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Next: 4.2.4 Subpixel Testing Up: 4.2 Basic Rendering Previous: 4.2.2 Sequential Rendering

4.2.3 Pixel Testing

 

From the interval inclusion property we know that

math25261

so setting tex2html_wrap_inline37727 to tex2html_wrap_inline37765 , for each pixel tex2html_wrap_inline31435 , will generate a rendering of tex2html_wrap_inline37715 .

An example tex2html_wrap_inline32653 rendering follows:

figure25271

The solid line depicts the associated graph.   tex2html_wrap_inline37775 denotes a rendering produced using pixel testing. For each pixel tex2html_wrap_inline31435 , tex2html_wrap_inline37779 is computed and tex2html_wrap_inline37727 is then set accordingly. An example evaluation follows:

math25409

Another example tex2html_wrap_inline32653 rendering follows:

figure25431

An example tex2html_wrap_inline32653 evaluation of S, using tex2html_wrap_inline35053 notation, follows:

math25492

We have chosen to take into account the lack of knowledge of tex2html_wrap_inline33463 when evaluating tex2html_wrap_inline37823 ; an alternative evaluation of S, which defers this accounting, follows:

math25523

Although the evaluation result is tex2html_wrap_inline32719 , we do not know tex2html_wrap_inline37855 , so we must assume that tex2html_wrap_inline37857 . We must therefore set tex2html_wrap_inline37727 to tex2html_wrap_inline32751 using either accounting approach. The two approaches differ when rendering

math25553

The two renderings follows:

figure25557

An example evaluation, using early accounting, follows:

math25715

the corresponding evaluation using deferred accounting is as follows:

math25745

Two tex2html_wrap_inline35053 renderings, corresponding to the preceding tex2html_wrap_inline32653 renderings, follow:

figure25775

Superior inequality renderings are possible using tex2html_wrap_inline35053 instead of tex2html_wrap_inline32653 .

For discontinuous equations, tex2html_wrap_inline35051 renderings are often superior to tex2html_wrap_inline35053 renderings. Consider the following renderings:

figure25921

With discontinuous specifications, set-based interval arithmetics may sharply bound discontinuous pieces; without this ability to use several bounds, the discontinuous pieces must be bound with a single interval.


next up previous notation contents
Next: 4.2.4 Subpixel Testing Up: 4.2 Basic Rendering Previous: 4.2.2 Sequential Rendering
Jeff TupperMarch 1996