Geometric Computations with Interval and New Robust Methods: by Helmut Ratschek, Jon Rokne

By Helmut Ratschek, Jon Rokne

This undergraduate and postgraduate textual content will familiarise readers with period mathematics and comparable instruments to realize trustworthy and verified effects and logically right judgements for a number of geometric computations, and the ability for relieving the consequences of the mistakes. It additionally considers computations on geometric point-sets, that are neither powerful nor trustworthy in processing with ordinary tools. The authors offer potent instruments for acquiring right effects: (a) period mathematics, and (b) ESSA the recent robust set of rules which improves many geometric computations and makes them rounding errors loose.

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Additional info for Geometric Computations with Interval and New Robust Methods: Applications in Computer Graphics, GIS and Computational Geometry

Example text

All kinds of errors can be controlled, especially rounding errors, trun­ cation errors, etc. • B . infinite data sets can be processed. These two reasons are now discussed in some detail: A. Present-day computers mainly employ an arithmetic called fixed length floating point arithmetic or short, floating point arithmetic for calculations in engineering and the natural sciences. In this arithmetic real numbers are approximated by a subset of the real numbers called the machine representable numbers (abbreviated: machine numbers or floating point numbers when dis­ cussing implementation details).

Instead of approximating a real value a; by a machine num­ ber, the usually unknown real value χ is approximated by an interval X having machine number upper and lower boundaries. The interval X contains the value x. The width of this interval may be used as measure for the quality of the approximation. The calculations therefore have to be executed using inter­ vals instead of real numbers and hence the real arithmetic have to be replaced by interval arithmetic. When computing with the usual machine numbers χ there is no direct estimate of the error | x - x |.

One could imagine another conceptually very simple way to determine an inner approximation of the range. It would consist of evaluating the functions at the 4 corners of the rectangle (or 8 corners of the box) with machine interval arithmetic, and to derive the inner approximation from this information. An algorithmic description of such a procedure is, already in the case of 2 variables, more involved than the algorithm. The reason is that the computation might have to be split up into cases in order to figure out which of the corners are to be used for the inner approximation, especially, if α or 6 contains zero.

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