Trilateration when only combinations of distance are available











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My problem setup is as shown below:



enter image description here



I know the location (x,y) of fixed points p1+, p1-, p2+, and p2-, and want to find the position marked "o" by trilateration. However, I do not know the individual distances r1+, r1-, r2+ and r2-. Rather, I have the following system of equations available:



$begin{align}
frac{1}{r_{1+}}-frac{1}{r_{1-}}&=V_1\
frac{1}{r_{2+}}-frac{1}{r_{2-}}&=V_2\
frac{1}{r_{3+}}-frac{1}{r_{3-}}&=V_3\
frac{1}{r_{4+}}-frac{1}{r_{4-}}&=V_4\
frac{1}{r_{5+}}-frac{1}{r_{5-}}&=V_5
end{align}$



I couldn't figure out an analytical solution to this system of equations, even though there are really only 3 unknowns -- (x,y) of the the point marked o, and thought the solution needs to be found numerically...



Thus my question is, is my conclusion correct in that analytical solution isn't possible?










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  • What are $r5+$ and $r5-$ ? Are they ascribed to a particular length visible in this figure ?
    – Jean Marie
    2 hours ago










  • Yes, I didn't mention that there may be more than two pairs of known points on the circle -- i.e. there are p3+,p3-, p4+,p4-, and p5+, p5- whose positions are known.
    – Asy
    1 hour ago















up vote
0
down vote

favorite












My problem setup is as shown below:



enter image description here



I know the location (x,y) of fixed points p1+, p1-, p2+, and p2-, and want to find the position marked "o" by trilateration. However, I do not know the individual distances r1+, r1-, r2+ and r2-. Rather, I have the following system of equations available:



$begin{align}
frac{1}{r_{1+}}-frac{1}{r_{1-}}&=V_1\
frac{1}{r_{2+}}-frac{1}{r_{2-}}&=V_2\
frac{1}{r_{3+}}-frac{1}{r_{3-}}&=V_3\
frac{1}{r_{4+}}-frac{1}{r_{4-}}&=V_4\
frac{1}{r_{5+}}-frac{1}{r_{5-}}&=V_5
end{align}$



I couldn't figure out an analytical solution to this system of equations, even though there are really only 3 unknowns -- (x,y) of the the point marked o, and thought the solution needs to be found numerically...



Thus my question is, is my conclusion correct in that analytical solution isn't possible?










share|cite|improve this question
























  • What are $r5+$ and $r5-$ ? Are they ascribed to a particular length visible in this figure ?
    – Jean Marie
    2 hours ago










  • Yes, I didn't mention that there may be more than two pairs of known points on the circle -- i.e. there are p3+,p3-, p4+,p4-, and p5+, p5- whose positions are known.
    – Asy
    1 hour ago













up vote
0
down vote

favorite









up vote
0
down vote

favorite











My problem setup is as shown below:



enter image description here



I know the location (x,y) of fixed points p1+, p1-, p2+, and p2-, and want to find the position marked "o" by trilateration. However, I do not know the individual distances r1+, r1-, r2+ and r2-. Rather, I have the following system of equations available:



$begin{align}
frac{1}{r_{1+}}-frac{1}{r_{1-}}&=V_1\
frac{1}{r_{2+}}-frac{1}{r_{2-}}&=V_2\
frac{1}{r_{3+}}-frac{1}{r_{3-}}&=V_3\
frac{1}{r_{4+}}-frac{1}{r_{4-}}&=V_4\
frac{1}{r_{5+}}-frac{1}{r_{5-}}&=V_5
end{align}$



I couldn't figure out an analytical solution to this system of equations, even though there are really only 3 unknowns -- (x,y) of the the point marked o, and thought the solution needs to be found numerically...



Thus my question is, is my conclusion correct in that analytical solution isn't possible?










share|cite|improve this question















My problem setup is as shown below:



enter image description here



I know the location (x,y) of fixed points p1+, p1-, p2+, and p2-, and want to find the position marked "o" by trilateration. However, I do not know the individual distances r1+, r1-, r2+ and r2-. Rather, I have the following system of equations available:



$begin{align}
frac{1}{r_{1+}}-frac{1}{r_{1-}}&=V_1\
frac{1}{r_{2+}}-frac{1}{r_{2-}}&=V_2\
frac{1}{r_{3+}}-frac{1}{r_{3-}}&=V_3\
frac{1}{r_{4+}}-frac{1}{r_{4-}}&=V_4\
frac{1}{r_{5+}}-frac{1}{r_{5-}}&=V_5
end{align}$



I couldn't figure out an analytical solution to this system of equations, even though there are really only 3 unknowns -- (x,y) of the the point marked o, and thought the solution needs to be found numerically...



