Geometric Computation
Topic Overview 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 ▪ ...