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

first circle: (x-k)^2 + (y-C)^2 = r^2

second circle: (x-a)^2 + (y-b)^2 = R^2

solves 2nd circle for x:

x = sqrt( R^2 - (y-b)^2) + a

substitutes into first circle equation:

[sqrt( R^2 - (y-b)^2) + (a-k)]^2 + (y-C)^2 = r^2

FOILS left side

(R^2 - (y-b)^2) + 2 * (a-k)*sqrt(R^2 - (y-b)^2 + (y-b)^4) + (y-C)^2 = r^2

(R^2 - (y-b)^2) + 2 * (a-k)*sqrt(R^2 - (y-b)^2 + (y-b)^4) + (y-C)^2 - r^2 = 0

You can now do newton's method on this equation