fitcircle - find mean position and pole of best-fit great [or small]
      circle to points on a sphere.


      fitcircle [ xyfile ] -Lnorm [ -H[nrec] ] [ -S ] [ -V ] [ -: ] [
      -bi[s][n] ]


      fitcircle reads lon,lat [or lat,lon] values from the first two columns
      on standard input [or xyfile].  These are converted to cartesian
      three-vectors on the unit sphere.  Then two locations are found:  the
      mean of the input positions, and the pole to the great circle which
      best fits the input positions.  The user may choose one or both of two
      possible solutions to this problem.  The first is called -L1 and the
      second is called -L2.  When the data are closely grouped along a great
      circle both solutions are similar.  If the data have large dispersion,
      the pole to the great circle will be less well determined than the
      mean.  Compare both solutions as a qualitative check.
      The -L1 solution is so called because it approximates the minimization
      of the sum of absolute values of cosines of angular distances.  This
      solution finds the mean position as the Fisher average of the data,
      and the pole position as the Fisher average of the cross-products
      between the mean and the data.  Averaging cross-products gives weight
      to points in proportion to their distance from the mean, analogous to
      the "leverage" of distant points in linear regression in the plane.
      The -L2 solution is so called because it approximates the minimization
      of the sum of squares of cosines of angular distances.  It creates a 3
      by 3 matrix of sums of squares of components of the data vectors.  The
      eigenvectors of this matrix give the mean and pole locations.  This
      method may be more subject to roundoff errors when there are thousands
      of data.  The pole is given by the eigenvector corresponding to the
      smallest eigenvalue; it is the least-well represented factor in the
      data and is not easily estimated by either method.

      -L   Specify the desired norm as 1 or 2, or use -L or  -L3 to see both


           ASCII [or binary, see -b] file containing lon,lat [lat,lon]
           values in the first 2 columns.  If no file is specified,
           fitcircle will read from standard input.

      -H   Input file(s) has Header record(s).  Number of header records can
           be changed by editing your .gmtdefaults file.  If used, GMT
           default is 1 header record.

      -S   Attempt to fit a small circle instead of a great circle.  The
           pole will be constrained to lie on the great circle connecting
           the pole of the best-fit great circle and the mean location of
           the data.

      -V   Selects verbose mode, which will send progress reports to stderr
           [Default runs "silently"].

      -:   Toggles between (longitude,latitude) and (latitude,longitude)
           input/output.  [Default is (longitude,latitude)].

      -bi  Selects binary input.  Append s for single precision [Default is
           double].  Append n for the number of columns in the binary
           file(s).  [Default is 2 input columns].


      Suppose you have lon,lat,grav data along a twisty ship track in the
      file ship.xyg.  You want to project this data onto a great circle and
      resample it in distance, in order to filter it or check its spectrum.

      fitcircle ship.xyg -L2

      project ship.xyg -Oox/oy -Ppx/py -S -pz | sample1d -S-100 -I1 >

      Here, ox/oy is the lon/lat of the mean from fitcircle, and px/py is
      the lon/lat of the pole.  The file has distance, gravity
      data sampled every 1 km along the great circle which best fits


      gmt, project, sample1d

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