Coordinate metrology employs a discrete sampling of data points to verify the size, form, orientation, and location of features contained in parts. Usually data points are collected intuitively with simple schemes that attempt to cover the surface of the features as best as possible. Data fitting methods are used to determine the zones of deviations about the ideal feature. A multitude of linear and nonlinear optimization procedures and the least squares method have been used to estimate the tolerance zone for straightness, flatness, circularity, and cylindricity. More complex forms such as conicity have been largely ignored in the literature, in spite of the sufficient need to inspect them in parts such as nozzles and tapered rollers in bearings.

In summary, an orderly procedure for sampling and fitting cones is developed which can lead to the development of comprehensive standards and solutions for industry.

This dissertation attempts to develop suitable guidelines for inspection of cones and conical frustums using probe-type coordinate measuring machines. The sampling problem, the path determination, and fitting of form zones are each addressed in great detail. Moreover, an integrative approach is taken for form verification and detailed experimental analysis is conducted as a pilot study for demonstrating the need for the same. Three separate sampling methods are derived: Hammersley, Halton-Zaremba, and Aligned Systematic; at various sample sizes using sampling theory and prior work in two dimensional sampling. A path plan is developed to illustrate the complexity of employing these sampling strategies for data sampling in cones. Linear and nonlinear deviations are formulated using optimization and least-squared methods and solved to yield competitive solutions. Comprehensive experimental analysis investigated issues of model adequacy, nesting, interactions, and individual effects, while studying conicity as a response variable in the light of sampling strategies, sizes, cone surface areas, and fitting methods.