how to compute the intersection of two hyperbola branches
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I am trying to develop an idea to determine the location of a cell phone using OTDOA (obseved time differences of arrival). The base stations emit electromagnetic wave to the cell phone, the cell phone records the time differences(OTDOAs) of different base stations, for example, $T_1$ is the time difference of station $1$ to station $0$, $T_2$ is the time difference of station $2$ to station $0$.
Here's what we know:
$X_i,Y_i$ of $3$ to many base stations, where $X_0,Y_0$ is the reference station.
$T_i$ of OTDOA for station $i$ to reference station $0$.
$C$ is the speed of light or electromagnetic wave.
We only do the computation on a plane, no $Z$ needed.
Here's what we want:
$x,y$ of the cell phone.
This eventually comes to two (or more) branches of two (or more) hyperbolas, because for every $T_i$ we can draw a branch of a hyperbola.
How can I compute the $x,y$ for the cell phone with known information?
Thank you so much!
May be useful:
Intersection of two hyperbolas
Equation of one branch of a hyperbola in general position
hyperbolic-geometry applications
edited Aug 13 at 7:28
tarit goswami
505116
asked Aug 13 at 5:59
baisong
32
 |Â
show 3 more comments
up vote
0
down vote
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I am trying to develop an idea to determine the location of a cell phone using OTDOA (obseved time differences of arrival). The base stations emit electromagnetic wave to the cell phone, the cell phone records the time differences(OTDOAs) of different base stations, for example, $T_1$ is the time difference of station $1$ to station $0$, $T_2$ is the time difference of station $2$ to station $0$.
Here's what we know:
$X_i,Y_i$ of $3$ to many base stations, where $X_0,Y_0$ is the reference station.
$T_i$ of OTDOA for station $i$ to reference station $0$.
$C$ is the speed of light or electromagnetic wave.
We only do the computation on a plane, no $Z$ needed.
Here's what we want:
$x,y$ of the cell phone.
This eventually comes to two (or more) branches of two (or more) hyperbolas, because for every $T_i$ we can draw a branch of a hyperbola.
How can I compute the $x,y$ for the cell phone with known information?
Thank you so much!
May be useful:
Intersection of two hyperbolas
Equation of one branch of a hyperbola in general position
hyperbolic-geometry applications
edited Aug 13 at 7:28
tarit goswami
505116
asked Aug 13 at 5:59
baisong
32
Welcome to math.SE. You should use MathJax/LaTeX to format your post, it makes it much more readable, the help center has some links to get you started. You should include more of your own thoughts/work, for instance: why do we get a hyperbola for each $T_i$?
– Henrik
Aug 13 at 6:10
Why do you want to use herperbola's ? Otherwise, the problem is "quite" simple usinng cartesian coordinates..
– Claude Leibovici
Aug 13 at 6:18
Hi Henrik, will try to use MathJax/LaTaX next time.
– baisong
Aug 13 at 6:44
Hi Claude, Could you provide more details, it's a big headache for me. The reason I am using hyperbola is every Ti leads to a branch of a hyperbola, because the distance difference is constant = Ti * C, this leads to a branch of a hyperbola.
– baisong
Aug 13 at 6:47
1
@taritgoswami Thanks for the editing. According to the definition of hyperbola, we can see that it's a collection of points that have a constant distance difference between focus A and B (in my example they are reference station 0 and anther station i), i.e., distance(cell phone to station 0) - distance(cell phone to station i) = constant = 2a, where a is a parameter of the hyperbola equation.
– baisong
Aug 14 at 5:18
 |Â
show 3 more comments
up vote
0
down vote
favorite
up vote
0
down vote
favorite
I am trying to develop an idea to determine the location of a cell phone using OTDOA (obseved time differences of arrival). The base stations emit electromagnetic wave to the cell phone, the cell phone records the time differences(OTDOAs) of different base stations, for example, $T_1$ is the time difference of station $1$ to station $0$, $T_2$ is the time difference of station $2$ to station $0$.
