Function Repository Resource:

WorldTravellerSignpostSolve

Find the location of a signpost with given distances to a set of cities

Contributed by: Sander Huisman

ResourceFunction["WorldTravellerSignpostSolve"][{{loc₁, dist₁},{loc₂, dist₂},…}]

given the locations loc_i and distances dist_i find the location of the signpost.

Details

The algorithm finds the location p for which the sum of distance errors (distance from p to loc_i minus dist_i) squared is minimized: ∑i=1n(||p-loc_i||-dist_i)² All locations loc_i are included in the calculation. The error between the distance from p to loc_i and dist_i can be anything such that even crude values will give a satisfactory answer. In general the crudeness/precision of the distances given gives the precision of the found location.

Examples

Basic Examples (1)

Given a set of cities and their distance, what is the location of the signpost:

In[1]:=

locs = {
{Entity["City", {"LosAngeles", "California", "UnitedStates"}], Quantity[9060, "Kilometers"]},
{Entity["City", {"NewYork", "NewYork", "UnitedStates"}], Quantity[6013, "Kilometers"]},
{Entity["City", {"Berlin", "Berlin", "Germany"}], Quantity[431, "Kilometers"]},
{Entity["City", {"Oslo", "Oslo", "Norway"}], Quantity[901, "Kilometers"]}
};
ResourceFunction["WorldTravellerSignpostSolve"][locs]

Out[2]=

Scope (1)

Numbers are interpreted as kilometers:

In[3]:=

locs = {
{Entity["City", {"LosAngeles", "California", "UnitedStates"}], 9000},
{Entity["City", {"NewYork", "NewYork", "UnitedStates"}], 6000},
{Entity["City", {"Berlin", "Berlin", "Germany"}], Quantity[431, "Kilometers"]},
{Entity["City", {"Oslo", "Oslo", "Norway"}], Quantity[901, "Kilometers"]}
};
ResourceFunction["WorldTravellerSignpostSolve"][locs]

Out[4]=

Neat Examples (2)

Given the signpost, find the location:

In[5]:=

locs = {
{Entity["Neighborhood", "Waikiki::Honolulu::Hawaii::UnitedStates"],
Quantity[9920, "Miles"]},
{Entity["Country", "SouthAfrica"], Quantity[2774, "Miles"]},
{Entity["City", {"HongKong", "HongKong", "HongKong"}], Quantity[4373, "Miles"]},
{Entity["Country", "Jamaica"], Quantity[9113, "Miles"]},
{Entity["City", {"Prague", "Prague", "CzechRepublic"}], Quantity[4542, "Miles"]},
{Entity["Country", "Barbados"], Quantity[7970, "Miles"]},
{Entity["City", {"NewYork", "NewYork", "UnitedStates"}], Quantity[8442, "Miles"]},
{Entity["AdministrativeDivision", {"California", "UnitedStates"}], Quantity[10189, "Miles"]},
{Entity["City", {"Prague", "Prague", "CzechRepublic"}], Quantity[4542, "Miles"]},
{Entity["City", {"Barcelona", "Barcelona", "Spain"}], Quantity[4626, "Miles"]}
};
loc = ResourceFunction["WorldTravellerSignpostSolve"][locs]

Out[6]=

Find the country:

In[7]:=

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Visualize the lines and the found minimum:

In[8]:=

Out[8]=

Investigate the effect of the precision of the distances on the final location. Given a location, and some random cities, find the estimates for 'rounded' distances:

In[9]:=