Built-in Mathematica Symbol

Plot3D

Plot3D[f, {x, x_min, x_max}, {y, y_min, y_max}]
generates a three-dimensional plot of f as a function of x and y.

Plot3D[{f₁, f₂, ...}, {x, x_min, x_max}, {y, y_min, y_max}]
plots several functions.

MORE INFORMATION

Holes are left at any positions where the f_i evaluate to None, or anything other than real numbers.

Plot3D treats the variables x and y as local, effectively using Block.

Plot3D has attribute HoldAll, and evaluates f only after assigning specific numerical values to x and y.

In some cases it may be more efficient to use Evaluate to evaluate f symbolically before specific numerical values are assigned to x and y.

Plot3D has the same options as Graphics3D, with the following additions and changes:

Axes	True	whether to draw axes
BoundaryStyle	Automatic	how to draw boundary lines for surfaces
BoxRatios	{1,1,0.4}	bounding 3D box ratios
ClippingStyle	Automatic	how to draw clipped parts of surfaces
ColorFunction	Automatic	how to determine the color of surfaces
ColorFunctionScaling	True	whether to scale arguments to ColorFunction
EvaluationMonitor	None	expression to evaluate at every function evaluation
Exclusions	Automatic	x, y curves to exclude
ExclusionsStyle	None	what to draw at excluded curves
Filling	None	filling under each surface
FillingStyle	Opacity[0.5]	style to use for filling
MaxRecursion	Automatic	the maximum number of recursive subdivisions allowed
Mesh	Automatic	how many mesh lines in each direction to draw
MeshFunctions	{#1&,#2&}	how to determine the placement of mesh lines
MeshShading	None	how to shade regions between mesh lines
MeshStyle	Automatic	the style for mesh lines
Method	Automatic	the method to use for refining surfaces
NormalsFunction	Automatic	how to determine effective surface normals
PerformanceGoal	$PerformanceGoal	aspects of performance to try to optimize
PlotPoints	Automatic	the initial number of sample points in each direction
PlotRange	{Full,Full,Automatic}	the range of z or other values to include
PlotStyle	Automatic	graphics directives for the style for each surface
RegionFunction	(True&)	how to determine whether a point should be included
WorkingPrecision	MachinePrecision	the precision used in internal computations

Interactive labeling can be specified for surfaces using Tooltip, StatusArea or Annotation.

Plot3D[Tooltip[{f₁, f₂, ...}], ...] specifies that the f_i should be displayed as tooltip labels for the corresponding surfaces.

Tooltip[f, label] specifies an explicit tooltip label for a surface.

PlotStyle->None draws no surface, so effectively does not eliminate hidden surfaces.

Plot3D initially evaluates each function at a grid of equally spaced sample points specified by PlotPoints. Then it uses an adaptive algorithm to choose additional sample points, subdividing at most MaxRecursion times.

You should realize that with the finite number of sample points used, it is possible for Plot3D to miss features in your functions. To check your results, you should try increasing the settings for PlotPoints and MaxRecursion.

With the setting Mesh->All, Plot3D draws mesh lines to show all subdivisions it makes.

The default setting MeshFunctions->{{#1&, #2&}} draws an x, y mesh on each surface.

The arguments supplied to functions in MeshFunctions and RegionFunction are x, y, z. Functions in ColorFunction are by default supplied with scaled versions of these arguments.

ColorFunction, MeshFunctions and RegionFunction are all evaluated over each surface.

With the default settings Exclusions->Automatic and ExclusionsStyle->None, Plot3D breaks surfaces at discontinuity curves it detects. Exclusions->None joins across discontinuities.

By default, surfaces are treated as uniform white diffuse reflectors, corresponding to ColorFunction->(White&).

Plot3D returns Graphics3D[GraphicsComplex[data]].

EXAMPLES

CLOSE ALL

Basic Examples (4)

Plot a function:

Plot several functions:

Restrict the domain:

Plot functions with branch cuts:

Plot a function:

In[1]:=

Out[1]=

Plot several functions:

Restrict the domain:

Plot functions with branch cuts:

In[1]:=

Out[1]=

Scope (15)

More points are sampled where the function changes quickly:

The plot range is selected automatically:

Areas where the function becomes nonreal are excluded:

The surface is split when there are discontinuities in the function:

Use PlotPoints and MaxRecursion to control adaptive sampling:

Use PlotRange to focus in on areas of interest:

Use Exclusions to remove curves or split the resulting surface:

Use RegionFunction to restrict the surface to a region given by inequalities:

