Is there a way to force ContourPlot re-check all the points on the each stage of it's recursion algorithm?

Posted by Alexey Popkov on Stack Overflow See other posts from Stack Overflow or by Alexey Popkov
Published on 2011-02-05T07:09:39Z Indexed on 2011/02/05 7:26 UTC
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Hello,

Thanks to this excellent analysis of the Plot algorithm by Yaroslav Bulatov, I now understand the reason why Plot3D and ContourPlot fail to draw smoothly functions with breaks and discontinuities.

For example, in the following case ContourPlot fails to draw contour x^2 + y^2 = 1 at all:

ContourPlot[Abs[x^2 + y^2 - 1], {x, -1, 1}, {y, -1, 1}, Contours -> {0}]

It is because the algorithm does not go deeply into the region near x^2 + y^2 = 1. It "drops" this region on an initial stage and do not tries to investigate it further. Increasing MaxRecursion does nothing in this sense. And even undocumented option Method -> {Refinement -> {ControlValue -> .01 \[Degree]}} does not help (but makes Plot3D a little bit smoother).

The above function is just a simple example. In real life I'm working with very complicated implicit functions that cannot be solved analytically.

Is there a way to get ContourPlot to go deeply into such regions near breaks and discontinuities?

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