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4.2.8 Sequential Rendering

As before, a rendering is built pixel by pixel. The four tests described in the previous section may be utilized with linear interval arithmetics. The tests simply utilize linear interval arithmetic in place of constant interval arithmetic.

For line-like renderings, subpixel testing is not always required when using a linear interval arithmetic. Consider the following two renderings:

figure26649

An example evaluation follows, with S = (g = 0) and tex2html_wrap_inline38093 :

math26754

From the evaluation we know that

math26805

and that g is continuous over tex2html_wrap_inline37563 ; it follows that

math26815

since

math26819

We may therefore set tex2html_wrap_inline37727 to tex2html_wrap_inline32719 .

Of course, subpixel evaluation is still needed to combat the interval over-estimation usually present in large specifications. Continuity information is usually needed when rendering specifications involving equality.

The following two renderings were produced using constant interval arithmetic and all of the subpixel tests described:

figure26824

The light grey pixels will not be resolved using any constant interval arithmetic. This is clear after noticing

math27023

and

math27039

for tex2html_wrap_inline38115 .

The light grey pixels may be resolved with linear interval arithmetic since operations may consider the dependence of the interval arguments upon the system parameters, namely x and y. For our preceding examples, note that

math27049

and

math27065

The following two renderings were rendered using linear interval arithmetic:

figure27088

Of course, a symbolic optimizer may transform the equations to avoid evaluation difficulties when presented with the simple cases shown.


next up previous notation contents
Next: 4.3 Optimization: Function Rendering Up: 4.2 Basic Rendering Previous: 4.2.7 Linear Interval Arithmetic
Jeff TupperMarch 1996