Trilateration when only combinations of distance are available
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0
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My problem setup is as shown below:
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
add a comment |
up vote
0
down vote
favorite
My problem setup is as shown below:
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
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 arep3+
,p3-
,p4+
,p4-
, andp5+
,p5-
whose positions are known.
– Asy
1 hour ago
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
My problem setup is as shown below:
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
My problem setup is as shown below:
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
geometry circle triangulation
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 arep3+
,p3-
,p4+
,p4-
, andp5+
,p5-
whose positions are known.
– Asy
1 hour ago
add a comment |
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 arep3+
,p3-
,p4+
,p4-
, andp5+
,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
add a comment |
1 Answer
1
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oldest
votes
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.
add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
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.
add a comment |
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.
add a comment |
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.
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.
edited 2 hours ago
answered 16 hours ago
Jean Marie
28k41848
28k41848
add a comment |
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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-
, andp5+
,p5-
whose positions are known.– Asy
1 hour ago