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Next: 4.2.2 Sequential Rendering Up: 4.2 Basic Rendering Previous: 4.2 Basic Rendering

4.2.1 Constant Interval Arithmetic

Let tex2html_wrap_inline31473 denote a constant interval arithmetic, such as tex2html_wrap_inline32653 or tex2html_wrap_inline35053 . We will not ensure that R represents tex2html_wrap_inline37549 directly; we will instead work with   tex2html_wrap_inline37713 . The interval specification tex2html_wrap_inline37715 is computed by evaluating the specification S using the interval arithmetic tex2html_wrap_inline31473 . The interval inclusion property assures us that

math25234

Let   tex2html_wrap_inline37721 describe tex2html_wrap_inline37563 , using an element of tex2html_wrap_inline31441 :

math25241

We may then determine tex2html_wrap_inline37727 by considering

math25246

for tex2html_wrap_inline37729 . The remaining sections detail how tex2html_wrap_inline37727 may be determined.

As we only assume that tex2html_wrap_inline37715 may be computed, S may be partial: the domain of S must be taken into account. With interval arithmetics that track tex2html_wrap_inline33463 , such as tex2html_wrap_inline35053 , the domain of tex2html_wrap_inline37715 is bounded, and the domain of S may be accounted for. With interval arithmetics that do not track tex2html_wrap_inline33463 , such as tex2html_wrap_inline32653 , we have no information as to the domain of tex2html_wrap_inline37715 . Two approaches may be taken with such arithmetics. This lack of information may be accounted for when performing the interval comparisons which occur while evaluating tex2html_wrap_inline37715 , or after the evaluation of tex2html_wrap_inline37715 has completed. We call the former approach ``early accounting'', and the latter approach ``deferred accounting''. The latter approach is preferable, as it allows for better renderings. The two approaches are compared in section gif.


next up previous notation contents
Next: 4.2.2 Sequential Rendering Up: 4.2 Basic Rendering Previous: 4.2 Basic Rendering
Jeff TupperMarch 1996