MeshFunctions

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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

open allclose all

Basic Examples (3)Summary of the most common use cases

Put 5 mesh lines in the direction:

In[1]:=1

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https://wolfram.com/xid/0tzmru6b-uyj

Out[1]=1

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

In[1]:=1

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https://wolfram.com/xid/0tzmru6b-nfck6z

Out[1]=1

Show intersection points:

In[1]:=1

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https://wolfram.com/xid/0tzmru6b-fx47rn

Out[1]=1

Applications (1)Sample problems that can be solved with this function

Define two polynomials:

In[1]:=1

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https://wolfram.com/xid/0tzmru6b-b9pxcj

Use MeshFunctions to find the intercepts:

In[2]:=2

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https://wolfram.com/xid/0tzmru6b-k1esxs

Out[2]=2

Use MeshFunctions to find the intersections between two functions:

In[3]:=3

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https://wolfram.com/xid/0tzmru6b-bw8e9l

Out[3]=3

Neat Examples (2)Surprising or curious use cases

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

In[1]:=1

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https://wolfram.com/xid/0tzmru6b-d8tl

Out[1]=1

Real and imaginary parts as mesh functions:

In[1]:=1

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https://wolfram.com/xid/0tzmru6b-z5brn

In[2]:=2

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https://wolfram.com/xid/0tzmru6b-i34ds

In[3]:=3

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https://wolfram.com/xid/0tzmru6b-jubzfm

Out[3]=3

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