Hi, I would need the algebraicly solved two equations for X and Y of the intersection points of two circles to write them as algorithms in a graphics language. Thanks

I understand the two circles to be:

(x-x1)^2+(y-y1)^2=r1^2 (x-x2)^2+(y-y2)^2=r2^2

to be multiplied out into: (1) x^2-2*x*x1+x1^2+y^2-2*y*y1+y1^2=r1^2 (2) x^2 -2*x*x2+x2^2+y^2-2*y*y2+y2^2=r2^2

subtract (2) from (1): (3) -2*x*x1+2*x*x2+x1^2-x2^2-2*y*y1+2*y*y2+y1^2+y2^2=r1^2-r2^2

I understand that (3) should be solved for x to make the equation (4) then x in (1) to be substituted with the result and (1) solved for y, then this result again plugged into (3) to again be solved for x?

anyway, I would need the two algebraic solutions, for x and for y

thank you Karl

[Dyuman Joshi]

Yes, you can write equation (3) by separating the variables (y on the lhs and x on the rhs, as you want to solve for x) and substituting it in either (1) or (2) to get the value(s) of x.

Then, substitute the value(s) of x in any of the equation to get the corresponding value(s) of y.

Answer 1

[Torsten]

syms x y x1 y1 x2 y2 r1 r2 real
eqn1 = (x-x1)^2+(y-y1)^2==r1^2;
eqn2 = (x-x2)^2+(y-y2)^2==r2^2;
sol = solve([eqn1,eqn2],[x y])

Warning: Solutions are only valid under certain conditions. To include parameters and conditions in the solution, specify the 'ReturnConditions' value as 'true'.

sol = struct with fields:
    x: [2×1 sym]
    y: [2×1 sym]

sol.x

ans = 

sol.y

ans =