An Equation to determine the (x,y,z) coordinates of a point

Max_Light

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Jan 22, 2013
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Right now I'm working on an Equation to determine the (x,y,z) coordinates of a point give that the distance away from (a,b,c) is j, the distance away from (d,e,f) is k and the distance away from (g,h,i) is l.
The equations I have so far are
1: j=(((x-a)^2)+((y-b)^2)+((z-c)^2))^1/2
2: k=(((x-d)^2)+((y-e)^2)+((z-f)^2))^1/2
3: l=(((x-g)^2)+((y-h)^2)+((z-i)^2))^1/2
and it's getting too complicated to solve.

I started out using Maple to solve equation 1 in terms of x, which I then substituted into 2 and told it to solve for y. (I used the positive roots, I'm not sure if that made much of a difference). When I tried to sub those x and y into equation 3, Maple kept calculating for almost 25 minutes, which is when I quite the program. Does anyone know how to solve this?
What I'm trying to do is create an equation that you can sub j,k and l into, with (a,b,c),(d,e,f), and (g,h,i) staying constant, which will give you the position (x,y,z).
 
Right now I'm working on an Equation to determine the (x,y,z) coordinates of a point give that the distance away from (a,b,c) is j, the distance away from (d,e,f) is k and the distance away from (g,h,i) is l.
The equations I have so far are
1: j=(((x-a)^2)+((y-b)^2)+((z-c)^2))^1/2
2: k=(((x-d)^2)+((y-e)^2)+((z-f)^2))^1/2
3: l=(((x-g)^2)+((y-h)^2)+((z-i)^2))^1/2
and it's getting too complicated to solve.

I started out using Maple to solve equation 1 in terms of x, which I then substituted into 2 and told it to solve for y. (I used the positive roots, I'm not sure if that made much of a difference). When I tried to sub those x and y into equation 3, Maple kept calculating for almost 25 minutes, which is when I quite the program. Does anyone know how to solve this?
What I'm trying to do is create an equation that you can sub j,k and l into, with (a,b,c),(d,e,f), and (g,h,i) staying constant, which will give you the position (x,y,z).

This is classic problem of triangulation - principle which drives the GPS system.

Solve the 2-D problem first - i.e. work with (x,y),(a,b),(d,e) and lengths of j & k.

That will give you some idea about the process to extend this to 3-D (which is several magnitude more complicated).

Think how would you do this geometrically (2-D problem) - with compass and ruler?
 
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