Given the radius and x,y coordinates of the center point of two circles how can I calculate their points of intersection if they have any?
This can be done without any trigonometry at all. Let the equations of the circles be
By subtracting (2) from (1) and then expanding, we in fact obtain a linear equation for x and y; after a little rearranging it becomes
(If the circles intersect, this is the equation of the line that passes through the intersection points.) This equation can be solved for one of x or y; let’s suppose y1−y2≠0 so that we can solve for y:
Substituting this expression for y into (1) or (2) gives a quadratic equation in only x. Then the x-coordinates of the intersection points are the solutions to this; the y-coordinates can be obtained by plugging the x-coordinates into (3).