Provide an explicit PlotStyle for the surface:

Provide separate styles for different surfaces:

Add labels:

Color the surface by height:

Style the areas between mesh lines:

Provide an interactive Tooltip for a surface:

Fill below a surface:

Options (79)

Use labels based on variables specified in Plot3D:

Use colored backgrounds:

Use a black boundary around the edges of the surface:

Use a thick boundary around the edges of the surface:

Use a thick red boundary around the edges of the surface:

Do not use any boundary:

BoundaryStyle applies to holes cut by RegionFunction:

BoundaryStyle does not apply to holes cut by Exclusions:

Automatic uses the natural scale from PlotRange:

Use BoxRatios to emphasize some particular feature, in this case a saddle surface:

Color clipped regions like the rest of the surface:

Do not draw clipped regions:

Make clipped regions partially transparent:

Color clipped regions red at the bottom and blue at the top:

Color according to the x and y coordinates:

Color by scaled

coordinate:

Use ColorData for predefined color gradients:

Named color gradients color in the

direction:

ColorFunction has higher priority than PlotStyle:

ColorFunction has lower priority than MeshShading:

Use unscaled coordinates:

Use scaled coordinates in the

direction and unscaled coordinates in the

and

directions:

Show where Plot3D samples a function:

Count how many times

is evaluated:

This uses automatic methods to compute exclusions, in this case from branch cuts:

Indicate that no exclusions should be computed:

Give a set of exclusions as an equation:

Give two sets of exclusions:

Use a condition with the exclusion equation:

Use both automatically computed and explicit exclusions:

Style the boundary with a thick blue line:

Style the boundary with a thick blue line and the surface in between transparent:

Use a transparent surface in the exclusion cuts:

Fill to the bottom:

Filling occurs along the region cut by the RegionFunction:

Fill to both top and bottom:

Fill surface 1 to the bottom with blue and surface 2 to the top with red:

Fill to the bottom with a variety of styles:

Fill to the plane

with red below and blue above:

Fill to the plane

from below only:

Refine the surface where it changes quickly:

Use no mesh:

Show the initial and final sampling meshes:

Use 5 mesh lines in each direction:

Use 3 mesh lines in the x direction and 6 mesh lines in the y direction:

Use mesh lines at specific values:

Use different styles for different mesh lines:

Use the z value as the mesh function:

Use mesh lines in the

and

directions:

Use mesh lines corresponding to fixed distances from the origin:

Use None to remove regions:

Lay a checkerboard pattern over a surface:

MeshShading has a higher priority than PlotStyle:

MeshShading has a higher priority than ColorFunction:

Use red mesh lines:

Use red mesh lines in the

direction and thick mesh lines in the

direction:

Normals are automatically calculated:

Use None to get flat shading for all the polygons:

Vary the effective normals used on the surface:

Generate a higher-quality plot:

Emphasize performance, possibly at the cost of quality:

Use more initial points to get a smoother surface:

Use 20 initial points in the

direction and 5 in the

direction:

Automatically compute the z range:

Use all points to compute the range:

Show the surface over the full x, y range:

Automatically compute the x, y range:

Use an explicit z range to emphasize features:

Color a surface with diffuse orange:

Use Specularity to get highlights:

Use Opacity to get transparent surfaces:

Use separate styles for each of the surfaces:

Produce a wire mesh:

Plot over an annulus region in

and

:

Filling will fill from the region boundary:

Regions do not have to be connected:

Use any logical combination of conditions:

Evaluate functions using machine-precision arithmetic:

Evaluate functions using arbitrary-precision arithmetic:

Applications (8)

This shows the solution to the heat equation in 1 dimension:

Plot a saddle surface; the mesh curves show where the function is zero:

Use a cutout to understand limit behavior:

There are different limits when approaching along different lines:

Make the surface transparent to see its inner structure:

The

,

and

norms, with the unit norm mesh line:

Show the real and imaginary parts of

:

Show the different complex components for a function:

Plot an iterated logistic map as a function of parameter and initial value:

Properties & Relations (7)

Plot3D samples more points where it needs to:

Plot3D is a special case of ParametricPlot3D:

Use ListPlot3D for plotting data:

Use Plot for univariate functions:

Use ParametricPlot for plane parametric curves and regions:

Use ContourPlot3D and RegionPlot3D for implicit surfaces and regions:

Use DensityPlot and ContourPlot for densities and contours:

Neat Examples (2)

The branch cuts of inverse trigonometric functions:

Real and imaginary parts as mesh functions: