Establishment of a pair of concentric circles with the minimum radial separation for assessing roundness error U Roy and X Zhang
The paper presents a computational-geometry-based method of determining the roundness error of a measured workpiece. A set ofn points (obtained from the measured workpiece) in a plane being given, it is required that the center and the radii of a pair of concentric circles be found such that no point is exterior to the space bounded by the circles, with the condition that the radial separation between the circles is minimum. The paper addresses the mathematical formalization of the problem. The properties of convex-hull and Voronoi diagrams have been exploited to develop a faster algorithm for establishing the circles. The methodolooy has been implemented, and the results have been presented to validate the computational effectiveness of the approach.
computational geometry, geometric tolerance, automatic part inspection
PROBLEM SPECIFICATION
According to the tolerance standards of References 1 and 2, a roundness error is evaluated with reference to ideal geometric features (i.e. a pair of ideal concentric circles), which must be established from actual measurements. The problem is defined as follows.
A set S of n points (Px, Pz ..... P.) in a plane being given, for n >~ 4 (for n < 4, the minimum separation of the pair of concentric circles from the n points can always be found to be zero), find a pair of concentric circles Ca and C2 with the minimum radial separation SEl" such that no point is exterior to the space bounded by the two circles.