Geometric regions such as points, curves, surfaces, volumes, and their higher-dimensional analogs occur in a variety of contexts, including mathematics, engineering, science, computer games, and geography. The Wolfram Language provides fully integrated capabilities for creating, analyzing, solving over, and visualizing regions. Regions can be created by using common special regions, from formulas, as meshes of simple regions, and by combining or modifying existing regions. Regions can be analyzed by computing standard properties such as dimension, measure (length, area, volume, etc.), nearest points, etc. Regions can be used as constraints to a large number of solvers, including equation solving, optimization, and solving partial differential equations. Regions are first-class citizens in the Wolfram Language; they can be used as input and output, and they are deeply integrated into the rest of the Wolfram Language.

Properties and Measures »

RegionNearest — find the nearest point in a region

RegionMeasure  ▪  RegionCentroid  ▪  RegionDistance  ▪  ...

Solvers over Regions »

Integrate, NIntegrate — integrate over regions

NDSolve — solve partial differential equations over regions

Solve  ▪  NSolve  ▪  Minimize  ▪  NMinimize  ▪  ...

Region — representation and automatic visualization of regions

CanonicalizeRegion — canonicalize the region representation

Special Regions »

Ball — a ball given by center and radius

Simplex  ▪  Hexahedron  ▪  InfiniteLine  ▪  ...

Polygon Regions »

Polygon — a general polygon

CrossingPolygon  ▪  RandomPolygon  ▪  PolygonDecomposition  ▪  ...

Polyhedron Regions »

Polyhedron — a general polyhedron

Dodecahedron  ▪  Icosahedron  ▪  DualPolyhedron  ▪  ...

Formula Regions

ImplicitRegion — a region given by combinations of inequalities and equalities

ParametricRegion — a region given by parametric functions

Mesh Regions »

MeshRegion — a region specified by a collection of mesh cells

BoundaryMeshRegion — a region boundary specified by a collection of mesh cells

HighlightMesh  ▪  MeshCellStyle  ▪  MeshCellLabel  ▪  ...

Derived Regions »

RegionUnion — the union of regions

TransformedRegion — a region as the image of a transformation

RegionIntersection  ▪  BooleanRegion  ▪  RegionBoundary  ▪  ...

Subdivision Regions

SubdivisionRegion — limiting region obtained by a subdivision process

Data Regions »

DelaunayMesh — a Delaunay triangulation mesh constructed from points

VoronoiMesh  ▪  ConvexHullMesh  ▪  ArrayMesh  ▪  ImageMesh  ▪  ...

Region Converters »

RegionConvert — convert to implicit or parametric representations of regions

DiscretizeGraphics, BoundaryDiscretizeGraphics — discretize 2D or 3D graphics

DiscretizeRegion, BoundaryDiscretizeRegion — discretize a region

Curated Regions »

PolyhedronData — polyhedron regions and properties

ExampleData  ▪  GeoElevationData  ▪  ChemicalData  ▪  AnatomyData  ▪  KnotData  ▪  LaminaData  ▪  SolidData  ▪  PlaneCurveData  ▪  SpaceCurveData  ▪  SurfaceData

Importing and Exporting Regions »

"STL" — import and export regions as STL

"PLY"  ▪  "OBJ"  ▪  "OFF"  ▪  "DAE"  ▪  ...

3D Printing »

Printout3D — print a 3D model using a print service or local printer

FindMeshDefects  ▪  RepairMesh  ▪  ShellRegion  ▪  ...

Synthetic Geometry »

GeometricScene  ▪  RandomInstance  ▪  FindGeometricConjectures  ▪  ...