I want to be able to localice the position of a number of elements hiden in a 3d space. I have a tool that give me the distance to every one from the point I am standing but since I cant get my own coordinates I cant use my own position as a known value.
I want to be able to create a map of where are those elements A,B,C and D
So I know where are neither the 4 points nor the places from with the distances are captured.
So There are A,B,C,and D points
and I have all the measurements I want 1,2,3,4,5...
for every measurement I know the distance from every point to my current position(unknow coordinates) You can know that what is the A point distance to every measurement for instance.
A1,B1,C1,D1,
A2,B2,C2,D2,
A3,B3,C3,D3,
A4,B4,C4,D4,
A5,B5,C5,D5,
...
So I want to know the distance of the points A,B,C and D to each other. Is that Posible?
As an extra help I could fix the distance from one measurment to the next (by doing periodic measurements while moving but I dont know my speed , but not the direction unless it is straight) If this help.
To simplify the problem consider everything to be in a 2 dimension world. No other restriction
Thanks
Consider for instance the first measurement you can take $$ \left( {P_{1,\,x} - A_{\,x} } \right)^{\,2} + \left( {P_{1,\,y} - A_{\,y} } \right)^{\,2} + \left( {P_{1,\,z} - A_{\,z} } \right)^{\,2} = A_{\,1} ^{\,2} $$ you have $1$ equation in $6$ unknowns: forget for a moment that they are quadratic.
When you add the measurement taken from another point $P_2$ to $A$ you are adding $1$ equation and $3$ more unknowns.
Adding a measument from the same point $P_1$ to $B$ $$ \left( {P_{1,\,x} - B_{\,x} } \right)^{\,2} + \left( {P_{1,\,y} - B_{\,y} } \right)^{\,2} + \left( {P_{1,\,z} - B_{\,z} } \right)^{\,2} = B_{\,1} ^{\,2} $$ you add again $1$ equation and $3$ more unknowns.
Suppose you know that the $P_n$ are along a straight line. Then from $P_3$ onwards, the measurements wrt the same point $A$ will keep adding $1$ equation and $1$ unknown more :the position of $P_n$ along the line through $P_1,\,P_2$.
So, in any case, there will be more variables than equations.