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Re: Plotting piecewise in 2D

On 5/14/10 at 5:32 AM, lambaugh at (Jim Lambaugh) wrote:

>Please take a look at

>DensityPlot[If[x == 0 && y == 0, 1000, 0], {x, -1, 1}, {y, -1, 1}]

>This just gives me a uniform plot. Why does the peak at (0,0) not
>show up?

There are a variety of issues. First, the plot routines in
Mathematica (or any software for that matter) obviously cannot
sample your function at all possible points since it is defined
for all real values of x, y. And unless your function is sampled
at {0,0} the output is a constant resulting in a uniform plot.

Next, even if your function were sampled at {0,0} you likely
would still see an uniform plot. By default, Mathematica scales
plots to show what it considers the interesting portion of the
plot. This results in a scaling that will not show one extreme
point. As a simple example, try the following:

data = Join[RandomReal[1, {100}], {1000}, RandomReal[1, {100}]];

The first plot will not show the single extreme data point at
1000. The second plot will show the extreme but loses all detail
shown by the first plot in the bulk of the data.

=46inally, even if the sampling included the value at {0,0} and
the plot were scaled correctly to display the function value, it
still might not be apparent in the plot. Depending on other
factors such as plot size, plot resolution etc., it might be
possible the value at {0,0} is represented by a single pixel on
your display. Since it would be surrounded by many pixels with
different values, the single pixel might well be effectively invisible.

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