Here's what we know:
$X_i,Y_i$ of $3$ to many base stations, where $X_0,Y_0$ is the reference station.
$T_i$ of OTDOA for station $i$ to reference station $0$.
$C$ is the speed of light or electromagnetic wave.
We only do the computation on a plane, no $Z$ needed.
Here's what we want:
$x,y$ of the cell phone.
This eventually comes to two (or more) branches of two (or more) hyperbolas, because for every $T_i$ we can draw a branch of a hyperbola.
How can I compute the $x,y$ for the cell phone with known information?
Thank you so much!
May be useful:
Intersection of two hyperbolas
Equation of one branch of a hyperbola in general position
hyperbolic-geometry applications
edited Aug 13 at 7:28
tarit goswami
505116
asked Aug 13 at 5:59
baisong
32
I am trying to develop an idea to determine the location of a cell phone using OTDOA (obseved time differences of arrival). The base stations emit electromagnetic wave to the cell phone, the cell phone records the time differences(OTDOAs) of different base stations, for example, $T_1$ is the time difference of station $1$ to station $0$, $T_2$ is the time difference of station $2$ to station $0$.
Here's what we know:
$X_i,Y_i$ of $3$ to many base stations, where $X_0,Y_0$ is the reference station.
$T_i$ of OTDOA for station $i$ to reference station $0$.
$C$ is the speed of light or electromagnetic wave.
We only do the computation on a plane, no $Z$ needed.
Here's what we want:
$x,y$ of the cell phone.
This eventually comes to two (or more) branches of two (or more) hyperbolas, because for every $T_i$ we can draw a branch of a hyperbola.
How can I compute the $x,y$ for the cell phone with known information?
Thank you so much!
May be useful:
Intersection of two hyperbolas
Equation of one branch of a hyperbola in general position
hyperbolic-geometry applications
edited Aug 13 at 7:28
tarit goswami
505116
asked Aug 13 at 5:59
baisong
32
edited Aug 13 at 7:28
tarit goswami
505116
edited Aug 13 at 7:28
tarit goswami
505116
edited Aug 13 at 7:28
tarit goswami
505116
505116
asked Aug 13 at 5:59
baisong
32
asked Aug 13 at 5:59
baisong
32
asked Aug 13 at 5:59
baisong
32
32
Welcome to math.SE. You should use MathJax/LaTeX to format your post, it makes it much more readable, the help center has some links to get you started. You should include more of your own thoughts/work, for instance: why do we get a hyperbola for each $T_i$?
– Henrik
Aug 13 at 6:10
Why do you want to use herperbola's ? Otherwise, the problem is "quite" simple usinng cartesian coordinates..
– Claude Leibovici
Aug 13 at 6:18
Hi Henrik, will try to use MathJax/LaTaX next time.
– baisong
Aug 13 at 6:44
Hi Claude, Could you provide more details, it's a big headache for me. The reason I am using hyperbola is every Ti leads to a branch of a hyperbola, because the distance difference is constant = Ti * C, this leads to a branch of a hyperbola.
– baisong
Aug 13 at 6:47
1
@taritgoswami Thanks for the editing. According to the definition of hyperbola, we can see that it's a collection of points that have a constant distance difference between focus A and B (in my example they are reference station 0 and anther station i), i.e., distance(cell phone to station 0) - distance(cell phone to station i) = constant = 2a, where a is a parameter of the hyperbola equation.
– baisong
Aug 14 at 5:18
 |Â
show 3 more comments
Welcome to math.SE. You should use MathJax/LaTeX to format your post, it makes it much more readable, the help center has some links to get you started. You should include more of your own thoughts/work, for instance: why do we get a hyperbola for each $T_i$?
– Henrik
Aug 13 at 6:10
Why do you want to use herperbola's ? Otherwise, the problem is "quite" simple usinng cartesian coordinates..
– Claude Leibovici
Aug 13 at 6:18
Hi Henrik, will try to use MathJax/LaTaX next time.
– baisong
Aug 13 at 6:44
Hi Claude, Could you provide more details, it's a big headache for me. The reason I am using hyperbola is every Ti leads to a branch of a hyperbola, because the distance difference is constant = Ti * C, this leads to a branch of a hyperbola.
