The minimum zone fitting and error evaluation for the logarithmic curve has important applications. Based on geometry optimization approximation algorithm whilst considering geometric characteristics of logarithmic curves, a new fitting and error evaluation method for the logarithmic curve is presented. To this end, two feature points, to serve as reference, are chosen either from those located on the least squares logarithmic curve or from amongst measurement points. Four auxiliary points surrounding each of the two reference points are then arranged to resemble vertices of a square. Subsequently, based on these auxiliary points, a series of auxiliary logarithmic curves (16 curves) are constructed, and the normal distance and corresponding range of values between each measurement point and all auxiliary logarithmic curves are calculated. Finally, by means of an iterative approximation technique consisting of comparing, evaluating, and changing reference points; determining new auxiliary points; and constructing corresponding auxiliary logarithmic curves, minimum zone fitting and evaluation of logarithmic curve profile errors are implemented. The example results show that the logarithmic curve can be fitted, and its profile error can be evaluated effectively and precisely using the presented method.
A tight approximation algorithm for the cluster vertex deletion problem
