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We investigate the propagation of errors through geometric transformations, such as re°ections, rotations, similarity transformations, and projections, and also the scalar product of vectors. This means computing tolerance zones of points which undergo such transformations, if these points and the transformations themselves are given by toleranced input data.

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The polygonal approximation problem is a primary problem in computer graphics, pattern recognition, CAD/CAM, etc. In $R^2$, the cone intersection method (CIM) is one of the most eÆcient algorithms for approximating polygonal curves. With CIM Eu and Toussaint, by imposing an additional constraint and changing the given error criteria, resolve the three-dimensional weighted minimum number polygonal approximation problem with the parallel-strip error criterion (PS-WMN) under $L_2$ norm. In this paper, without any additional constraint and change of the error criteria, a CIM solution to the same problem with the line segment error criterion (LS-WMN) is presented, which is more frequently encountered than the PS-WMN is. Its time complexity is $O(n^3)$, and the space complexity is $O(n^2)$. An approximation algorithm is also presented, which takes $O(n^2)$ time and $O(n)$ space. Results of some examples are given to illustrate the eÆciency of these algorithms.

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