Algorithm to find all points on a 2D grid some distance away from another point

Question

I have some point on a 2D grid (x, y) and I need to find all points that are n distance away from that point. The way I'm measuring distance is by using the distance formula between the two points. Anyone know how to do this?

Edit: Just for reference, what I'm trying to do is to write some AI path finding that will maintain some distance away from a target in a system that uses grid based locations. Currently I'm using A* path finding, but I'm not sure if that matters or makes a difference since I'm kind of new to this stuff.

Answer 1

Here's what I would do:

1. First filter out all points that are further than D on either x or y. These are certainly outside the circle of radius D. This is a much simpler computation, and it can quickly eliminate a lot of work. This is a outer bounding-box optimization.

2. You can also use an inner bounding-box optimization. If the points are closer than D * sqrt(2)/2 on either x or y, then they're certainly within the circle of radius D. This is also cheaper than calculating the distance formula.

3. Then you have a smaller number of candidate points that may be within the circle of radius D. For these, use the distance formula. Remember that if D = sqrt(Δx²+Δy²), then D² = Δx²+Δy².
   So you can skip the cost of calculating square root.

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So in pseudocode, you could do the following:

for each point
begin
    if test 1 indicates the point is outside the outer bounding box, 
    then skip this point

    if test 2 indicates the point is inside the inner bounding box, 
    then keep this point

    if test 3 indicates the point is inside the radius of the circle, 
    then keep this point
end