Line and Interval are full-dimensional regions in 1D:

An ImplicitRegion is 1D if it has one variable:

The discretization can be clipped to a specified range:

A ParametricRegion is 1D if it has only one function:

The discretization can be clipped to a specified range:

Because this region is unbounded, clip it to discretize:

A BooleanRegion in 1D:

Boundary discretization can only represent full-dimensional region components:

Use DiscretizeRegion to discretize lower-dimensional components as well:

Rectangle, Disk, and Simplex are special regions that can be full dimensional in 2D:

Disk:

Simplex:

An ImplicitRegion is 2D if it has two variables:

For an unbounded region, clip the discretization to a specified range:

A ParametricRegion is in 2D if it has two functions:

Clip the region:

A region in 2D with parameters constrained to a unit disk:

Parameters constrained to a rectangle:

Given two exact regions, ParametricRegion can be used to represent their Minkowski sum:

A RegionUnion in 2D:

A region can include components of different dimensions:

The boundary discretization can only represent full-dimensional components, however:

A polygon with GeoGridPosition:

Cuboid, Ellipsoid, and Simplex are special regions that can be full dimensional in 3D:

Ellipsoid:

Simplex:

An ImplicitRegion is 3D if it has precisely three variables:

Clip a region:

A ParametricRegion is in 3D if it has precisely three functions:

A solid in 3D generated with the parameters constrained to the unit ball:

A region can include components of different dimensions:

The boundary discretization can only represent full-dimensional components, however:

The measure of cells in the discretization can be controlled using MaxCellMeasure:

By default, when given as a number, it applies to the boundary dimension:

With area a, a length l is computed so that triangles with sides of length l will have area a:

Using TriangulateMesh with the same specification will maintain quality near the edge:

For nonlinear regions, the measure of boundary cells depends on several options:

The length of any segment may be controlled by MaxCellMeasure:

The default PrecisionGoal is chosen to be a value so that curves appear as visually smooth:

MaxCellMeasure->∞ may be used to base the boundary measure on precision:

PrecisionGoal->None may be used to base the boundary measure on MaxCellMeasure:

AccuracyGoal->a may be used to specify an absolute tolerance :

The default is for MaxCellMeasure to apply to the boundary dimension:

The measure on the boundary may be further restricted by approximation requirements:

The measure of cells in the discretization can be controlled using MaxCellMeasure:

By default, this controls the measure of the edges:

Use AccuracyGoal to ensure the discretized boundary is close to the exact boundary:

The discretization with the higher AccuracyGoal is closer to the true boundary:

Use PrecisionGoal to ensure the discretized boundary is close to the exact boundary:

The discretization with the higher PrecisionGoal is closer to the true boundary:

Set PerformanceGoal to "Quality" for a high-quality discretization:

Or to "Speed" for a faster discretization that may be of lower quality:

Ensure that the discretized boundary representation is close to the boundary:

The largest deviation from the true disk is at the centers of the segments:

With a larger accuracy goal, the error is reduced by using more points:

With MaxCellMeasure->m, boundary cell size is less than or equal to :

This gives the lengths of the segments:

In 3D, the area of faces is controlled by MaxCellMeasure:

This gives the areas of the faces:

The lengths of edges can be controlled by setting a max measure for "Length":

This gives the lengths of the edges:

MeshCellHighlight allows you to specify highlighting for parts of a BoundaryMeshRegion:

By making faces transparent, the internal structure of a 3D BoundaryMeshRegion can be seen:

Individual cells can be highlighted using their cell index:

Or by the cell itself:

MeshCellLabel can be used to label parts of a BoundaryMeshRegion:

Label the vertices and edges of a rectangle:

Individual cells can be labeled using their cell index:

Or by the cell itself:

MeshCellMarker can be used to assign values to parts of a BoundaryMeshRegion:

Use MeshCellLabel to show the markers:

MeshCellShapeFunction allows you to specify functions for parts of a BoundaryMeshRegion:

Individual cells can be drawn using their cell index:

Or by the cell itself:

MeshCellStyle allows you to specify styling for parts of a BoundaryMeshRegion:

By making faces transparent, the internal structure of a 3D BoundaryMeshRegion can be seen:

Individual cells can be highlighted using their cell index:

Or by the cell itself:

The "Continuation" method uses a curve continuation method that can in many cases resolve corners, cusps, and sharp changes quite well:

The "RegionPlot" method is based on improving output from RegionPlot and can sometimes be faster:

The "Boolean" method is optimized for Boolean regions:

The "DiscretizeGraphics" method is optimized for graphics primitives:

The "RegionPlot3D" method for 3D regions is based on RegionPlot3D:

The "ContourPlot3D" method for 3D regions is based on ContourPlot3D:

Use a theme with grid lines and a legend:

Use a theme to draw a wireframe:

Ensure that the discretized boundary representation is close to the boundary:

The largest deviation from the true disk is at the centers of the segments:

With a larger accuracy goal, the error is reduced by using more points: