[Math] Finding the intersection points of two circles – where one circle has a diff x and y coord than the other

circlesgeometry

Following this link:
http://mathworld.wolfram.com/Circle-CircleIntersection.html

I now understand how to calculate the offset of the radical line from circle_a (a)

However:

Let two circles of radii  and  and centered at   and  intersect in a region shaped like an asymmetric lens. The equations of the two circles are

x^2 + y^2 = R^2
(x-d)^2 + y^2 = R^2

So this methods assumes that the y coordinates of both circles are the same?

How do I calculate the intersection points where both the x and y coordinates are not the same then?

Thanks.

Best Answer

You can always assume that the y-coordinates of both circles are the same because you can rotate both circles in a way that they lie on a line that are parallel to the x-axis. Then you can translate both circles so the first one is in the origin.

Using that rotation the $d$ in your formula will be the distance of both centers and your formulas will look a lot better. To undo this you have to undo the translation and then the rotation. Either you use rotation matrices or you use a complete solution like for example this one.

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