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Asked: 2026-05-19T17:31:03+00:00

Thanks to this excellent analysis of the Plot algorithm by Yaroslav Bulatov, I now

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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 does not try 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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1 Answer

  1. Editorial Team

    These are the standard Plots .

    enter image description here

    enter image description here

    And the Contour at 10^-20

    enter image description here

    They are pretty clear for me.

    Do you expect something more accurate?

    Edit

    If you want to use the Contours->{0} Option, you may use:

    enter image description here

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