entity heads in the direction directly to the target) to get a weighting from 1 to zero for the other directions.

Now I was wondering if there is some other way to get the weights without having to calculate the distance for each direction. Maybe there is some sort of relation between the angle from the ideal direction and the distance that could be used to calculate the weights instead? So far I've figured out that at least there is no no linear relation between the angle and the distance.

Here is an image to hopefully clear things up a bit:

Answer 1

What you want is the dot-product between each direction vector and the vector to the target. This value will be maximised by the direction which will take you closest to the target.

If you are not familiar with the dot product:

u . v = sum(u_i * v_i)

where the sum is over the components of the vectors.

Edit: Please do ensure that the vectors are normalised (i.e. have magnitude = 1) before conducting the dot product. You can normalise a vactor by dividing by its magnitude.

Answer 2

The distance between the centre of the circle (x1,y1) and the target (x2,y2) is:

D = sqrt((x1-x2)² + (y1-y2)²)

So the distance between the target and any point on the circle varies between D - r and D + r.

For a given angle a with the horizontal axis, the position of a point on the circle is:

x = x1 + r·cos(a)
y = y1 + r·sin(a)

And so the distance between (x,y) and (x2,y2) is:

d = sqrt((x1 + r·cos(a) - x2)² + (y1 + r·sin(a) - y2)²)

All these values can be combined together to compute a weight that only depends on the value of a and goes from 0 (worst choice for a) to 1 (best choice for a):

W = (D + r - d) / 2r

I hope this can be helpful.