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1.1 Sampling

There is the question as to how many times, and for which arguments, the function is computed. Consider the following equation:

figure3792

Computing q(x) generates the sample (x,q(x)), of the graph of y=q(x). Sampling, as we did before, generates the following table:

figure3804

Table of q(x)

Surprisingly, the table matches our ealier one. It is not surprising that the plotted points match.

figure3822

Continuing our procedure by rote, the same graph is generated.

figure3924

By adding more samples to our table, we see that the previous graph is incorrect.

figure4011

Richer Table of q(x)

Using our richer table, we again plot the points which we know satisfy our equation.

figure4043

We have lost confidence in the line which connects the points. Without warning, it has failed us. There is hope that we may be able to predict its failure for polynomials, or other classes of functions, but we aim to graph general equations.

Although calculating a large number of samples guarantees to consume vast resources, it does not guarantee that a more reliable graph is generated. Consider the equation

figure4266

Using over a million uniformly spaced samples of (x,r(x)), from [-2,2], results in the following graph:

figure4274

The actual graph follows, which may be reliably computed using a handful of samples.

figure4368

The actual graph is a very sharp hyperbola, and can not be generated by following our procedure, as the graph is composed of two disjoint curves.


next up previous notation contents
Next: 1.2 Implicit Equations Up: 1 Motivation Previous: 1 Motivation
Jeff TupperMarch 1996