Thus my question is, is my conclusion correct in that analytical solution isn't possible?







geometry circle triangulation






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edited 16 hours ago

























asked 17 hours ago









Asy

133




133












  • What are $r5+$ and $r5-$ ? Are they ascribed to a particular length visible in this figure ?
    – Jean Marie
    2 hours ago










  • Yes, I didn't mention that there may be more than two pairs of known points on the circle -- i.e. there are p3+,p3-, p4+,p4-, and p5+, p5- whose positions are known.
    – Asy
    1 hour ago


















  • What are $r5+$ and $r5-$ ? Are they ascribed to a particular length visible in this figure ?
    – Jean Marie
    2 hours ago










  • Yes, I didn't mention that there may be more than two pairs of known points on the circle -- i.e. there are p3+,p3-, p4+,p4-, and p5+, p5- whose positions are known.
    – Asy
    1 hour ago
















What are $r5+$ and $r5-$ ? Are they ascribed to a particular length visible in this figure ?
– Jean Marie
2 hours ago




What are $r5+$ and $r5-$ ? Are they ascribed to a particular length visible in this figure ?
– Jean Marie
2 hours ago












Yes, I didn't mention that there may be more than two pairs of known points on the circle -- i.e. there are p3+,p3-, p4+,p4-, and p5+, p5- whose positions are known.
– Asy
1 hour ago




Yes, I didn't mention that there may be more than two pairs of known points on the circle -- i.e. there are p3+,p3-, p4+,p4-, and p5+, p5- whose positions are known.
– Asy
1 hour ago










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accepted










I realize now that I haven't taken into account the fact that the values $r1+, r1-$, etc. are only known through the system of equations you give. But nevertheless, I am going to show that even if we know these four values, we are practically unable to find an exact solution.



I recall first that an ellipse with focii $F$ and $F'$ is the locus of points $M$ such that the sum of distances $MF+MF'$ is a constant $k$.



Here, point $O$ belongs in particular to




  • the ellipse with focii $p1-$ and $p2-$ and constant $k=r1-+r2-$, and to


  • the ellipse with focii $p1+$ and $p2+$ and constant $k=r1++r2+$.



Otherwise said, point $O$ is an intersection point of these two ellipses. Knowing that an ellipse has a second degree equation, the coordinates of $O$ involve a $2^2=4$th degree equation (a rapid check : indeed, two ellipses can have up to 4 intersection points).



Thus, theoretically, you could get an (awfully complicated) analytical answer, but practically speaking, you have to look for a numerical method.



Remarks :



1) One could have chosen other ellipses such as the one with focii $p2+$ and $p1-$. There is no guarantee that the $binom{4}{2}=6$ ellipses and the $binom{6}{2}=15$ intersection points (at least) you can get in this way will intersect in a very same point due to measuring errors. Therefore, the numerical method you will use needs a "least squares" step.



2) one could have used hyperbolas instead of ellipses.



3) the fact that the 4 points $p1+, ...$ belong to a same circle hasn't been used.






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    I realize now that I haven't taken into account the fact that the values $r1+, r1-$, etc. are only known through the system of equations you give. But nevertheless, I am going to show that even if we know these four values, we are practically unable to find an exact solution.



    I recall first that an ellipse with focii $F$ and $F'$ is the locus of points $M$ such that the sum of distances $MF+MF'$ is a constant $k$.



    Here, point $O$ belongs in particular to




    • the ellipse with focii $p1-$ and $p2-$ and constant $k=r1-+r2-$, and to


    • the ellipse with focii $p1+$ and $p2+$ and constant $k=r1++r2+$.



    Otherwise said, point $O$ is an intersection point of these two ellipses. Knowing that an ellipse has a second degree equation, the coordinates of $O$ involve a $2^2=4$th degree equation (a rapid check : indeed, two ellipses can have up to 4 intersection points).



    Thus, theoretically, you could get an (awfully complicated) analytical answer, but practically speaking, you have to look for a numerical method.



    Remarks :



    1) One could have chosen other ellipses such as the one with focii $p2+$ and $p1-$. There is no guarantee that the $binom{4}{2}=6$ ellipses and the $binom{6}{2}=15$ intersection points (at least) you can get in this way will intersect in a very same point due to measuring errors. Therefore, the numerical method you will use needs a "least squares" step.



    2) one could have used hyperbolas instead of ellipses.



    3) the fact that the 4 points $p1+, ...$ belong to a same circle hasn't been used.






    share|cite|improve this answer



























      up vote
      0
      down vote



      accepted










      I realize now that I haven't taken into account the fact that the values $r1+, r1-$, etc. are only known through the system of equations you give. But nevertheless, I am going to show that even if we know these four values, we are practically unable to find an exact solution.



