Re: 3quadratic eq and 3 unkowns

syms a b c X A B C x y z
a=[a1,a2,a3];
b=[b1,b2,b3];
c=[c1,c2,c3];
X=[x,y,z];
A^2=((X-a).*(X-a));
B^2=((X-b).*(X-b));
C^2=((X-c).*(X-c));
sols=Solve((A^2,B^2,C^2),X)
points=X./sols

this is how i am starting but i keep on getting errors stating that
the equations to the left is not valid. i am starting to get lost.

John D'Errico wrote:
>
>
> In article <ef11846.-1@xxxxxxxxxxxxxxxx>,
> "Hector G" <guerra@xxxxxxxxxxxx> wrote:
>
>> >I am looking for the trigonometric / algebreic formulae one
> would
>> use
>> >to triangulate an unknown position, knowing 3 fixed points
and
> the
>> >respective distances to the unknown point. (make sence?)
>> >Oh - in 3 dimensional space.
>>
>> I am surveying a very inregular object. Its the intersection of
3
>> spheres. Please help!!!!!! please
>
> I assume that you want to compute the points of
> intersection?
>
> Lets consider the solution for the intersection points
> of two circles. (x1,y1), (x2,y2), and r1, r2 are all
> given.
>
> (x-x1)^2 + (y-y1)^2 = r1^2
> (x-x2)^2 + (y-y2)^2 = r2^2
>
> The solution, if it exists, must satisfy both equations.
> Expand and subtract the pair. This is linear in the
> variables x and y. It defines a line in the x-y plane.
>
> -2*x*x1 + 2*x*x2 - 2*y*y1 + 2*y*y2 =
> r1^2 - r2^2 - x1^2 + y2^2
>
> We could now substitute back into either of the circle
> equations and solve. We will get either zero, one or
> two roots, since the result will be quadratic. The
> single root case implies the two circles are touching.
> (What happens if one circle is fully inside the other?)
>
> How can we use this same approach for the 3d case?
>
> [1] (x-x1)^2 + (y-y1)^2 + (z-z1)^2 = r1^2
> [2] (x-x2)^2 + (y-y2)^2 + (z-z2)^2 = r2^2
> [3] (x-x3)^2 + (y-y3)^2 + (z-z3)^2 = r3^2
>
> Subtract pairs of these equations. [1] - [2]
> and [2] - [3]. Each time we do so, we get a linear
> equation in the unknowns (x,y,z). Essentially each
> equation defines a plane in the 3-d space. Find
> the intersection of these two planes. If the
> intersection exists (i.e., the planes are not
> parallel) then it will define a line.
>
> Find the intersection of that line with the original
> spheres. Again, there should be zero, one or two
> solutions.
>
> A little algebra never hurt anybody. Try it and see.
>
> HTH,
> John D'Errico
>
>
> --
> The best material model of a cat is another, or
> preferably the same, cat.
> A. Rosenblueth, Philosophy of Science, 1945
>
.

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