how can I write in a equation the intersection "x" of two arbitrary circles in terms of their ratios and centers?

Question

I tried with analytic geometry, but the intersection of two arbitrary circles with equations $(x-a)^2 + (y-b)^2 = r^2$ and $(x-c)^2 + (y-d)^2 = s^2$ respectively is the solution of this system with variables, $x,y$. but $x$ is very large, very big, big, and its impossible to work with this very big point(s). Are there a small very small equation to find $x$ in terms of the centers $(a,b), (c,d)$ and the ratios $r, s$ respectively? what is it?

How can I write in a equation for the intersection "$x$" of two arbitrary circles in terms of their ratios and centers?, which is the smallest form to write the intersection?

You can translate and rotate so that $(a,b)=(0,0)$ and $(c,d)=(c,0)$. Then, once you have the answer, invert to operation to find the original $x,y$. — Thomas Andrews, Oct 03 '16 at 19:58
Maybe you could write using dollar signs around math equations, and with proper capitalization and line breaks, etc. to improve readability. — Caleb Stanford, Oct 03 '16 at 20:01
What exactly do you mean by “too big?” The transformations Thomas Andrews suggest involve multiplying by sines and cosines (all with absolute value at most one) and adding and subtracting some constants. — amd, Oct 03 '16 at 20:25
i want to say that the intersection (x,y) of the two circles is very big and large because when i write it in term of a, b, c, d, r, s, the equation is very large — Sofi Lescas, Oct 03 '16 at 20:46

score 1 · Answer 1 · edited Apr 13 '17 at 12:19

As mentioned, the resultant equation will be “too big”.

That is why some of the textbooks will simplify your question a bit by letting the second circle be the standard one ($x^2 + y^2 = s^2$). This leaves us with four variables only, namely a, b, r, and s.

The above can be done by just say “without loss of generosity”, or can be achieved by translation of axes.

We can further cut the number of variables down by:-

(1) re-scale the whole picture by letting that circle be the unit circle so that s = 1; or/and

(2) place the center of the second circle on the x-axis; (i. e. b = 0).

If keeping all 6 variables is a must, the “big” formula can be found in http://2000clicks.com/MathHelp/GeometryConicSectionCircleIntersection.aspx or How can I find the points at which two circles intersect?

how can I write in a equation the intersection "x" of two arbitrary circles in terms of their ratios and centers?

1 Answers1