BoundaryDiscretizeRegion
discretizes the region reg into a BoundaryMeshRegion.
BoundaryDiscretizeRegion[reg,{{x_{min},x_{max}},…}]
restricts to the bounds .
Details and Options
 BoundaryDiscretizeRegion is also known as boundary evaluation.
 BoundaryDiscretizeRegion effectively discretizes the boundaries of the fulldimensional parts of a region reg.
 The region reg can be anything that is ConstantRegionQ and RegionEmbeddingDimension less or equal to 3.
 BoundaryDiscretizeRegion has the same options as BoundaryMeshRegion, with the following additions and changes:

AccuracyGoal Automatic digits of accuracy sought MaxCellMeasure Automatic maximum cell measure Method Automatic method to use PerformanceGoal $PerformanceGoal whether to consider speed or quality PrecisionGoal Automatic digits of precision sought  With AccuracyGoal>a and PrecisionGoal>p, an attempt will be made to keep the maximum distance between the region reg or the discretized region dreg and any point in RegionSymmetricDifference[reg,dreg] to less than , where is the length of the diagonal of the bounding box.
 With MaxCellMeasure>m where m>0, the cell measure in the boundary dimension d1 where d is the region dimension will be limited to m. Measure limits for specific dimensions may be specified with MaxCellMeasure>{…,d_{i}>m_{i},…}.
Examples
open allclose allBasic Examples (2)
Scope (24)
Regions in 1D (5)
Line and Interval are fulldimensional 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 fulldimensional region components:
Use DiscretizeRegion to discretize lowerdimensional components as well:
Regions in 2D (8)
Rectangle, Disk, and Simplex are special regions that can be full dimensional in 2D:
Disk:
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:
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 fulldimensional components, however:
A polygon with GeoGridPosition:
Regions in 3D (5)
Cuboid, Ellipsoid, and Simplex are special regions that can be full dimensional in 3D:
An ImplicitRegion is 3D if it has precisely three variables:
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 fulldimensional components, however:
Detail (2)
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:
Quality (4)
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 highquality discretization:
Or to "Speed" for a faster discretization that may be of lower quality:
Options (24)
AccuracyGoal (1)
MaxCellMeasure (2)
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":
MeshCellHighlight (3)
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:
MeshCellLabel (3)
MeshCellLabel can be used to label parts of a BoundaryMeshRegion:
Label the vertices and edges of a rectangle:
MeshCellMarker (1)
MeshCellMarker can be used to assign values to parts of a BoundaryMeshRegion:
Use MeshCellLabel to show the markers:
MeshCellShapeFunction (2)
MeshCellShapeFunction allows you to specify functions for parts of a BoundaryMeshRegion:
MeshCellStyle (3)
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:
Method (6)
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:
Applications (2)
Properties & Relations (5)
The output of BoundaryDiscretizeRegion is a BoundaryMeshRegion:
Given a boundary discretization, TriangulateMesh can discretize the interior:
The boundary discretization represents the regular closure of a region:
The lowerdimensional component is lost, but can be represented by DiscretizeRegion:
BoundaryDiscretizeRegion can discretize a region with holes:
BoundaryDiscretizeRegion can discretize a region with disjoint components: