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RegionEqual

RegionEqual[reg₁,reg₂]

returns True if the regions reg₁ and reg₂ are equal.

RegionEqual[reg₁,reg₂,reg₃,…]

returns True if the regions reg₁, reg₂, reg₃, … are all equal.

Details and Options

The regions reg₁ and reg₂ are equal if all points in reg₁ are points in reg₂ and vice versa.

If all reg_i are parameter-free regions, i.e. ConstantRegionQ[reg_i] is True, the regions are point sets, and typically True or False is returned.
If some reg_i depend on parameters, i.e. ConstantRegionQ[reg_i] is False, then reg_i represents a family of regions, and RegionEqual will attempt to compute conditions on parameters such that the regions are equal.
The following options can be given:
Assumptions $Assumptions assumptions to make about parameters

GenerateConditions False whether to generate conditions on parameters

Examples

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Basic Examples (2)

Test whether two regions are equal:

Generate conditions for which two regions are equal:

Scope (16)

Basic Uses (4)

Test for equality:

Show two regions are not equal:

Find conditions that make regions equal:

Compare three regions at once:

Basic Regions (4)

Regions in including Line and Interval:

Regions in including Point:

Disk, Ball and Ellipsoid:

Rectangle and RegularPolygon:

Regions in including Point:

Cuboid and Hexahedron:

Ball and Ellipsoid:

Tetrahedron and Simplex:

Regions in including Cuboid and Parallelepiped in :

Ellipsoid and Ball in :

Formula Regions (4)

Implicit regions:

Parametric regions:

Compare two formula regions:

Nonconstant formula regions:

Mesh Regions (3)

Compare MeshRegion in :

In :

In :

Compare BoundaryMeshRegion in :

In :

In :

Compare MeshRegion with BoundaryMeshRegion in :

In :

Derived Regions (1)

Compare BooleanRegion:

Options (2)

Assumptions (1)

Find all radii that describe the unit circle:

Find only the positive radii:

GenerateConditions (1)

Find when the unit disk is equal to an implicitly described annulus:

Show the conditions for which the result is valid:

Explicitly allow for degenerate cases:

Applications (6)

Find when two lamina are equal:

Create the unit disk by covering the hole of an annulus with a disk. Find all possible radii:

Show an instance of this:

Create the unit disk by covering the hole of an annulus with a square. Find all possible side lengths:

Show an instance of this:

Find all ways to express the unit rectangle in terms of Parallelogram:

Verify the results:

Express an implicitly defined ellipse through Ellipsoid:

Find a value for each parameter:

Visualize both representations:

Demonstrate De Morgan's law, :

Properties & Relations (4)

Equivalent regions have all points in common:

RegionEqual can be expressed with RegionWithin:

For non‐empty regions, RegionDisjoint returns False when RegionEqual returns True:

Use FindInstance to find points that lie in one region but not the other:

Use RandomPoint to find a uniform sampling of points that lie in one region but not the other:

Use Reduce to find where two regions differ:

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