Friday, November 2, 2012

More on the Least Squares Estimate of a Circle's Center

  The formula for the least squares estimate for the center of a circle that was used in a recent blog uses the difference in the square of the distance from the center for pairs of points as the difference whose square is to be optimized. One can study how the formula works for different sets of pairs of points. A path can be drawn between the original five points that will link all the 10 pairs of points in a sequence. The first two pair of points with the largest separation gives the center of the circumscribed circle through the set of three points. The same is true for the three pair pairs of points for the triangle with the largest sides. As one adds more pairs of points the estimated center is less dependent on the individual points used and tends to converge but there is some residual difference between the distance of a point from the center for the pairs. Squares of the distance of a point from the center were used to compute the differences to avoid the use of square roots in deriving the formula. The method is similar to using hyperbolic navigation to determine one's position.
  Supplemental (Nov 2): I got residuals that differed in the third decimal place from the radial residuals for the 10 pair of points above using a method for determining position based on differences in signal arrival times. The data points for the circle were treated as signal sources whose positions were known and the differences in arrival times were set to zero. The estimated position of the center also differed in the third decimal place.

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