Finding all points common to two circles

Question

In Python, how would one find all integer points common to two circles?

For example, imagine a Venn diagram-like intersection of two (equally sized) circles, with center-points (x1,y1) and (x2,y2) and radii r1=r2. Additionally, we already know the two points of intersection of the circles are (xi1,yi1) and (xi2,yi2).

How would one generate a list of all points (x,y) contained in both circles in an efficient manner? That is, it would be simple to draw a box containing the intersections and iterate through it, checking if a given point is within both circles, but is there a better way?

Answer 1

Keep in mind that there are four cases here.

1. Neither circle intersects, meaning the "common area" is empty.
2. One circle resides entirely within the other, meaning the "common area" is the smaller/interior circle. Also note that a degenerate case of this is if they are the same concentric circle, which would have to be the case given the criteria that they are equal-diameter circles that you specified.
3. The two circles touch at one intersection point.
4. The "general" case where there are going to be two intersection points. From there, you have two arcs that define the enclosed area. In that case, the box-drawing method could work for now, I'm not sure there's a more efficient method for determining what is contained by the intersection. Do note, however, if you're just interested in the area, there is a formula for that.