Computing the Angularity Tolerance

In computational metrology one needs to compute whether an object satis es speci cations of shape within an acceptable tolerance. To this end positions on the object are measured, resulting in a collection of points in space. From this collection of points one wishes to extract information on atness, roundness, etc. of the object. In this paper we study one particular feature of objects, the angularity. The angularity indicates how well a plane makes a speci ed angle with another plane. We study the problem in 2-dimensional space (where the planes become lines) and in 3-dimensional space. In 2-dimensional space the problem is equivalent to computing the smallest wedge of the a given angle that contains all the points. We give an O(n logn) algorithm for this problem. In 3-dimensional space we study the more restricted problem where one of the planes is known (a datum plane). In this case the problem is equivalent to asking for the smallest width 3-dimensional strip that contains all the points and makes a given angle with the datum plane. We give an O(n logn) algorithm to solve this version. We also show that in the case of uncertainty in the measured points, upperbounds and lowerbounds on the width can be computed in similar time bounds.