MeshFunctions—Wolfram Documentation [proxy]

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.

Examples

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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.html

BibTeX

@misc{reference.wolfram_2025_meshfunctions, author="Wolfram Research", title="{MeshFunctions}", year="2007", howpublished="\url{https://reference.wolfram.com/language/ref/MeshFunctions.html}", note=[Accessed: 03-March-2026]}

BibLaTeX

@online{reference.wolfram_2025_meshfunctions, organization={Wolfram Research}, title={MeshFunctions}, year={2007}, url={https://reference.wolfram.com/language/ref/MeshFunctions.html}, note=[Accessed: 03-March-2026]}