RegionPlot3D

RegionPlot3D[pred,{x,x_min,x_max},{y,y_min,y_max},{z,z_min,z_max}]

makes a plot showing the three-dimensional region in which pred is True.

RegionPlot3D[{pred₁,pred₂,…},…]

plots several regions corresponding to the pred_i.

Details and Options

The predicate pred can be any logical combination of inequalities.

The region plotted by RegionPlot3D can contain disconnected parts.
RegionPlot3D treats the variables x, y and z as local, effectively using Block.
RegionPlot3D has attribute HoldAll and evaluates pred after assigning numerical values to x, y and z. In some cases, it may be more efficient to use Evaluate to evaluate pred symbolically first.
RegionPlot3D has the same options as Graphics3D, with the following additions and changes:

Axes	True	whether to draw axes
BoundaryStyle	Automatic	how to draw boundaries of regions
BoxRatios	{1,1,1}	bounding 3D box ratios
ColorFunction	Automatic	how to color surfaces
ColorFunctionScaling	True	whether to scale arguments to ColorFunction
EvaluationMonitor	None	expression to evaluate at every function evaluation
MaxRecursion	Automatic	the maximum number of recursive subdivisions allowed
Mesh	Automatic	how many mesh lines in each direction to draw
MeshFunctions	{#1&,#2&,#3&}	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
PlotLegends	None	legends for surfaces
PlotPoints	Automatic	the initial number of sample points in each direction
PlotRange	Full	the range of values to include in the plot
PlotStyle	Automatic	graphics directives for the style of the surface of each region
PlotTheme	$PlotTheme	overall theme for the plot
ScalingFunctions	None	how to scale individual coordinates
TextureCoordinateFunction	Automatic	how to determine texture coordinates
TextureCoordinateScaling	True	whether to scale arguments to TextureCoordinateFunction
WorkingPrecision	MachinePrecision	the precision used in internal computations

RegionPlot3D initially evaluates pred at a 3D grid of equally spaced sample points specified by PlotPoints. Then it uses an adaptive algorithm to subdivide at most MaxRecursion times, attempting to find the boundaries of all regions in which pred is True.
You should realize that since it uses only a finite number of sample points, it is possible for RegionPlot3D to miss regions in which pred is True. To check your results, you should try increasing the settings for PlotPoints and MaxRecursion.
With the default setting PlotRange->Full, RegionPlot3D will explicitly include the full ranges x_min to x_max, etc.
With Mesh->All, RegionPlot3D will explicitly draw mesh lines to indicate the subdivisions it used to find each region.
RegionPlot3D can in general only find regions of positive measure; it cannot find regions that are just lines or points.
The arguments supplied to functions in MeshFunctions are x, y, and z. Functions in ColorFunction and TextureCoordinateFunction are by default supplied with scaled versions of these arguments.
RegionPlot3D returns Graphics3D[GraphicsComplex[data]].
Possible settings for ScalingFunctions include:
{s_x,s_y,s_z} scale x, y and z axes
Common built-in scaling functions s include:

	"Log"		log scale with automatic tick labeling
	"Log10"		base-10 log scale with powers of 10 for ticks
	"SignedLog"		log-like scale that includes 0 and negative numbers
	"Reverse"		reverse the coordinate direction

Examples

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Basic Examples (4)

Plot a 3D region:

Plot multiple regions:

Plot 3D regions defined by logical combinations of inequalities:

Use simple styling of region boundaries:

Scope (12)

Sampling (3)

Areas where the function is not True are excluded:

Use logical combinations of regions:

Regions do not have to be connected:

Presentation (9)

Provide an explicit PlotStyle for the region:

Specify styles for each region:

Add labels:

Use an overlay mesh:

Style the areas between mesh lines:

Color the region with an overlay density:

Use a theme with simple ticks in a bright color scheme:

Use a monochrome theme:

Apply a log scale to the z axis:

Options (54)

BoundaryStyle (3)

Boundary lines are black by default:

Use None to not draw any boundary lines:

Use red boundary lines:

BoxRatios (2)

Regions are shown in a cube by default:

Use the natural scale of the region:

ColorFunction (5)

Color regions by scaled , , and values:

Named color functions use the scaled direction:

Color regions according to a function of unscaled , , and values:

ColorFunction has higher priority than PlotStyle:

ColorFunction has lower priority than MeshShading:

ColorFunctionScaling (1)

Color regions according to a function of unscaled , , and values:

Mesh (7)

Show the sampling mesh:

Show no mesh:

Use 5 mesh lines in each direction:

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

Use mesh lines at specific values:

Use different styles for different mesh lines:

Mesh lines apply to the whole region, not to each component:

MeshFunctions (2)

Mesh lines in the , , and directions:

Mesh lines at fixed radii from the origin:

MeshShading (5)

Alternate red and blue sections:

MeshShading has higher priority than ContourStyle for styling:

Use the PlotStyle for some segments by setting MeshShading to Automatic:

MeshShading can be used with ColorFunction:

Fill between regions defined by multiple mesh functions:

MeshStyle (2)

Use a dashed mesh in the direction:

Use a dashed mesh in the direction and a blue mesh in the direction:

NormalsFunction (4)

Normals are automatically calculated:

Use None to get flat shading for all the polygons:

Vary the effective normals used on the surface:

The NormalsFunction does not get applied to clipped regions:

PerformanceGoal (2)

Generate a higher-quality plot:

Emphasize performance, possibly at the cost of quality:

PlotLegends (3)

Identify regions with a legend:

Use legends for color gradients:

Use Placed to put legends above the plot:

PlotPoints (1)

Use more initial points to get a smoother region:

PlotStyle (5)

Regions are shown as solids:

Use None to show a wireframe of the bounding surfaces:

Use an orange surface:

ColorFunction has higher priority than PlotStyle:

MeshShading has higher priority than PlotStyle:

PlotTheme (2)

Use a highly stylized theme:

Remove the mesh lines:

ScalingFunctions (5)

By default, plots have linear scales in all directions:

Apply a log scale to the z axis:

Use a shifted log scale to show a function with negative values:

Use ScalingFunctions to reverse the coordinate direction in the direction:

Use a scale defined by a function and its inverse:

TextureCoordinateFunction (4)

Textures use scaled and coordinates by default:

Use the and coordinates:

Use unscaled coordinates:

Use textures to highlight how parameters map onto a surface:

TextureCoordinateScaling (1)

Use scaled or unscaled coordinates for textures:

Applications (3)

Find the intersection of two half-spaces:

Simple regions including a cube:

Half of a cube shell:

Ball:

Half of a spherical shell:

Ellipsoid:

Half of an ellipsoidal shell:

Spherical wedge:

Combine PolyhedronData regions with other inequalities:

Properties & Relations (8)

Use RegionPlot for areas:

Use ContourPlot and ContourPlot3D for systems of equalities:

Use ComplexRegionPlot for regions in the complex plane:

Use RegionFunction to constrain other plots:

Use ParametricPlot3D for parametric curves and surfaces:

Use Integrate or NIntegrate to integrate over regions:

The integration region:

Use Maximize, NMaximize, or FindMaximum to optimize over regions:

Use Reduce to get a cylindrical representation of the region:

Use FindInstance to find specific samples in regions:

Neat Examples (2)

The region between norm balls:

Plot a scalar field over a 3D region:

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RegionPlot3D

Details and Options

Examples

Basic Examples (4)