– baisong
Aug 13 at 6:47
1
@taritgoswami Thanks for the editing. According to the definition of hyperbola, we can see that it's a collection of points that have a constant distance difference between focus A and B (in my example they are reference station 0 and anther station i), i.e., distance(cell phone to station 0) - distance(cell phone to station i) = constant = 2a, where a is a parameter of the hyperbola equation.
– baisong
Aug 14 at 5:18
Welcome to math.SE. You should use MathJax/LaTeX to format your post, it makes it much more readable, the help center has some links to get you started. You should include more of your own thoughts/work, for instance: why do we get a hyperbola for each $T_i$?
– Henrik
Aug 13 at 6:10
Welcome to math.SE. You should use MathJax/LaTeX to format your post, it makes it much more readable, the help center has some links to get you started. You should include more of your own thoughts/work, for instance: why do we get a hyperbola for each $T_i$?
– Henrik
Aug 13 at 6:10
Why do you want to use herperbola's ? Otherwise, the problem is "quite" simple usinng cartesian coordinates..
– Claude Leibovici
Aug 13 at 6:18
Why do you want to use herperbola's ? Otherwise, the problem is "quite" simple usinng cartesian coordinates..
– Claude Leibovici
Aug 13 at 6:18
Hi Henrik, will try to use MathJax/LaTaX next time.
– baisong
Aug 13 at 6:44
Hi Henrik, will try to use MathJax/LaTaX next time.
– baisong
Aug 13 at 6:44
Hi Claude, Could you provide more details, it's a big headache for me. The reason I am using hyperbola is every Ti leads to a branch of a hyperbola, because the distance difference is constant = Ti * C, this leads to a branch of a hyperbola.
– baisong
Aug 13 at 6:47
Hi Claude, Could you provide more details, it's a big headache for me. The reason I am using hyperbola is every Ti leads to a branch of a hyperbola, because the distance difference is constant = Ti * C, this leads to a branch of a hyperbola.
– baisong
Aug 13 at 6:47
1
1
@taritgoswami Thanks for the editing. According to the definition of hyperbola, we can see that it's a collection of points that have a constant distance difference between focus A and B (in my example they are reference station 0 and anther station i), i.e., distance(cell phone to station 0) - distance(cell phone to station i) = constant = 2a, where a is a parameter of the hyperbola equation.
– baisong
Aug 14 at 5:18
@taritgoswami Thanks for the editing. According to the definition of hyperbola, we can see that it's a collection of points that have a constant distance difference between focus A and B (in my example they are reference station 0 and anther station i), i.e., distance(cell phone to station 0) - distance(cell phone to station i) = constant = 2a, where a is a parameter of the hyperbola equation.
– baisong
Aug 14 at 5:18
 |Â
show 3 more comments
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Welcome to math.SE. You should use MathJax/LaTeX to format your post, it makes it much more readable, the help center has some links to get you started. You should include more of your own thoughts/work, for instance: why do we get a hyperbola for each $T_i$?
– Henrik
Aug 13 at 6:10
Why do you want to use herperbola's ? Otherwise, the problem is "quite" simple usinng cartesian coordinates..
– Claude Leibovici
Aug 13 at 6:18
Hi Henrik, will try to use MathJax/LaTaX next time.
– baisong
Aug 13 at 6:44
Hi Claude, Could you provide more details, it's a big headache for me. The reason I am using hyperbola is every Ti leads to a branch of a hyperbola, because the distance difference is constant = Ti * C, this leads to a branch of a hyperbola.
– baisong
Aug 13 at 6:47
1
@taritgoswami Thanks for the editing. According to the definition of hyperbola, we can see that it's a collection of points that have a constant distance difference between focus A and B (in my example they are reference station 0 and anther station i), i.e., distance(cell phone to station 0) - distance(cell phone to station i) = constant = 2a, where a is a parameter of the hyperbola equation.
– baisong
Aug 14 at 5:18