Evaluation of form and profile error in the measurement of discrete parts

The minimax (Chebyshev or minimum zone) criterion is the only criterion accepted by ANSI and ISO for form and profile errors evaluation. Evaluation of form and profile errors is a non-linear and non-convex problem which is difficult to handle using the minimax criterion. For this reason, this criterion has seldom been implemented to solve general form and profile error problems. The objective of this research is to develop a series of theorems and mathematical models based on the minimax criterion to evaluate general form and profile errors.; The Voronoi diagram is the basic tool used to develop algorithms leading to optimum results. Exact solutions for circularity and sphericity are obtained from developed models. This research also introduces the concepts of virtual circle and virtual sphere using residual vectors resulting from measurements. The virtual circle and virtual sphere are critical components of 2-D and 3-D profile errors and satisfy circular and spherical properties. These concepts are then used to show that general 2-D and 3-D profile error evaluations can be mapped into circularity and sphericity problems, and this development provides a novel way to treat profile error verification. Finally, a validation is carried out to illustrate the efficiency and effectiveness of the developed models.