      I recall first that an ellipse with focii $F$ and $F'$ is the locus of points $M$ such that the sum of distances $MF+MF'$ is a constant $k$.



      Here, point $O$ belongs in particular to




      • the ellipse with focii $p1-$ and $p2-$ and constant $k=r1-+r2-$, and to


      • the ellipse with focii $p1+$ and $p2+$ and constant $k=r1++r2+$.



      Otherwise said, point $O$ is an intersection point of these two ellipses. Knowing that an ellipse has a second degree equation, the coordinates of $O$ involve a $2^2=4$th degree equation (a rapid check : indeed, two ellipses can have up to 4 intersection points).



      Thus, theoretically, you could get an (awfully complicated) analytical answer, but practically speaking, you have to look for a numerical method.



      Remarks :



      1) One could have chosen other ellipses such as the one with focii $p2+$ and $p1-$. There is no guarantee that the $binom{4}{2}=6$ ellipses and the $binom{6}{2}=15$ intersection points (at least) you can get in this way will intersect in a very same point due to measuring errors. Therefore, the numerical method you will use needs a "least squares" step.



      2) one could have used hyperbolas instead of ellipses.



      3) the fact that the 4 points $p1+, ...$ belong to a same circle hasn't been used.






      share|cite|improve this answer

























        up vote
        0
        down vote



        accepted







        up vote
        0
        down vote



        accepted






        I realize now that I haven't taken into account the fact that the values $r1+, r1-$, etc. are only known through the system of equations you give. But nevertheless, I am going to show that even if we know these four values, we are practically unable to find an exact solution.



        I recall first that an ellipse with focii $F$ and $F'$ is the locus of points $M$ such that the sum of distances $MF+MF'$ is a constant $k$.



        Here, point $O$ belongs in particular to




        • the ellipse with focii $p1-$ and $p2-$ and constant $k=r1-+r2-$, and to


        • the ellipse with focii $p1+$ and $p2+$ and constant $k=r1++r2+$.



        Otherwise said, point $O$ is an intersection point of these two ellipses. Knowing that an ellipse has a second degree equation, the coordinates of $O$ involve a $2^2=4$th degree equation (a rapid check : indeed, two ellipses can have up to 4 intersection points).



        Thus, theoretically, you could get an (awfully complicated) analytical answer, but practically speaking, you have to look for a numerical method.



        Remarks :



        1) One could have chosen other ellipses such as the one with focii $p2+$ and $p1-$. There is no guarantee that the $binom{4}{2}=6$ ellipses and the $binom{6}{2}=15$ intersection points (at least) you can get in this way will intersect in a very same point due to measuring errors. Therefore, the numerical method you will use needs a "least squares" step.



        2) one could have used hyperbolas instead of ellipses.



        3) the fact that the 4 points $p1+, ...$ belong to a same circle hasn't been used.






        share|cite|improve this answer














        I realize now that I haven't taken into account the fact that the values $r1+, r1-$, etc. are only known through the system of equations you give. But nevertheless, I am going to show that even if we know these four values, we are practically unable to find an exact solution.



        I recall first that an ellipse with focii $F$ and $F'$ is the locus of points $M$ such that the sum of distances $MF+MF'$ is a constant $k$.



        Here, point $O$ belongs in particular to




        • the ellipse with focii $p1-$ and $p2-$ and constant $k=r1-+r2-$, and to


        • the ellipse with focii $p1+$ and $p2+$ and constant $k=r1++r2+$.



        Otherwise said, point $O$ is an intersection point of these two ellipses. Knowing that an ellipse has a second degree equation, the coordinates of $O$ involve a $2^2=4$th degree equation (a rapid check : indeed, two ellipses can have up to 4 intersection points).



        Thus, theoretically, you could get an (awfully complicated) analytical answer, but practically speaking, you have to look for a numerical method.



        Remarks :



        1) One could have chosen other ellipses such as the one with focii $p2+$ and $p1-$. There is no guarantee that the $binom{4}{2}=6$ ellipses and the $binom{6}{2}=15$ intersection points (at least) you can get in this way will intersect in a very same point due to measuring errors. Therefore, the numerical method you will use needs a "least squares" step.



        2) one could have used hyperbolas instead of ellipses.



        3) the fact that the 4 points $p1+, ...$ belong to a same circle hasn't been used.







        share|cite|improve this answer














        share|cite|improve this answer



        share|cite|improve this answer








        edited 2 hours ago

























        answered 16 hours ago









        Jean Marie

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