Re: [Maxima-discuss] trying to solve for intersection of 2 circles

[Michel T. <ta...@lp...>]

Le 25/03/2022 à 18:33, Richard Fateman a écrit :
> Solving for y in one equation, substituting in the other eq, and 
> solving for x..
> I got a number of solutions.  Just picking one for x and the 
> corresponding y..
> here we have (one) solution for the two equations.
>
>  [x=-(sqrt((-b^2+a^2+2)*sqrt(-b^2+2*a*b-a^2+2)*sqrt(b^2+2*a*b+a^2-2)+4*a^2)-2^(3/2))/2^(3/2),y=-sqrt(sqrt(-b^2+2*a*b-a^2+2)*(b^2-a^2-2)*sqrt(b^2+2*a*b+a^2-2)+4*a^2)/2^(3/2)
>
> There are nested sqrts in there. Does algsys give up too easily? 
> to_poly_solve seemed to get a better form.
> RJF

I think that what you are doing here is the algorithm that solve uses. 
At least it is documented that way. Choosing

the "simplest" equation, here they are the same, and substituting in the 
other ones, etc. Then solve tries to check

that the solutions are correct, and it is perhaps here that things go 
awry because computing with nested radicals

is not a strong point of CAS systems and notably maxima.  As Barton said 
the correct procedure is to try to triangularize

the system (for example using Grobner). In the present case it is 
particularly trivial, since both equations begin by

x^2+y^2, so a simple subtraction leaves a linear equation. It is then 
easy to solve and substitute in one of the circle equations,

leaving a second order equation and not a fourth degree equation. So no 
nested radicals, and 2 points of intersection.

Using maxima:


(%o1) false
(%i2) e1:(x-1)^2+y^2-a^2;

(%o2) y^2+(x-1)^2-a^2
(%i3) e2:x^2+(y-1)^2-b^2;

(%o3) (y-1)^2+x^2-b^2
(%i4) solve([e1,expand(e1-e2)],[x,y]);

(%o4) [[x = -(sqrt((-b^4)+(2*a^2+4)*b^2-a^4+4*a^2-4)-b^2+a^2-2)/4,
         y = -(sqrt((-b^4)+(2*a^2+4)*b^2-a^4+4*a^2-4)+b^2-a^2-2)/4],
        [x = (sqrt((-b^4)+(2*a^2+4)*b^2-a^4+4*a^2-4)+b^2-a^2+2)/4,
         y = (sqrt((-b^4)+(2*a^2+4)*b^2-a^4+4*a^2-4)-b^2+a^2+2)/4]]



-- 
Michel Talon