navigation math

   Anyone know the method for calculating a reciever's position from the time 
difference between three rf pulse transmiters of known positions? This has 
apparantly been in use since the second world war but a description of the 
mathematics involved is hiding. Maybe a text on navagation methods?

Hul

On 11/19/21 2:28 PM, Hul Tytus wrote:
>     Anyone know the method for calculating a reciever's position from the time
> difference between three rf pulse transmiters of known positions? This has
> apparantly been in use since the second world war but a description of the
> mathematics involved is hiding. Maybe a text on navagation methods?
> 
> Hul
> 
The search term is LORAN. I think the last ones were decommissioned 
years ago - replaced by GPS.

On 2021-11-19, Hul Tytus <ht@panix.com> wrote:

> Anyone know the method for calculating a reciever's position from the time 
> difference between three rf pulse transmiters of known positions?

Yes. If you know the delta between the distances to two known
locations, that places you on a hyperbola whose focii are those two
known points. Plot the hyperbola on your map.

Repat for the other two pairs of points. Hopefully there's one point where all three
intersect.


> This has apparantly been in use since the second world war but a
> description of the mathematics involved is hiding. Maybe a text on
> navagation methods?

https://www.sparknotes.com/math/precalc/conicsections/section4/

You can do it analytically instead of graphically if you want:

https://www.analyzemath.com/HyperbolaProblems/hyperbola_intersection.html
https://math.stackexchange.com/questions/1920147/intersection-of-two-hyperbolas

On 2021-11-19, Grant Edwards <invalid@invalid.invalid> wrote:
> On 2021-11-19, Hul Tytus <ht@panix.com> wrote:
>
>> Anyone know the method for calculating a reciever's position from the time 
>> difference between three rf pulse transmiters of known positions?
>
> Yes. If you know the delta between the distances to two known
> locations, that places you on a hyperbola whose focii are those two
> known points. Plot the hyperbola on your map.

Actually coastal navigational maps had/have those LORAN hyperbolas
printed on them. So all you really had to do was read the delta value
off the receiver, and then interpolate between two lines with
dividers... rinse, repeat.

https://www.penobscotmarinemuseum.org/pbho-1/collection/loran-lines-penobscot-bay-chart

Modern LORAN receivers know where the transmitters are, do all the
math internally, and just show you longitude/lattitude or display a
map just like a GPS receiver.

--
Grant

On 11/19/2021 1:28 PM, Hul Tytus wrote:
>     Anyone know the method for calculating a reciever's position from the time
> difference between three rf pulse transmiters of known positions? This has
> apparantly been in use since the second world war but a description of the
> mathematics involved is hiding. Maybe a text on navagation methods?

You will be dealing with families of "concentric" hyperbolae
(as the equation for a hyperbola involves maintaining a constant
difference between lengths of vectors to foci).

LORAN was renowned for using this -- on a global scale.  It has
since been decommissioned in the wild but there's an abundance
of information regarding its use and deployment.

Note, however, that there are many subtleties buried in the
LORAN implementation that make it differ from a theoretical
approach.  E.g., there are intentional delays introduced
to make the numbers cleaner.

If you are truly looking to navigate on a *large* scale
(hundreds of miles), then you will have to consider things
like changes in propagation delays over different types
of terrain and the "shape" of that terrain (e.g., the Earth
is an oblate sphere).  Again, LORAN has these covered but
you'll have to dig for details.

Similar problems exist "in the small" for position
resolution within a structure!  (I use similar technology
to determine where, in an "arena" -- home or office, in
my case -- the user is sited)

[If you look at a preprinted maritime map augmented with
LORAN "lines of constant time difference", you'd see
that they differ from what you would otherwise expect
from a more naive mathematical/geometric treatment]

On 11/19/2021 3:54 PM, Grant Edwards wrote:
> On 2021-11-19, Hul Tytus <ht@panix.com> wrote:
> 
>> Anyone know the method for calculating a reciever's position from the time
>> difference between three rf pulse transmiters of known positions?
> 
> Yes. If you know the delta between the distances to two known
> locations, that places you on a hyperbola whose focii are those two
> known points. Plot the hyperbola on your map.
> 
> Repat for the other two pairs of points. Hopefully there's one point where all three
> intersect.

There may be *two* places where the hyperbolae intersect.
If you understand the geometry of the pairs of foci, you
may be able to rule out one case as "not possible" (in
your application).


> 
>> This has apparantly been in use since the second world war but a
>> description of the mathematics involved is hiding. Maybe a text on
>> navagation methods?
> 
> https://www.sparknotes.com/math/precalc/conicsections/section4/
> 
> You can do it analytically instead of graphically if you want:
> 
> https://www.analyzemath.com/HyperbolaProblems/hyperbola_intersection.html
> https://math.stackexchange.com/questions/1920147/intersection-of-two-hyperbolas
> 
> 
> 

Hul Tytus <ht@panix.com> writes:
>    Anyone know the method for calculating a reciever's position from
> the time difference between three rf pulse transmiters of known
> positions?

See if this helps:

https://en.wikipedia.org/wiki/True-range_multilateration

On Fri, 19 Nov 2021 20:28:37 -0000 (UTC), Hul Tytus <ht@panix.com>
wrote:

>Anyone know the method for calculating a reciever's position from the time 
>difference between three rf pulse transmiters of known positions?

The word to look for is multilateration. Wikipedia has an article on
it:

https://en.wikipedia.org/wiki/Multilateration

Calculating the time differences based on a known position is
relatively simple. Once you try to solve the equations to go the other
way, the math gets ugly.

Years ago, when I was playing with this, I resorted to a numerical
solution.
-- 
RoRo

Thanks Dennis. That points me in the right direction.

Hul

Dennis <dennis@none.none> wrote:
> On 11/19/21 2:28 PM, Hul Tytus wrote:
> >     Anyone know the method for calculating a reciever's position from the time
> > difference between three rf pulse transmiters of known positions? This has
> > apparantly been in use since the second world war but a description of the
> > mathematics involved is hiding. Maybe a text on navagation methods?
> > 
> > Hul
> > 
> The search term is LORAN. I think the last ones were decommissioned 
> years ago - replaced by GPS.

Thanks Grant - that's whats needed.

Hul

Grant Edwards <invalid@invalid.invalid> wrote:
> On 2021-11-19, Hul Tytus <ht@panix.com> wrote:

> > Anyone know the method for calculating a reciever's position from the time 
> > difference between three rf pulse transmiters of known positions?

> Yes. If you know the delta between the distances to two known
> locations, that places you on a hyperbola whose focii are those two
> known points. Plot the hyperbola on your map.

> Repat for the other two pairs of points. Hopefully there's one point where all three
> intersect.


> > This has apparantly been in use since the second world war but a
> > description of the mathematics involved is hiding. Maybe a text on
> > navagation methods?

> https://www.sparknotes.com/math/precalc/conicsections/section4/

> You can do it analytically instead of graphically if you want:

> https://www.analyzemath.com/HyperbolaProblems/hyperbola_intersection.html
> https://math.stackexchange.com/questions/1920147/intersection-of-two-hyperbolas