I'm trying to find a real world coordinate where 3 spheres collide and interact. At the moment I have been able to set up my triangulation equations so that I can work out the 2D position of where my 3 spheres collide but I'm stuck trying to find the height of where they collide. What would the equation for this be?
Following the rather obtuse comment I'll provide images to try and clear things up.
The picture above represents my triangulation of 3 intersecting spheres in 2D space. This is the known. What I'm trying to do is find is now find the height in which all 3 intersect. The below picture is my best attempt at re-creating this.
There are no known variables in this equation. Given that I have already worked out the 2D triangulation of my intersecting points, what do I need to do so I can work out the 3D coordinates of my spheres?
You have three equations to solve (assuming you know the centre and radius of each sphere).
$(x-a)^2+(y-b)^2+(z-c)^2=r^2\\(x-A)^2+(y-B)^2+(z-C)^2=R^2\\(x-\alpha)^2+(y-\beta)^2+(z-\gamma)^2=\rho^2$
By subtracting one equation from the other two, you get two linear equations in $x,y,z$. The solution to that is generally a straight line, that can be parametrized by a single variable $t$. Put that solution back into any of the three quadratic equations, and you get one quadratic equation in $t$.