4.2.3 Pixel Testing

 

From the interval inclusion property we know that

so setting to , for each pixel , will generate a rendering of .

An example rendering follows:

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

Another example rendering follows:

An example evaluation of S, using notation, follows:

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

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

The two renderings follows:

An example evaluation, using early accounting, follows:

the corresponding evaluation using deferred accounting is as follows:

Two renderings, corresponding to the preceding renderings, follow:

Superior inequality renderings are possible using instead of .

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

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.

Jeff Tupper

March 1996