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

Midpoint  ▪  TriangleCenter

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

ShortestCurveDistance — the minimal distance between two points on a region

Curvature Measures

RegionGaussianCurvature — Gaussian curvature at a point on a region

RegionMeanCurvature — mean curvature at a point of a region

RegionMinCurvature — minimum curvature at a point of a region

RegionMaxCurvature — maximum curvature at a point of a region

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

CoplanarPoints  ▪  NegativelyOrientedPoints