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

## NAME

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

## SYNOPSIS

**fitcircle** [ *xyfile* ] **-L***norm* [ **-H**[*nrec*] ] [ **-S** ] [ **-V** ] [ **-:** ]
[ **-bi**[**s**][*n*] ]

## DESCRIPTION

**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 solutions.

## OPTIONS

*xyfile*- 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)]. Applies to geographic coordinates only.
**-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].

## EXAMPLES

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.
Try:
**fitcircle** ship.xyg **-L**2
**project** ship.xyg **-C***ox*/*oy* **-T***px*/*py* **-S** **-pz** | sample1d **-S**-100 **-I**1
> output.pg
Here, *ox*/*oy* is the lon/lat of the mean from **fitcircle**, and *px*/*py* is the lon/lat
of the pole. The file output.pg has distance, gravity data sampled every 1 km along the great circle which best fits ship.xyg