# distance_to_position

distance_to_position, a MATLAB code which estimates the positions of cities given a city-to-city distance table.

In 2D, the problem is singular. In particular, the position of one city is completely arbitrary, and one component of a second city is completely arbitrary (and a third city's position can be "flipped" about the line connecting cities one and two). To remove some of this singularity, the program assigns city #1 the position (0,0) and city #2 is given a zero Y coordinate.

In N-dimensional space, a similar set of constraints must be placed on the first N cities, or the least squares solver is likely to fail. (That is, we specify all N components of the first city to be 0, N-1 components of the second one, and so on, up to the N-th city which has a single zero component).

The computations carried out by this program assume that the cities lie on a plane. If the distance data is for international cities, the effect of spherical geometry may make the planar approximation very bad.

Once the nonlinear least squares problem is set up, MATLAB's LSQNONLIN function is called to seek a solution.

### Usage:

distance_to_position ( 'distance.txt' )
reads the distance information in 'distance.txt', estimates the positions of the cities, and writes out a position table in distance.coord.txt.

### Languages:

distance_to_position is available in a FORTRAN90 version and a MATLAB version.

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distance_to_position_sphere, a MATLAB code which estimates the positions of cities on a sphere (such as the earth) based on a city-to-city distance table.

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### Source Code:

Last revised on 07 January 2019.