As we saw in the first part of this article, several different algorithms are available for fitting circles to CMM point data.  This article will examine the calculations that are used for additional algorithms commonly available in CMM software.

Chebychev / Minimum Zone Circle

• This algorithm is included in most CMM softwares.  Different names are used – Minimum Zone or various spellings of Chebychev (e.g. Tchebyshev) who is credited with the algorithm

• Finds a circle that goes halfway between the “peaks and valleys” of the data points

• Corresponds to form tolerancing.  The center of the Minimum Zone circle coincides with the smallest Circularity zone that the points will conform to.

• The optimization criterion is that the range of deviations (difference between largest and smallest) is minimized, with no other constraints.

• Because the result is determined by the most extreme “high” and “low” data points, the Minimum Zone algorithm is sensitive to outliers Location Constrained Maximum Inscribed Circle

• Finds the largest circle that will fit completely inside the data points, with the additional constraint that the location in the X direction is fixed

• Corresponds to the largest location constrained gage pin that will fit into a hole.  This situation can occur if the hole is a tertiary datum feature.

• In most cases, there are 2 points of contact where the deviation is zero.

• Maximizing the size of the circle minimizes the area between the circle and the surface, as well as the average deviation

• The optimization criteria are that the average absolute deviation is minimized, with the constraints that the minimum deviation is zero and the circle’s X coordinate is fixed at zero.

• Because the result is determined by the most extreme “low” data points, the Maximum Inscribed algorithm is sensitive to outliers Location Constrained Minimum Circumscribed Circle

• Finds the smallest circle that will fit completely outside the data points, with the additional constraint that the location in the X direction is fixed

• Corresponds to the smallest ring gage that will fit over a pin.  This situation can occur if the pin is a tertiary datum feature.

• In most cases, there are 2 points of contact where the deviation is zero

• Minimizing the size of the circle minimizes the area between the circle and the surface, as well as the average deviation

• The optimization criteria are that the average absolute deviation is minimized, with the constraints that the maximum deviation is zero and the circle’s X coordinate is fixed at zero.

• Because the result is determined by the most extreme “high” data points, the Maximum Inscribed algorithm is also sensitive to outliers Summary

Circle fitting algorithms all involve optimizing a certain aspect of the circle within particular constraints.  This can be used to calculate circles corresponding to various different functional requirements.

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