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MeshFunctions—Wolfram Language Documentation

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BUILT-IN SYMBOL

MeshFunctions

is an option for plotting functions that specifies functions to use to determine the placement of mesh divisions.

Details

In Plot3D, the default setting MeshFunctions->{#1&,#2&} specifies that meshes corresponding to x and y coordinates should be constructed.
With the setting MeshFunctions->{m₁,m₂,…}, each function m_i defines a family of mesh divisions.
By default, the mesh divisions are taken to lie at positions giving equally spaced values of m_i[…].
The arguments supplied to the m_i and the default MeshFunctions settings are as follows:
Plot and ListLinePlot x, y {#1&} ParametricPlot x, y, u or x, y, u, v {#3&} or {#3&,#4&} PolarPlot and ListPolarPlot x, y, θ, r (#3&) RegionPlot x, y {#1&,#2&} ContourPlot and ListContourPlot x, y, f {} DensityPlot and ListDensityPlot x, y, f {#1&,#2&} ContourPlot3D and ListContourPlot3D x, y, z, f {#1&,#2&,#3&} Plot3D and ListPlot3D x, y, z {#1&,#2&} ListSurfacePlot3D x, y, z {#1&,#2&,#3&} ParametricPlot3D x, y, z, u or x, y, z, u, v {#4&} or {#4&,#5&} RegionPlot3D x, y, z {#1&,#2&,#3&}
Each m_i effectively defines a foliation.
The m_i should normally be chosen to be continuous monotonic functions.

Examplesopen allclose all Basic Examples (3)

Put 5 mesh lines in the direction:

Show curves of constant real and imaginary parts of a function:

Show intersection points:

Applications (1)

Define two polynomials:

Use MeshFunctions to find the intercepts:

Use MeshFunctions to find the intersections between two functions:

Neat Examples (2)

A case where Fubini's theorem does not hold [more info]:

Real and imaginary parts as mesh functions:

Wolfram Research (2007), MeshFunctions, Wolfram Language function, https://reference.wolfram.com/language/ref/MeshFunctions.html. Text Wolfram Research (2007), MeshFunctions, Wolfram Language function, https://reference.wolfram.com/language/ref/MeshFunctions.html.CMS Wolfram Language. 2007. "MeshFunctions." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/MeshFunctions.html.APA Wolfram Language. (2007). MeshFunctions. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/MeshFunctions.htmlBibTeX @misc{reference.wolfram_2025_meshfunctions, author="Wolfram Research", title="{MeshFunctions}", year="2007", howpublished="\url{https://reference.wolfram.com/language/ref/MeshFunctions.html}", note=[Accessed: 12-July-2025 ]}BibLaTeX @online{reference.wolfram_2025_meshfunctions, organization={Wolfram Research}, title={MeshFunctions}, year={2007}, url={https://reference.wolfram.com/language/ref/MeshFunctions.html}, note=[Accessed: 12-July-2025 ]}

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