Geometric Computation

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.

(OrangeLink[Properties and Measures », paclet:guide/GeometricPropertiesAndMeasures])

(RefLink[RegionNearest, paclet:ref/RegionNearest]) — find the nearest point in a region

(OrangeLink[Solvers over Regions », paclet:guide/GeometricSolvers])

(RefLink[Integrate, paclet:ref/Integrate]), (RefLink[NIntegrate, paclet:ref/NIntegrate]) — integrate over regions

(RefLink[NDSolve, paclet:ref/NDSolve]) — solve partial differential equations over regions

(RefLink[Region, paclet:ref/Region]) — representation and automatic visualization of regions

(OrangeLink[Special Regions », paclet:guide/GeometricSpecialRegions])

(RefLink[Ball, paclet:ref/Ball]) — a ball given by center and radius

(OrangeLink[Polygon Regions », paclet:guide/Polygons])

(RefLink[Polygon, paclet:ref/Polygon]) — a general polygon

(OrangeLink[Polyhedron Regions », paclet:guide/Polyhedra])

(RefLink[Polyhedron, paclet:ref/Polyhedron]) — a general polyhedron

Formula Regions

(RefLink[ImplicitRegion, paclet:ref/ImplicitRegion]) — a region given by combinations of inequalities and equalities

(RefLink[ParametricRegion, paclet:ref/ParametricRegion]) — a region given by parametric functions

(OrangeLink[Mesh Regions », paclet:guide/MeshRegions])

(RefLink[MeshRegion, paclet:ref/MeshRegion]) — a region specified by a collection of mesh cells

(RefLink[BoundaryMeshRegion, paclet:ref/BoundaryMeshRegion]) — a region boundary specified by a collection of mesh cells

(OrangeLink[Derived Regions », paclet:guide/DerivedRegions])

(RefLink[RegionUnion, paclet:ref/RegionUnion]) — the union of regions

(RefLink[TransformedRegion, paclet:ref/TransformedRegion]) — a region as the image of a transformation

(OrangeLink[Data Regions », paclet:guide/MeshRegions])

(RefLink[DelaunayMesh, paclet:ref/DelaunayMesh]) — a Delaunay triangulation mesh constructed from points

(OrangeLink[Region Converters », paclet:guide/MeshRegions])

(RefLink[DiscretizeGraphics, paclet:ref/DiscretizeGraphics]), (RefLink[BoundaryDiscretizeGraphics, paclet:ref/BoundaryDiscretizeGraphics]) — discretize 2D or 3D graphics

(RefLink[DiscretizeRegion, paclet:ref/DiscretizeRegion]), (RefLink[BoundaryDiscretizeRegion, paclet:ref/BoundaryDiscretizeRegion]) — discretize a region

(OrangeLink[Curated Regions », paclet:guide/MathematicalData])

(RefLink[PolyhedronData, paclet:ref/PolyhedronData]) — polyhedron regions and properties

(OrangeLink[Importing and Exporting Regions », paclet:guide/3DGeometryAndModelingFormats])

(StringTypeLink["STL", paclet:ref/format/STL]) — import and export regions as STL

(OrangeLink[3D Printing », paclet:guide/3DPrinting])

(RefLink[Printout3D, paclet:ref/Printout3D]) — print a 3D model using a print service or local printer

(OrangeLink[Synthetic Geometry, paclet:guide/SyntheticGeometry])