Trilaterate the two points given with 3 points and 3 distances
I have 3 given 3 points in an 3d area with given distance to an searched object. I want to find the two possible positions of this object.
My first thougt was, to intersect two globes so that I get an intersection circle. After that, I wanted to iterate over the circle until I match my third distance. Problem is, that I couldnt calculate the plane, in which the circle lies. The only information I have is the middle point, a normal vector and the radius. With that information I can't calculate a random circle point. Please correct me, if I am wrong.
So my second approach was to use trilateration. For that i wanted to use the formulas given on Wolfram Alpha
I built a plane with my given 3 points. For that I wanted to translate my first point into the origin of the coordinate system, then I would like to rotate my plane, that all z-values are zero. Then i could use the given formulas to calculate two different points, which one of them is my searched object position. Problem here is, that I don't know, how to do the rotation, that one of my points lies in the origin, another one on the x-axis and the z-value is always zero.
Im open for every advice to locate the object with the given 3 points and the 3 distances. I know that their are 2 solutions for the location. Maybe their is an idea which I can't see.
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