We describe a new method for numeric computations, which we call affine arithmetic (AA). This model is similar to standard interval arithmetic, to the extent that it automatically keeps track of rounding and truncation errors for each computed value. However, by taking into account correlations between operands and sub-formulas, AA is able to provide much tighter bounds for the computed quantities, with errors that are approximately quadratic in the uncertainty of the input variables. We also describe two applications of AA to computer graphics problems, where this feature is particularly valuable: namely, ray tracing and the construction of octrees for implicit surfaces.

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Introduction Interval arithmetic (IA), also known as interval analysis, is a technique for numerical computation where each quantity is represented by an interval of floatingpoint numbers. Those intervals are added, subtracted, multiplied, etc. in such a way that each computed interval is guaranteed to contain the (unknown) value of the quantity it represents [3, 4]. Since its introduction in the 60's by R. E. Moore, IA became widely appreciated for its ability to manipulate imprecise data, keep track automatically of truncation and round-off errors, and probe the behavior of functions efficiently and reliably over whole sets of arguments at once. The main weakness of IA is that it tends to be too conservative: the intervals it produces are often much wider than the true range of the corresponding quantities, often to the point of uselessness. This problem is particularly severe in long computation chains, where the intervals computed at one stage are inp