Region Properties and Measures
The Wolfram Language supports a broad range of standard properties and measures for geometric regions, including point membership tests; integral measures such as length, area, volume, and centroid; or optimization measures such as nearest point or distance.
Basic Predicates
RegionQ — test whether an object is a region
ConstantRegionQ — test whether a region is constant and free of symbolic parameters
BoundedRegionQ — test whether the region is bounded
ConvexRegionQ ▪ SolidRegionQ ▪ MeshRegionQ ▪ BoundaryMeshRegionQ ▪ CSGRegionQ
Membership Predicates
Element — assert that a symbolic point is an element of a region p∈reg
RegionMember — test membership or give conditions for membership
RegionMemberFunction — function used for repeated membership tests
RegionBounds — find the bounds for a region
BoundingRegion — simple bounding regions for point sets
Region Relations
RegionEqual — test whether two regions are equal
RegionDisjoint — test whether two regions are disjoint
RegionWithin — test whether a region is within another
RegionCongruent — test whether two regions are congruent
RegionSimilar — test whether two regions are similar
Generating Points
RandomPoint — generate points uniformly in a region
RegionCentroid — the centroid (center of mass etc.) of the region
Dimension Measures
RegionDimension — the geometric dimension of the region
RegionEmbeddingDimension — the embedding dimension of the region
Integral Measures
RegionMeasure — the measure (count, length, area, volume, etc.) of the region
RegionMoment — any order moment of the region
MomentOfInertia — moment of inertia and moment of inertia matrix for the region
ArcLength ▪ Area ▪ Volume ▪ Perimeter ▪ SurfaceArea
Distance Measures
RegionDistance — minimum distance from a point to a region
SignedRegionDistance — signed minimum distance from a point to a region
RegionNearest — the nearest point in a region
RegionDistanceFunction — function used for repeated distance computation
RegionNearestFunction — function used for repeated nearest computation
RegionHausdorffDistance — the separation distance between two regions
RegionFarthestDistance — the farthest distance between two regions
Angle Measures »
PlanarAngle — angle between lines in 2D, 3D
DihedralAngle — angle between planes in 3D
SolidAngle — solid angle subtended by a region
PolygonAngle, PolyhedronAngle — angles inside polygons, polyhedra
VectorAngle — angle between vectors
Point Orientations
PositivelyOrientedPoints — test whether a set of points is positively oriented
CollinearPoints — test whether a set of points is collinear