Summary. Construction of optimal triangular meshes for controlling the errors in gradient estimation for piecewise linear interpolation of data functions in the plane is discussed. Using an appropriate linear coordinate transformation, rigorously optimal meshes for controlling the error in quadratic data functions are constructed. It is shown that the transformation can be generated as a curvilinear coordinate transformation for any cr data function with nonsingular Hessian matrix. Using this transformation, a construction of nearly optimal meshes for general data functions is described and the error equilibration properties of these meshes discussed. In particular, it is shown that equilibration of errors is not a sufficient condition for optimality. A comparison of meshes generated under several different criteria is made, and their equilibrating properties illustrated.
Abstract.We present a modification and extension of the (linear time) visibility polygon algorithm of Lee. The algorithm computes the visibility polygon of a simple polygon from a viewpoint that is either interior to the polygon, or in its blocked exterior (the cases of viewpoints on the boundary or in the free exterior being simple extensions of the interior case). We show by example that the original algorithm by Lee, and a more complex algorithm by El Gindy and Avis, can fail for polygons that wind sumciently. We present a second version of the algorithm, which does not extend to the blocked exterior case. CR Cate#ories: F.2.2.Keywords: computational geometry, visibility. Introduction.We consider the following problem: Given a simple polygon P and a viewpoint z in the plane, find all points on the boundary of P that are 'visible' from z. The position of the viewpoint z determines three cases : z in the interior or exterior of P, or on the boundary of P. The exterior case can be further categorized as free exterior or blocked exterior depending on whether there exists a ray from z which does not intersect P or there is no such ray. The boundary and free exterior cases can be handled as simple extensions of the interior case. So from an algorithmic viewpoint, there are basically two cases, the interior case and the blocked exterior case.Two linear time algorithms for computing the visibility polygon of P from an interior viewpoint have been published, Lee [7] and E1 Gindy and Avis [1], and Lee also presented a modification for the blocked exterior case in his paper. Lee's algorithm is simpler in structure; in particular it requires only one stack which eventually yields the visibility polygon, as opposed to three in the E1
The problem of determining optimal incidences for triangulating a given set of vertices for the model problem of interpolating a convex quadratic surface by piecewise linear functions is studied. An exact expression for the maximum error is derived, and the optimality criterion is minimization of the maximum error. The optimal incidences are shown to be derivable from an associated Delaunay triangulation and hence are computable in O(N log N) time for N vertices.
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